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the solution of the @xmath7 worldsheet string theory is one of the cornerstones of the ads / cft duality program . despite continuous effort and progress on classical aspects in particular @xcite , and the generally accepted presence of both integrability and conformal invariance symmetries , most aspects of the quantum theory remain elusive . it is natural to try to understand some aspects of this quantum theory by first tackling simpler models with similar properties . the so called @xmath0 coset model - specifically , a sigma model on the supersphere @xmath8 - is a very attractive candidate for such an exercise : like the @xmath7 worldsheet theory it is conformal invariant and its target space is a supergroup coset . of course , it lacks other aspects such as the brst structure of the string theory . apart from the string theory motivation , models such as the @xmath0 coset model are extremely interesting from the pure conformal field theory point of view . indeed , they are sigma models which are massless without any kind of topological term , and for a large range of values of the coupling constant @xmath1 . to make things more precise let us briefly remind the reader of some generalities . supersphere sigma models have target super space the supersphere @xmath9 and can be viewed as a `` supersymmetric '' extension of the nonlinear @xmath10 sigma models ( which differs of course from the usual @xmath10 `` supersymmetric '' models ) . use as coordinates a real scalar field @xmath11 where the first @xmath12 components are bosons , the last @xmath13 ones fermions , and the invariant bilinear form @xmath14 where @xmath15 is the orthosymplectic metric @xmath16 @xmath17 denoting the identity . the unit supersphere is defined by the constraint @xmath18 the action of the sigma model ( conventions are that the boltzmann weight is @xmath19 ) reads @xmath20 the perturbative @xmath21 function depends only on @xmath22 to all orders ( see , e.g. , ref @xcite ) , and is the same as the one of the @xmath10 model with @xmath23 . physics can be reliably understood from the first order beta function @xmath24 the model for @xmath25 positive flows to strong coupling for @xmath26 . like in the ordinary sigma models case , the symmetry is restored at large length scales , and the field theory is massive . for @xmath27 meanwhile , the model flows to weak coupling , and the symmetry is spontaneously broken . one expects this scenario to work for @xmath1 small enough , and the corresponding goldstone phase to be separated from a non perturbative strong coupling phase by a critical point . the case we are interested in here is @xmath28 , where the @xmath21 function vanishes to all orders in perturbation theory , and the model is expected to be conformal invariant , at least for @xmath1 small enough , the goldstone phase being replaced by a phase with continuously varying exponents not unlike the low temperature kosterlitz thouless phase . how the group symmetry combines with the ( logarithmic ) conformal symmetry in such models is largely unknown . it is an essential question to be solved before any serious attempts to understanding universality classes in non interacting disordered 2d electronic systems can be contemplated @xcite . the @xmath0 coset model was considered in particular in two papers by mann and polchinski using the massless scattering and bethe ansatz approaches . this is indeed a natural idea , since supersphere sigma models are in general integrable , and , when massive ( ie @xmath26 ) can be described by a scattering theory involving particles in the fundamental representation of the group . the @xmath29 matrix is well known @xmath30 here , @xmath17 is the identity , @xmath31 is the graded permutation operator , and @xmath32 is proportional to the projector on the identity representation . for @xmath33 arbitrary , factorizability requires that @xmath34 where @xmath35 , while @xmath36 itself is determined , up to cdd factors , by crossing symmetry and unitarity . one immediately observes that when @xmath37 , the amplitude @xmath36 cancels out , leaving a scattering matrix with a simpler tensorial structure , since the @xmath31 operator disappears . this corresponds to a particular point @xcite on the sigma model critical line ( where , among other things , the symmetry is enhanced to @xmath38 ) , the rest of which is not directly accessible by this construction . is special and allows for an extension of the s matrix to the whole @xmath39 critical line . ] the idea used in @xcite is to consider an analytical continuation to @xmath33 real , and an approach to @xmath28 with proper scaling of the mass . though interesting results were obtained , the emphasis in these papers was not on conformal properties . another line of attack , more suited to the conformal aspects , was launched by read and saleur in 2001 @xcite , who proposed to use a lattice regularization to control the integrable features of the model . they obtained in this way the spectrum of critical exponents for several related sigma models on super target spaces , including the @xmath0 coset one at a particular ( critical ) value of the coupling @xmath1 . the results exhibited several mysterious features , including a pattern of large degeneracies , and a set of values of the exponents covering ( modulo integers ) all the rationals . in two subsequent papers @xcite , it was argued further that many algebraic properties of the conformal field theory could be obtained at the lattice level already . these include fusion , and the structure of conformal `` towers '' ( see below for further details ) .    the work we present in this paper and its companion is an attempt at understanding the conformal field theoretic description of the @xmath0 model for all values of the coupling by using a lattice regularization . foremost in the lattice approach is the understanding of the algebraic structure of the lattice model - the algebra defined by the local transfer matrices and its commutant . while in the cases discussed in @xcite most necessary results were already available in the mathematical literature , the situation here is much more complicated : in a few words , we have to deal , instead of the temperley lieb algebra , with the _ brauer algebra _ whose representation theory , in the non semi - simple case , is far from fully understood . an important part of our work has consisted in filling up the necessary gaps of the literature , sometimes rigorously , but sometimes at the price of some conjectures . this algebraic work is the subject of the first paper , which we realize might be a bit hard to read for a physics reader . we capitalize on the algebraic effort in the companion paper , where the boundary conformal field theory for the coset sigma model is analyzed thoroughly .    in the second section of this paper we discuss generalities about lattice regularizations of @xmath10 sigma models in 2 dimensions and define the model we shall be interested in . in section 3 , the transfer matrix , the loop reformulation and the associated brauer algebra are introduced and discussed . section 4 is the main section , where the full decomposition of the hilbert space of the lattice model under the action of @xmath0 and @xmath3 is obtained . our main result can be found in eqs .   and . section 5 discusses aspects of the hamiltonian limit and section 6 contains conclusions . technical aspects of representation theory are discussed further in the appendices .    for the reader s convenience , we provide here a list of notations used throughout the paper :    * @xmath40 is the lie superalgebra of the supergroup @xmath41 * @xmath42 is the brauer algebra on @xmath43 strings with fugacity for loops @xmath35 * @xmath44 is the mod 2 graded vector space @xmath45 with even part @xmath46 and odd part @xmath47 . we shall often drop the indices @xmath48 in @xmath49 . * @xmath2 is considered as a left @xmath40 and right @xmath42 bimodule * @xmath50 stands for `` @xmath51 is a partition of @xmath43 '' and @xmath52 is the partition @xmath51 transposed * @xmath53 and @xmath54 are the symmetric group on @xmath43 objects and its group algebra * @xmath55 is the temperley lieb algebra with fugacity for loops @xmath56 * @xmath57 and @xmath58 are generic elements of @xmath42 and @xmath41 * @xmath59\,\}$ ] is the set of partitions labeling the weights of @xmath42 . @xmath60 selects those of them which do realize on @xmath2 * associate weights @xmath61 are labels of @xmath41 irreps which are nonequivalent(identical ) and become(split into two ) isomorphic(nonequivalent ) irreps under the restriction to the proper subgroup @xmath62 of supermatrices with @xmath63 * @xmath64 is the set of hook shape partitions labeling the weights of @xmath40 irreps appearing in @xmath2 and @xmath65 * @xmath66 are standard or generically irreducible representations of @xmath42 * @xmath67 , @xmath68 , @xmath69 , @xmath70 are irreducible representations of @xmath71 , @xmath41 , @xmath42 and @xmath55 respectively . * @xmath72 are direct summands of @xmath2 as a @xmath40 , @xmath41 and @xmath42 module * @xmath73 are the supersymmetric generalization of @xmath10 symmetric functions * @xmath74 are characters of @xmath66 , @xmath69 and @xmath75 . lattice discretizations of @xmath10 sigma models have had a long history . the simplest way to go is obviously to introduce spins taking values in the target manifold - the sphere @xmath76 - on the sites of a discrete lattice , with an interaction energy of the heisenberg type @xmath77 ( where @xmath78 stands for the bilinear @xmath10-invariant quadratic form ) . this is however difficult to study technically , as the number of degrees of freedom on each site is infinite . a possible way to go is to discretize the target space , leading to various types of `` cubic models '' @xcite . another way which has proved especially fruitful in two dimensions has been to reformulate the problem of calculating the partition or correlation functions geometrically by using the techniques of high or low temperature expansions , thus obtaining graphs with complicated interaction rules and weights determined by properties of the underlying groups . the simplest of these formulations appeared in @xcite where the authors studied the @xmath10 model on the honeycomb lattice in two dimensions , and replaced moreover the term @xmath79 by its considerably simpler high temperature approximation @xmath80 , @xmath81 @xcite . expanding the brackets , in say the calculation of the partition function , one can draw graphs by putting a bond between neighboring sites @xmath82 and @xmath83 whenever the term @xmath84 is picked up . the integral over spin variables leaves only loops , with a fugacity equal to @xmath85 as there are @xmath85 colors one can contract . note that because of the very low coordination number of the honeycomb lattice , only self - avoiding loops are obtained . this leads to the well known self - avoiding loop gas partition function : @xmath86 where the sum is taken over all configurations @xmath87 of self avoiding , mutually avoiding closed loops in number @xmath43 , covering a total of @xmath32 edges . note that once an expression such as ( [ nien_triv ] ) is written down , it is possible to analytically continue the definition of the model for @xmath85 an arbitrary real number . barring the use of superalgebras , only @xmath85 integer greater or equal to one has a well defined meaning as a spin model ( the case @xmath88 coincides with the ising model   is different from @xmath89 because of the additional @xmath90 freedom in choosing the sign of the determinant . ] ) . in two dimensions , the mermin wagner theorem prevents spontaneous symmetry breaking , so for @xmath85 integer , critical behavior can only occur for @xmath91 . analysis of the same beta function suggests however that lattice models defined by suitable analytic continuation should have a goldstone low temperature phase for all @xmath92 , though it says nothing about whether this phase might end by a second or first order phase transition .    model ( [ nien_triv ] ) lacks interaction terms which would appear with less drastic choices of the lattice and the interactions : these are the terms where the loops intersect , either by going over the same edge , or over the same vertex , maybe many times . it has often been argued that such terms are irrelevant for the study of the critical points of the @xmath10 models in two dimensions . most of the interest has focused on such critical points for @xmath93 $ ] , which have geometrical applications - in particular the case @xmath94 is related with the physics of self - avoiding walks . it turns out however that intersection terms are crucial for the understanding of low temperature phases . indeed , the model ( [ nien_triv ] ) _ does have _ a sort of goldstone phase for @xmath93 $ ] called the dense phase , but its properties are not generic , and destroyed by the introduction of a small amount of intersections . a simple way to see that the dense phase is not generic is that the exponents at @xmath37 are always those of the kosterlitz thouless transition point : model ( [ nien_triv ] ) does not allow one to enter the low temperature phase of the xy model . also , model ( [ nien_triv ] ) has a first order transition for @xmath95 , which is not the behavior expected from the sigma model analysis . it was suggested in @xcite that model ( [ nien_triv ] ) can be repaired by allowing for some intersections . the minimal scenario one can imagine is to define a similar model on the square lattice , and allow for self intersections at vertices only , so either none , one or two loops go through the same vertex . the resulting objects are often called trails . this gives the new partition function @xmath96 where the sum is taken over all configurations @xmath97 of closed loops , which visit edges of the lattice at most once , in number @xmath43 , covering a total of @xmath32 edges , with @xmath17 intersections . the phase diagram of this model has not been entirely investigated . it is expected that at least for @xmath98 small enough , the critical behavior obtained with @xmath99 is not changed ( though @xmath100 is ) , while the low temperature behavior @xmath101 will be .    in @xcite a yet slightly different version was considered corresponding , roughly , to the limit of very large @xmath102 , where all the edges of the lattice are covered . the partition function of this fully packed trails model then depends on only two parameters : @xmath103 and an example of allowed configuration is given in figure [ f : loop_cov ] . numerical and analytical arguments suggest strongly that this model has all the generic properties of the @xmath10 sigma model in the spontaneously broken symmetry phase ( such that one might derive from analytic continuation in @xmath85 of the rg equations ) , for all @xmath104 . [ f : loop_cov ]    ) , contractible ( @xmath105 ) , even ( @xmath32 ) and odd ( @xmath106 ) loops . periodic imaginary time runs vertically ]    note that expression ( [ ours ] ) can be obtained very naturally if instead of putting the degrees of freedom on the vertices , one puts them on the edges of the lattice . in this case , the minimal form of interaction involves two edges crossing at one vertex . invariance under the @xmath10 group allows for three invariant tensors as illustrated on the figure [ f : vert ] , while isotropy and invariance under an overall scale change of the boltzmann weights leaves one with a single free parameter , the crossing weight @xmath98 . graphical representation of the contractions on the invariant tensors reproduces eq . ( [ ours ] ) , as will be discussed below .    for @xmath92 , model ( [ ours ] ) flows to weak coupling in the ir , and therefore it is expected that the critical properties of the corresponding low temperature ( goldstone ) phase do not depend on @xmath98 , a fact checked numerically in @xcite . the case @xmath37 is expected to be different : as mentioned already in the introduction , the beta function of the corresponding sigma model is exactly zero so the coupling constant does not renormalize . it is indeed easy to see that the loop model ( [ ours ] ) with @xmath37 is equivalent to the 6 vertex model with @xmath107 . consider the vertices of the 6 vertex model as represented on figure [ f:6v ] . we chose isotropic weights @xmath108 . we can decide to split the vertices of a configuration into pieces of oriented loops as represented on figure [ f:6vo2map ] . for each vertex , there are two possible splittings , and we assume that they are chosen with equal probability . the loops obtained by connecting all the pieces together provide a dense covering of the lattice , and come with two possible orientations , hence a fugacity of two once the orientations are summed over . the loops can intersect , with a weight @xmath98 given from the obvious correspondence : @xmath109 and thus @xmath110     symmetric six vertex model ]        we note that there are indeed _ three _ invariant tensors for the case of @xmath39 . the corresponding projectors are @xmath111 . they project respectively on two one dimensional representations , and on a single two dimensional one . the parameter @xmath112 covers the interval @xmath113 $ ] as @xmath114 $ ] . changing @xmath112 is well known to change the exponents of 6 vertex model , and therefore eq .  ( [ ours ] ) for @xmath37 exhibits a critical line , which is in fact in the universality class of the continuous xy ( @xmath39 ) model in the low temperature ( kosterlitz thouless ) phase . all what was said so far can be easily generalized to the case of spins taking values on a supersphere @xmath115 . the fugacity of loops is now equal to @xmath22 : this combination is the number of bosonic minus the number of fermionic coordinates , and follows from the usual fact that when contracting fermions along a loop , a minus sign is generated   models to the case of orthosymplectic groups dates back to the work of parisi and sourlas @xcite . ] , see sec . [ sec:1749 ] . the loop model formulation therefore provides a convenient graphical representation of the discrete supersphere sigma models for all @xmath22 , in particular @xmath116 where interesting physics is expected to occur . this physics was explored in @xcite , and the expected results were obtained for @xmath27 . the purpose of this paper is to explore the more challenging @xmath28 case . of course , at the naive level of partition functions and without worrying about boundary conditions , it looks as if there is no difference between the @xmath10 spin model and its supersphere cousins provided @xmath117 . the point is that the _ observables _ of the models are different or , at the very least , come with different multiplicities . indeed , consider for instance correlation functions of spin variables . in the @xmath39 case , the spin has only two components @xmath118 , so one can not build a totally antisymmetric tensor on three indices . this means that the corresponding operator ( which has a nice geometrical interpretation to be given in the next paper ) will not be present in this case , though it will be in the @xmath0 model when @xmath119 . note that in general , correlators involving spins within the first @xmath12 bosonic and the first @xmath13 fermionic labels will be the same for _ any _ choice of group @xmath120 with @xmath121 and @xmath122 . bosonic and @xmath123 fermionic degrees of freedom are the same in both cases . ] this is immediately proved by performing a graphical expansion of the correlator : variables outside of the set of the first @xmath12 bosonic and the first @xmath13 fermionic labels are not getting contracted with the spins in the correlators , and cancel against each other in the loop contractions .    a standard trick to extract the full operator content of a model is to study the partition function with different boundary conditions . consider for instance the spin model on an annulus with some symmetry preserving boundary conditions in the space direction . with what we will call periodic boundary conditions ( corresponding to taking the supertrace of the evolution operator ) in the time direction , representations of @xmath0 will always be counted with their superdimension , and the partition function will be identical with the one of the @xmath39 case . but if we take antiperiodic boundary conditions , we will get a modified partition function ( in the sense of @xcite ) which is a _ trace _ over the hilbert space instead of a supertrace , counts all observables with the multiplicities ( not supermultiplicities ) , and will turn out to be a very complex object .    a good algebraic understanding of the lattice model will be essential to make further progress , and , since the area is largely unexplored , this will occupy us for most of the rest of this first paper . as discussed briefly in the introduction , the @xmath41 spin model we consider is most easily defined on a square lattice with degrees of freedom ( states ) on the edges and interactions taking place at vertices . the set of states on every edge is a copy of the base space @xmath124 of the fundamental @xmath41 representation . interactions at a vertex can be encoded in a local transfer matrix @xmath125 acting on @xmath126 and commuting with the @xmath41 supergroup action . we call @xmath125 an intertwiner and write @xmath127 . the boltzmann weights of the model are components of the transfer matrix along a basis of intertwiners . a natural choice of basis are the projectors onto @xmath41 irreducible representations appearing in the decomposition of the tensor product of two fundamental @xmath41 representations . to find them one can apply the same ( anti)symmetrization and trace substraction techniques used for reducing @xmath10 tensor representations . if @xmath128 is a mod 2 graded set of basis vectors in @xmath124 with grading @xmath129 , the decomposition of @xmath126 will read @xmath130 here @xmath131 is the @xmath41 invariant tensor , @xmath132 , and @xmath133 ( resp . @xmath134 ) if @xmath82 is fermionic ( resp . bosonic ) . each of the three terms on the l.h.s .  of transforms according to an irreps of @xmath41 , or , in other words , belongs to a simple @xmath41 module . introduce the identity @xmath17 , the graded permutation operator @xmath31 ( also known as braid operator ) , and @xmath32 the temperley lieb operator ( proportional to the projector on the trivial representation ) , @xmath135 in terms of projectors onto irreducible @xmath41 modules , eq . may be written in a more elegant way as @xmath136    let @xmath137 denote as usual the inversion of space , @xmath138 the inversion of time and @xmath139 the charge conjugation with the matrix @xmath15 . one can check directly from definition that @xmath31 is @xmath140 , @xmath137 and @xmath141 invariant , while @xmath32 is @xmath142 and @xmath143 invariant . moreover , @xmath32 and @xmath17 transform into each other under the @xmath144 rotation of the lattice @xmath145 , while @xmath32 and @xmath31 are related by the crossing symmetry @xmath146 @xmath147    take @xmath17 , @xmath32 and @xmath31 as basis of intertwiners in @xmath148 . the local transfer matrix generally depends on three independent weights @xmath149 , @xmath150 and @xmath151 . however , on a homogeneous and isotropic lattice one can normalize @xmath152 and leave only the weight @xmath153 . finally , the local transfer matrix takes the form @xmath154     and @xmath155 represented on a diagonal lattice of width 6 . ]    on a diagonal lattice with open boundaries represented in fig .  [ f : dl ] choose the time in vertical direction . the notation of sites at a fixed time is such that the left edge @xmath82 and right edge @xmath156 meet at vertex @xmath82 . let @xmath157 denote a transfer matrix acting nontrivially only at vertex @xmath82 according to eq .  . from the figure it is clear that odd and even times are inequivalent the transfer matrix @xmath158 , propagating one step forward at equivalent times , may be written as a product @xmath159 of one layer transfer matrices @xmath160}t_{2i},\quad y = \prod_{i=1}^{[l/2]}t_{2i-1},\ ] ] schematically shown in fig . [ f : dl ] . the simplest way to define a partition function that depends on the whole spectrum of the transfer matrix @xmath158 is by taking the _ trace _ of @xmath158 at a certain power @xmath21 . selecting other boundary conditions with some nontrivial symmetry generally amounts to restricting the whole space of states of the model to a subspace compatible with the symmetry of chosen boundary conditions . what exactly we mean by `` symmetry of boundary conditions '' will be explained later in sec . [ sec:1749 ] . for the moment let us just say that it is convenient to consider a more general class of boundary conditions , called quasiperiodic , in which @xmath161 is `` twisted '' by the action of an element @xmath58 of the supergroup . define the quasiperiodic partition function to be @xmath162 we must take the _ supertrace _ in eq . if we want the quasiperiodic partition function to be well defined . for instance , when @xmath163 we get the usual trace partition function and when @xmath58 equals to the identity matrix we get the supertrace partition function . + note that because @xmath58 is a supermatrix , the tensor product in @xmath164 has to be graded , that is @xmath165    after inserting the local transfer matrix from eq .   in eq .   and expanding the transfer matrix @xmath158 , the quasiperiodic partition function reads as a sum of weighted products of @xmath166 s and @xmath167 s . such linearly independent products must be considered as words of a _ transfer matrix algebra _ , while intertwiners @xmath166 and @xmath167 are _ generators _ of this algebra . in the next section we identify this algebra as a representation of the brauer algebra .      for an abstract introduction to the brauer algebra see ref . @xcite while in the context of @xmath40 centralizer algebra see ref . we collect in this section some well known facts about the brauer algebra we shall use in the next sections . let @xmath166 and @xmath167 , @xmath168 act nontrivially as @xmath32 and @xmath31 in eq .   only at the sites @xmath169 of @xmath2 . one can check that for @xmath167 and @xmath166 so defined the following relations hold : @xmath170 in the second line of these relations @xmath82 and @xmath83 are supposed to be nonadjacent sites .    relations ( [ e : def_rel ] ) ( is one of the many ways to ) define the @xmath42 brauer algebra ( also denoted sometimes by the names of braid - monoid algebra or degenerate birman - wenzel - murakami algebra @xcite ) . note that this algebra depends on a single , generally complex , parameter @xmath85 , and contains the maybe more familiar temperley lieb algebra , generated by @xmath166 s alone , and the symmetric group algebra , generated by @xmath167 alone .    for @xmath85 fixed and @xmath43 big enough , the @xmath41 spin models provide highly unfaithful representations of the brauer algebra @xmath42 . this is because , in @xmath2 , the generators @xmath167 and @xmath166 satisfy additional higher order relations @xmath171 on top of . symmetry in the fundamental representation , and corresponding quotients of the hecke algebra . ] for a simple example , consider the @xmath39 spin model on a lattice of width 3 . the projector onto the antisymmetric tensor of rank 3 is zero , thus , @xmath171 contains the additional relation @xmath172 . our spin models in general provide representations of the quotient algebras @xmath173 . the set of relations @xmath171 can be explicitly described for @xmath174 , see @xcite and references therein , and we have little to say about the case @xmath119 . the first step in understanding the spectrum of the transfer matrix @xmath158 brings up the question of @xmath42 irreducible representations , and of their multiplicities in @xmath158 for a particular choice of @xmath12 and @xmath29 . this leads us to discussing some results about the representation theory of the brauer algebra .     in @xmath175 . ] the most natural representation to begin with is the _ adjoint representation_. it admits a diagrammatic representation in terms of graphs on @xmath176 points in which every vertex has degree 1 . usually one orders the @xmath176 points on two horizontal parallel lines as shown in fig . [ f:1factor ] . let @xmath177 denote the vector space spanned on the @xmath178 such diagrams . the product @xmath179 of two diagrams @xmath180 and @xmath181 is performed by putting @xmath180 on top of @xmath181 and replacing each of the loops in the resulting diagram with @xmath85 . define diagrammatically the identity @xmath17 and the generators @xmath166 , @xmath167 as represented in fig . [ f : gen_diag ] . one can check that the graphical representation of generators with the multiplication @xmath182 of diagrams satisfy all of the eqs .  . the left action of generators on @xmath177 via the multiplication @xmath182 of diagrams provide the adjoint representation of @xmath42 .    , @xmath166 and @xmath167 . ]    from the graphical representation we see that @xmath177 has a series of invariant subspaces @xmath183 , where @xmath184 is spanned on diagrams with fewer than @xmath185 vertical lines and @xmath186 . the vector space spanned on diagrams with exactly @xmath185 vertical lines may be defined as a @xmath42 module by the coset @xmath187 . the left action of @xmath42 on this modules may be seen as a modified multiplication @xmath188 of diagrams @xmath189    under the left action of the algebra the position of horizontal lines in the bottom of a diagram does not change . for a given configuration of the horizontal lines in the top of a diagram and a given pattern of intersections of vertical lines there are @xmath190 possibilities of choosing the configuration of horizontal lines in the bottom of the diagram . this simply means that @xmath191 decomposes into a direct sum of @xmath192 equivalent modules . the coset representative @xmath193 of these equivalent left modules is spanned on @xmath194 graphs on @xmath43 points with every vertex having degree 0 or 1 and a labeling with numbers @xmath195 of free vertices . an example of such a labeled graph is shown in fig . [ f : lab_1fact ] . if the labellings are omitted the resulting graph is called a partial diagram . the labeling of the @xmath185 free points of a labeled graph is a permutation @xmath196 in the symmetric group @xmath197 . the labeled graphs will provide a representation of the brauer algebra , which is irreducible for generic values of @xmath85 , if we take the labellings @xmath196 in an irreducible representation of @xmath197 . we call such representations generically irreducible . let @xmath198 be a partition of @xmath185 , which we write as @xmath199 . in a more algebraic language the definition of generically irreducible left modules translates to @xmath200 where @xmath201 is an irreducible @xmath197 module . in view of later numerical analysis we give below a basis in @xmath66 and describe the action of @xmath42 on this basis .    . the labeling permutation is @xmath202 . ] let @xmath203 denote the labeling of a partial diagram @xmath204 with the permutation @xmath196 , @xmath205 be a set of basis vectors in @xmath201 and @xmath206 be the matrix of the permutation @xmath207 in the representation @xmath208 . a natural basis in @xmath66 is given by all pairs @xmath209 . the action of a diagram @xmath210 on a basis vector is @xmath211 where @xmath207 is the labeling of @xmath212 and @xmath213 is the partial diagram @xmath204 labeled with the identity permutation . the dimensions @xmath214 of @xmath66 is @xmath215 .    in simple words , a generically irreducible module is a span on graphs on @xmath43 points , obtained by choosing @xmath185 points among @xmath43 , pairing all the others ( this gives the multiplicity @xmath216 since intersections are allowed ) , choosing for the @xmath185 unpaired ones a representation of the permutation group and setting to zero the action of any brauer diagram that reduces the number @xmath185 of unpaired points . the generically irreducible representations labeled by @xmath217 , @xmath218 $ ] appear in the decomposition of the adjoint representation with multiplicity given by their dimension @xmath214 when @xmath42 is semisimple . let us conclude with a few words about the reducibility of generically irreducible modules @xmath66 . for integer @xmath85 and a number of strings @xmath219 the brauer algebra is not semisimple and , as a consequence , certain of the modules @xmath66 become reducible , though they remain _ indecomposable_. the irreducible components appearing in such reducible modules @xmath66 are far from being understood ( the situation is much worse than in the case of the nonsemisimple temperley lieb algebra @xcite ) . numerical computations based on the diagonalization of the transfer matrix in the diagrammatic representation of @xmath42 restricted to @xmath66 decreases in efficiency very fast with increasing @xmath43 , compared to the ideal case where the transfer matrix is restricted to an irreps of @xmath42 . this is because for big @xmath43 and @xmath198 fixed the number of irreducible components in @xmath66 `` goes wild '' and there are a lot of `` accidental degeneracies '' in the spectrum of the transfer matrix restricted to @xmath66 . however , a significant progress in this direction has been recently made in @xcite . let us note that , as described in @xcite , the content of ( at least some ) @xmath66 s can be computed by repeated applications of frobenius reciprocity applied to the short exact sequence of @xcite describing the structure of the induced modules @xmath220 .    in the end we recall the basic results for the temperley lieb algebra , to allow a quick comparison with brauer . temperley lieb algebra diagrams are a subset of brauer algebra diagrams subject to the constraint that no intersections between edges are allowed . the dimension of the algebra is given by the catalan numbers @xmath221 . the main line of reasoning for finding generically irreducible modules follows the same way , except there is no available action of the symmetric group on vertical lines . therefore , the analogue of the labeled graphs will be the partial diagrams , in which no free points may be trapped inside an edge . the number of such graphs is @xmath222 , where @xmath223 is the number of edges . the generically irreducible modules @xmath224 are parametrized by the number @xmath225 of free points in the graphs . the presented facts about the brauer algebra should be enough to understand the loop gas reformulation of @xmath41 spin model , which we give in the next section . the emergence of dense intersecting loops becomes transparent if we take the local transfer matrices in the adjoint representation of the brauer algebra . this simply amounts to replacing in eq . the generators @xmath17 , @xmath166 and @xmath167 defined by eq .   with the diagrams in fig . [ f : gen_diag ] . the adjoint local transfer matrix is represented in fig . [ f : vert ] . we now define a loop model on a diagonal lattice represented in fig . [ f : dl ] , with reflecting boundaries on the left and right ( ie , free boundary conditions in the space direction ) and identified boundaries on the top and bottom ( quasiperiodic boundary conditions in the time direction . ) the states of this model are coverings of the lattice with dense _ intersecting _ loops . dense means that every edge on the lattice necessarily belongs to a loop . avoiding loop vertices have weight 1 and intersections come with weight @xmath98 . there are two possible ways for a line to close in a loop . the first one comes from the graphical representation of the relation @xmath226 in fig . [ f : e2 ] . we call such loops _ bulk _ loops . clearly the fugacity of bulk loops is fixed to @xmath85 by the brauer algebra . the second possibility is that the ends of the line close in the identified points of the top and bottom boundaries of the lattice . we call such loops _ cycles_. the boundary condition we consider have an annulus geometry and , thus , a cycle can be either _ contractible _ or _ uncontractible_. the fugacity of cycles is not fixed by the algebra . in fact , as we explain below , this is exactly the degree of freedom allowing for multiple mappings from the @xmath41 spin models with @xmath117 fixed and the dense intersecting loop model with fugacity @xmath85 for loops . we start by evaluating the trace @xmath227 of a diagram @xmath57 in the spin representation and then we generalize the result for quasiperiodic boundary conditions given by @xmath228 . we follow the same line of reasoning as in @xcite .    a cycle in a diagram @xmath57 is the subgraph on the set of points belonging to a loop if we identify its top and bottom vertices . by an abuse of language we call the corresponding loop also cycle . if we put a diagram @xmath180 to the left of a diagram @xmath181 we get a new diagram which we denote @xmath229 . let @xmath230 be the cycles in @xmath57 . we can separate them by permuting the top and bottom vertices of @xmath57 with the same permutation @xmath196 @xmath231 thus the trace of a diagram depends only on the weights of cycles @xmath232 more than that , the weight of a cycle depends only on how many times it winds the annulus . indeed , if a cycle on @xmath233 points has no horizontal lines , then , by applying the same permutation to the top and bottom vertices we can bring it to the cycle @xmath234 . this is because permutations with one cycle are conjugate in @xmath197 . if a cycle @xmath235 has a horizontal edge between the first and the second vertex in the top then it has the same weight as a certain cycle @xmath236 on four points less then @xmath235 @xmath237 if we compare the @xmath235 on the left with @xmath236 on the right        it is clear that , in the end of the iterative application of eq .  , the final cycle can be interpreted as being the initial cycle @xmath235 maximally contracted on the annulus .    in the end , the only weights we need to compute explicitly are that of the cycles @xmath32 and @xmath234 @xmath238    for boundary conditions twisted by the matrix @xmath239 the generalized weights are computed to be @xmath240    in the fundamental representation , every supermatrix @xmath58 is diagonalizable . the diagonal form of @xmath241 in the fundamental representation is determined by exponentiating elementary weights @xmath242 and @xmath243 introduced in sec . [ sec : superalg ] . thus , @xmath58 restricted to @xmath244 has eigenvalues @xmath245 and restricted to @xmath246 has eigenvalues @xmath247 . the braces in @xmath248 mean that @xmath249 appears for odd @xmath12 only . eq .   can now be rewritten @xmath250 for @xmath251 only the eigenvalues in @xmath244 change with respect to the previous case . there are of the form @xmath252 for @xmath12 odd , while for @xmath12 even @xmath253 and @xmath254 . instead of eq . one has now @xmath255    to summarize the basic results in this section , let @xmath87 be a dense loop covering of the lattice , @xmath17 be the number of intersections , @xmath256 be the number of bulk loops , @xmath105 be the number of contractible loops ( cycles ) and @xmath32(@xmath106 ) be the number of loops winding the annulus an even(odd ) number of times .    on the annulus the trace partition function ( which would correspond to antiperiodic boundary conditions in the ( imaginary ) time direction ) of the @xmath41 spin model may be reformulated as a dense intersecting loop model in the following way @xmath257 we see that it does depend on @xmath33 separately and not only on @xmath85 .    meanwhile the supertrace partition function ( which would correspond to periodic couplings ) reads @xmath258 where @xmath259 is the total number of `` loops '' and , since it depends on @xmath85 only , is the same as for the @xmath10 model . one can say that taking the supertrace in the partition function is equivalent to restricting the @xmath41 supersymmetry of the spin model to a , smaller , @xmath10 symmetry . denote the spectrum of the transfer matrix of the @xmath41 spin model by @xmath260 . we have the following important inclusion property @xmath261 the only difference between the @xmath41 and @xmath262 quasiperiodic partition functions is the weight of uncontractible cycles . for @xmath239 a matrix with eigenvalues @xmath263 and @xmath264 , except @xmath265 , the weight of uncontractible cycles is , according to eq .   either @xmath85 or @xmath266 . notice that these are exactly the weights of uncontractible cycles in the trace partition function for the @xmath262 spin model , which proves eq .  . we will use the inclusion property in the next section to derive some information about the indecomposable representations of @xmath41 appearing in the decomposition of @xmath2 . assume the transfer matrix be a generic element of @xmath42 . the action of @xmath42 on the tensor space @xmath2 was defined in the beginning of sec .  [ sec : br_alg ] . the complete picture of the reducibility of the transfer matrix can be conveniently encoded in the decomposition of @xmath267 into a direct sum of @xmath42 indecomposable modules @xmath268 where @xmath269 denotes the multiplicity of isomorphic indecomposable @xmath42 modules @xmath270 ( it does not depend on @xmath43 ) , and the set @xmath271 is defined implicitly by the formula , and will be defined explicitly below . we remind the reader that @xmath2 is not necessarily a semisimple @xmath42 module if @xmath219 , so the modules @xmath270 appearing on the rhs of eq .   can be reducible . the question of computing degeneracies of eigenvalues of the spin transfer matrix is easier to treat by looking at the centralizer @xmath272 , which acts on @xmath2 from the left if one consider @xmath42 acting from the right . the dimension of indecomposable modules @xmath273 in the decomposition of @xmath2 as a @xmath274-module @xmath275 will give the desired degeneracies . this is due to the fact that @xmath276 are dimensions of simple @xmath42 modules @xmath69 appearing as constituents of @xmath270 in eq .  , while @xmath269 are dimensions of simple @xmath277 modules @xmath75 appearing as constituents of @xmath273 in eq .  . taking the character of both eqs . ( [ eq : bd],[eq : cd ] ) one can see that the number @xmath278 of irreducible components @xmath69 in @xmath270 is equal to the number @xmath279 of irreducible components @xmath75 in @xmath273 .    because the action of @xmath40 commutes with @xmath42 we have that @xmath280 . however , when @xmath2 is semisimple it follows from the wedderburn decomposition theorem that @xmath281 . in the following we suppose that there is still a schur duality between @xmath40 and the quotient of @xmath42 faithfully represented on @xmath2 . this allows us to give an algorithm to compute the lhs of eqs . ( [ eq : bd],[eq : cd ] ) for small tensor powers @xmath43 and get some intuition about the general structure of @xmath270 and @xmath273 . the set of partitions @xmath59\,\}$ ] labels @xmath42 irreps , while @xmath60 selects those of them which do realize on the tensor space @xmath2 . denote by @xmath282 the double sided ideal defined by @xmath283 . the annihilator @xmath284 is diagrammatically described in @xcite . under the homomorphism @xmath285 , the indecomposable modules @xmath66 give rise to induced modules @xmath286 . clearly , @xmath287 is a tensor representation and can be generated by trace substraction and symmetrization as @xmath288 where @xmath289 , @xmath290 extracts all the traces from the tensor space @xmath291 and @xmath292 acts nontrivially only on @xmath291 as a young symmetrizer . the double sided ideal @xmath293 is completely characterized by the set of weights @xmath294 . note that @xmath295 . the surviving indecomposable tensor modules @xmath287 are given by @xmath66 with irreducible components @xmath296 removed . the quotient @xmath297 can be carried out by imposing the vanishing of all words @xmath298 with @xmath299 . it is useful to notice that not all of these conditions are independent and as one can see from eq . @xmath300 if @xmath301 .    as discussed in sec . [ sec : superalg ] and [ sec : supergr ] , the @xmath40 irreducible components of @xmath273 are indexed ( up to an equivalence under the action of the outer automorphism @xmath302 induced by the symmetry of the dynkin diagram of @xmath40 when @xmath12 even ) by the set @xmath64 of hook shape partitions . representing the supergroup as a semidirect product @xmath303 , the elements of @xmath271 naturally acquire the structure of a couple of the form @xmath304 or @xmath305 if @xmath306 and @xmath307 if @xmath308 , where @xmath309 are the trivial , alternating ( superdeterminant ) and two dimensional representations of @xmath310 . thus , every @xmath311 gives rise to two @xmath41 inequivalent irreps with highest weights @xmath312 and the associate @xmath313 if @xmath306 and a single self - associate irreps of highest weight @xmath314 if @xmath308 . for typical @xmath311 , one can realize the @xmath41 irreps @xmath61 on tensors @xmath315 and describe their symmetry by some young tableaux . as discussed in details in sec . [ sec : tmb ] , the young tableau corresponding to @xmath316 can be constructed by adding a border strip to the young tableau of shape @xmath51 corresponding to @xmath317 . ultimately , this is justified by the fact that eq . ( [ eq : elchar],[eq : onchar ] ) gives the right characters for @xmath318 and that they coincide up to @xmath319 . although atypical representations can not be realized as tensor representations we represent the associate weight @xmath320 of an atypical weight @xmath51 by a young tableau such that @xmath321 is a skew partition described in sec . [ sec : tmb ] and sec . [ sec : mod_rules ] . the idea is to exploit the fact that the characters of indecomposable modules @xmath66 , given in @xcite , do not depend on the semisimplicity of @xmath42 . this and some properties of generalized schur functions , which are summarized in @xcite , can be used to prove that @xmath322 is true even for all @xmath43 . here @xmath323 is the character of @xmath210 in the representation provided by @xmath66 . the functions @xmath324 are polynomials in the eigenvalues of @xmath239 , which where introduced for the first time by bars in @xcite in an early attempt to describe the supercharacters of @xmath41 . they can be defined recursively as @xmath325 and @xmath326    for @xmath327 the brauer algebra @xmath42 is semisimple and @xmath328 . consequently , @xmath66 are irreducible and @xmath329 . because of the commuting actions of @xmath41 and @xmath42 one can naturally consider @xmath2 as a @xmath41-@xmath330-bimodule , with @xmath41 acting from the left and @xmath42 from the right . then , eq .   can be understood as a consequence of the decomposition @xmath331 with @xmath332 being actual characters of tensor irreducible modules @xmath333 as shown in @xcite .    for @xmath33 such that @xmath219 , the polynomials @xmath332 can not generally be interpreted as the character of some @xmath41 representation . as we have seen in sec . [ sec:1749 ] , @xmath334 can be brought to the form @xmath335 and eq . is not more then a simple equality between two polynomials in eigenvalues of @xmath58 . moreover , the two eqs . ( [ eq : cd],[eq : bf ] ) are still compatible , even if there are much more elements in @xmath336 then in @xmath337 . this is possible because @xmath332 are not functionally independent when @xmath219 . then , for @xmath338 the polynomials @xmath332 can be written in terms of functionally independent @xmath73 with @xmath339 by means of modification rules for characters @xmath340 given in @xcite and discussed in details in sec . [ sec : mod_rules ] . the fundamental eq . is useful for small widths @xmath43 , when it is possible to compute the number @xmath341 of irreducible components @xmath342 in @xmath66 either by repeated applications frobenius reciprocity , as explained in @xcite , or by numerically diagonalizing the transfer matrix of sec .   in the adjoint representation of @xmath42 and detecting the `` accidental degeneracies '' in its spectrum . indeed , from the explicit definition it is clear how to restrict the adjoint transfer matrix to indecomposable modules @xmath66 . after we described in details the action of generators on the basis of @xmath66 in sec . [ sec : br_alg ] , the algorithm of a numerical diagonalization is straightforward .    the information about the structure of @xmath66 and the modification rules in eq .   can now be used to bring eq .   to the form @xmath343 we see that @xmath344 iff  , @xmath345 are not irreducible @xmath40 characters , one can still use them as a basis for representing the character of any representation . ] there is at least one @xmath339 such that @xmath346 . to determine @xmath279 one has to decompose the factor of @xmath347 in eq .   as a sum of @xmath41 irreducible characters , which are explicitly known , as far as we know , only for @xmath348 and @xmath349 . given the huge order of the set of weights @xmath336 , it may seem that calculations according to eq . are extremely cumbersome already for small @xmath43 . the simplifying point is that @xmath350 ( or @xmath351 ) is non zero only if both weights are in the same equivalence class of @xmath271 ( or @xmath337 ) . the splitting of @xmath271 ( or @xmath337 ) into equivalence classes , called _ blocks _ and described in details in sec . [ sec : tmb ] , is with respect to an equivalence relation between irreducible components of indecomposable @xmath41 ( or @xmath42 ) modules . an important consequence of the fact that @xmath41 supertrace partition functions depends only on the @xmath10 part of the spectrum is the vanishing of the superdimension @xmath352 for all indecomposable modules with @xmath353 . a more restrictive criterion for @xmath273 supercharacters deriving from the full inclusion sequence in eq . can be derived by taking a matrix @xmath58 with eigenvalues @xmath354 and @xmath355 for @xmath356 , @xmath357 . then , it can be seen from eqs . ( [ eq : trp+],[eq : loop_pf ] ) or eqs . ( [ eq : elchar],[eq : onchar],[eq : bf ] ) that any @xmath41 quasiperiodic partition function will also be an @xmath262 quasiperiodic partition function . as a consequence @xmath358 and the supercharacters of @xmath273 vanish when @xmath359 and @xmath58 can be embedded in @xmath262 .    for @xmath174 the modules @xmath2 is semisimple . therefore @xmath360 . on the other hand , if @xmath29 is big enough @xmath361 . it could be interesting to understand the relation between the sequence @xmath362 and the cohomology of the radical @xmath363 . observe that a filtration similar to that of @xmath337 is available on @xmath271 by the degree of atypicality of its elements . in fact , we explain in sec .  [ sec : tmb ] of the appendix how the weights in a block of @xmath337 or @xmath271 can be organized by the number of _ removable balanced continuous border strips _ in the corresponding young tableau . this number can be interpreted as the degree of atypicality when the corresponding partition represents an @xmath41 weight . let @xmath124 be a two dimensional vector space endowed with an action of @xmath364 . the action of @xmath39 on the tensor space @xmath2 is @xmath365 @xmath3 acts according to the following definition of generators @xmath366 @xmath367 where @xmath368 have the following representation on @xmath126 @xmath369 ( note that @xmath32 differs in some essential way from the projection operator onto the singlet representation in the usual @xmath370 basis ) . the decomposition of @xmath2 as a @xmath39-@xmath3-bimodule is simply @xmath371 here @xmath372 the tensor representations @xmath373 are irreducible with dimensions @xmath374 and @xmath375 . the representation @xmath376 is the trivial one and @xmath377 is the associate one dimensional @xmath378 representation . at the restriction to @xmath89 the representations @xmath376 and @xmath377 become equivalent , while @xmath379 splits into two nonequivalent one dimensional representations @xmath380 where @xmath381 is the @xmath382 angle . the @xmath3 representations @xmath69 are irreducible as well and are constructed by acting with @xmath3 on the tensor module in eq .  . the dimensions of simple modules @xmath383 are easily computed by looking at the first row of @xmath384 , where @xmath105 is the fusion matrix @xmath385 described by a @xmath386 type dynkin diagram with labeling of the nodes shown in fig . [ fig : fusion ] . it is not hard to solve the recurrence relations satisfied by @xmath387 @xmath388 and get that @xmath389+k}$ ] except the case when @xmath43 is even and @xmath390 for which @xmath391 . these are , as expected , the number of eigenvalues of the transfer matrix for the 6 vertex model in the sector of spin @xmath392 .    . ]    we are interested in giving an algebraic description of the 6 vertex model transfer matrix algebra . in other words we want to identify the brauer algebra annihilator @xmath393 of @xmath2 and carry out the quotient @xmath394 . all the @xmath3 weights @xmath395 such that @xmath396 satisfy either @xmath397 or @xmath398 . thus , it is enough to consider @xmath399 and impose the vanishing of the double sided ideal of the word @xmath400 projecting onto @xmath401 . as explained in the beginning of the previous section @xmath402 , where @xmath403 extracts all the traces from @xmath404 and @xmath405 are the young symmetrizers corresponding to @xmath406 . the projector @xmath403 can be found by looking at the form of an arbitrary tensor @xmath407 after extracting all of its traces @xmath408 which gives @xmath409 and clearly @xmath410 . the young symmetrizer @xmath411 is @xmath412 and @xmath413 , where @xmath414 are the projectors onto the _ standard _ young tableau @xmath415 $ ] and @xmath416 $ ] . the two orthogonal projectors @xmath417 and @xmath418 are independent only if we restrict to the right @xmath3 action . in fact , the left ideal of the word @xmath419 is the same as the double sided ideal of the word @xmath420 .    the condition @xmath421 gives the following restriction @xmath422 on generators @xmath423 . putting @xmath424 one can see that eq .   implies that @xmath425 are temperley lieb operators with @xmath426 . there are no more restrictions that can be drawn from the conditions @xmath427 , because @xmath428 is a one dimensional projector .    before exploring the next vanishing condition let us revise the the defining relations of @xmath3 given in eq . @xmath429 which imply @xmath430 observe that although the algebra has now two temperley lieb operators their role is not symmetric yet at this stage . next , the condition @xmath420 implies @xmath431 which after inserting @xmath432 with the help of eqs . ( [ e : first_step],[e : first_step1 ] ) becomes @xmath433 multiplying eq .   by @xmath434 on the right we get : @xmath435 which can be used to rewrite eq .   as @xmath436 multiplying by @xmath437 on the left and on the right of eq . and using only the relations between @xmath425 , the relations between @xmath166 and eq .   one can get all the eqs . ( [ e : first_step][e : first_step10 ] ) and also @xmath438 which establish a complete symmetry between @xmath166 and @xmath425 .    the double sided ideal of @xmath420 is composed of four linearly independent words  two generated by the left action and other two generated by the right action of @xmath3 . it is useful to note that after taking the quotient of @xmath439 we are left with 10 independent words instead of 15 , which is exactly what we need for the 6 vertex local transfer matrix .    we give the following abstract definition to the 6 vertex model transfer matrix algebra @xmath440 in term of generators @xmath441 latexmath:[\ ] ] where @xmath592 and the content of @xmath587 is represented below @xmath589 .    apart the last irreducible module @xmath593 , we recognize in the terms of eq . the projective representations of the quiver @xmath594 in fig . [ fig : einfty ] , which describes the homomorphisms between the @xmath3 tensor modules @xmath531 realized on @xmath2 . it will turn out in our forthcoming analysis of conformal properties to be easier to study numerically the hamiltonian @xmath595 the expectation  which we will confirm in great details  is that this hamiltonian will be in the same universality class as the spin model we had started with . the hamiltonian @xmath596 is obviously local and has only nearest neighbour interactions if the @xmath32 s and @xmath31 s are taken in the spin representation provided by eq .  . however , this is no longer true if we think of @xmath596 as an element of the adjoint representation of @xmath3 . the lowest eigenvalue of @xmath596 belongs to the @xmath3 irreducible representation labeled by @xmath597 .    for generic @xmath112 it is nondegenerate if @xmath43 is even and has degeneracy @xmath598 if @xmath43 is odd . on the other hand , the highest eigenvalue belongs to the completely antisymmetric representation labeled by @xmath599 . in this representation the @xmath31 s act as -1 and the @xmath32 s as 0 . the hamiltonian @xmath596 is determined up to an arbitrary additive constant and multiplicative factor . for numerical diagonalization it is convenient to fix the additive constant such that the maximal eigenvalue of @xmath596 be zero . the multiplicative factor is fixed by requiring @xmath600 with @xmath601 as in eq .  , @xmath15 as in app . [ sec : superalg ] and @xmath602 being the _ xxz _ spin chain hamiltonian in its usual form @xmath603    the fact that the eigenvalues of the 6 vertex model appear as a subset of the eigenvalues of the transfer matrix for the @xmath0 model and @xmath604 carries over to a similar result for the hamiltonians . the velocity of sound for the massless excitations can thus be derived from its value for the _ xxz _ subset , which is well known from @xcite to be @xmath605 the hamiltonian @xmath606 is diagonalized numerically in the adjoint representation of the brauer algebra by studying its action on the diagrams just like for the transfer matrices . next , once the structure of indecomposable modules @xmath66 is known , eq .   can be used as explained in sec . [ sec : gf ] to select the part of the spectrum which does indeed appear for a fixed @xmath29 spin model . however , in the two special cases @xmath607 the hamiltonian @xmath596 greatly simplifies . in the following two subsections we discuss the behaviour of the spectrum of @xmath596 in the two limits @xmath608 . when @xmath609 the temperley lieb operators @xmath166 do not contribute to @xmath596 and , thus , the hamiltonian is no longer a generic element of the brauer algebra @xmath3 , but belongs instead to the subalgebra @xmath610 . this will translate to additional degeneracies in the spectrum of @xmath596 at the point @xmath611 compared to other points in the range @xmath612 . hamiltonians of type @xmath613 , with @xmath31 s in the representation provided by eq . , are integrable and have been studied in @xcite and @xcite . although the continuum limit of such spin chains is a gapless field theory , it fails to be conformal , because excitations have a @xmath614 scaling law in the thermodynamic limit . this can readily be seen from the vanishing of the sound velocity in eq .  . we will not enter into the details here , but just mention that the different systems of bethe ansatz equations are indexed by @xmath615-hook shape partitions @xmath50 . this is exactly the label of irreducible representations of the group algebra @xmath54 realizing in the centralizer of the spin chain @xmath2 , with @xmath124 being the fundamental representation of @xmath38 . we see that the symmetry of our spin model @xmath0 jumps to @xmath38 at the point @xmath611 . the additional degeneracies in the spectrum of the @xmath0 spin model at the point @xmath611 can be understood by looking at the decomposition of @xmath3 modules @xmath66 , into a direct sum of @xmath54 irreducible modules @xmath67 . let @xmath289 and @xmath50 , then it was shown in @xcite that the multiplicity of @xmath67 in the decomposition of @xmath66 is @xmath616 where @xmath617 are littlewood - richardson coefficients . alternatively , @xmath618 is the number of tensors of rank @xmath619 , with index symmetry of some fixed standard young supertableau of shape @xmath198 , that can be obtained from a tensor of rank @xmath43 , with index symmetry of some standard young supertableau of shape @xmath51 , by contracting @xmath620 indices in all the possible ways .    one can apply eq .   to understand the degeneracy of the lowest level of @xmath596 at @xmath611 . first , observe that @xmath613 is minimized in the sector @xmath621 ( where @xmath31 s acts as 1 ) . the _ only _ @xmath198 such that @xmath622 are one row partitions . thus , the lowest eigenvalues of @xmath596 restricted to @xmath623 for @xmath218 $ ] become all degenerate at @xmath611 .    arguments of this kind can be used to derive information about the critical exponents of the spin model in the limit @xmath624      the same reasoning can be applied to the point @xmath626 . at this point , the hamiltonian @xmath596 belongs to the temperley lieb subalgebra @xmath627 and the model can be considered as a spin chain @xmath628 where @xmath629 are the fundamental representation of @xmath38 and its conjugate . additional degeneracies can be understood by looking at the decomposition of @xmath3 modules @xmath66 as a direct sum of standard irreducible @xmath630 modules @xmath70 . let us compute the multiplicity @xmath631 of irreducible modules @xmath70 in the decomposition of @xmath66 with @xmath632 .    as explained in sec . [ sec : br_alg ] , @xmath66 has a natural basis composed of all possible pairings @xmath633 of partial diagrams @xmath204 with @xmath634 free points and basis vectors @xmath635 of @xmath201 . we say that a horizontal line of a partial diagram @xmath204 is _ intersected _ either if it intersects another horizontal line or if there is a free point in @xmath204 between the two ends of the horizontal line . let us associate to each partial diagram @xmath204 the number of intersected horizontal lines @xmath502 in @xmath204 . it is not hard to see that the span on the basis vectors @xmath209 , with @xmath204 s having at most @xmath502 horizontal intersected lines , is a @xmath630 submodule in @xmath66 . if we denote this submodule by @xmath636 there is an obvious filtration @xmath637 of @xmath66 . consider the natural action of @xmath630 on the quotient modules @xmath638 . observe that the action of @xmath630 changes the labeling @xmath639 of free points in a labeled graph @xmath203 if and only if it also reduces the number of horizontal intersected lines . therefore , @xmath640 is isomorphic to a direct sum of @xmath641 modules @xmath642 . obviously @xmath643 and , therefore , we get @xmath644 for @xmath645 , @xmath646 for @xmath647 and finally @xmath648 .    thus , our problem effectively reduces to understanding the action of @xmath630 on the module @xmath649 , which is composed of partial diagrams @xmath204 on @xmath43 points with @xmath185 unlabeled free points .    at a closer look , one can see that the action of @xmath630 on partial diagrams keeps the reciprocal configuration of intersected lines and free points intact . in other words , if @xmath650 is a map that eliminates all the nonintersected horizontal lines from a partial diagram and acts as identity otherwise , then @xmath650 defines an invariant of @xmath630 , that is @xmath651    to understand the meaning of this invariant let us define a local map @xmath381 between partial diagrams which sends intersected horizontal lines to free points as depicted in fig . [ fig : themapphi ] and acts as identity otherwise . the local map @xmath381 is applied repeatedly until there are no more horizontal intersected lines left .    . ]    it is not hard to see that @xmath381 extends to a homomorphism of @xmath630 modules @xmath652 in fact , the role of the map @xmath381 is to show that @xmath653 is composed of a direct sum of isomorphic @xmath654 modules , while that of the map @xmath650 is to distinguish between these modules . the set of partial diagrams @xmath204 in @xmath653 splits into subsets of constant @xmath655 and each of these subsets is isomorphic to @xmath654 as a @xmath630 module .    according to what was said before , we get that @xmath656 equals to the number of graphs on @xmath657 vertices and @xmath502 intersected edges . it follows that @xmath656 does not actually depend on @xmath43 and we drop the index @xmath43 in the following . this fact allows , in principle , for an iterative computation of @xmath658 by simply computing the dimensions of the left and right hand sides of the decomposition formula @xmath659 that is @xmath660 successively for @xmath661 or @xmath662 . one can give an explicit expression for @xmath658 with a little more combinatorial work . we call a horizontal line an _ empty cup _ if its ends are adjacent and simply a _ cup _ if there are separated by free points . observe that all the lines in a partial diagram are intersected if and only if there is at least one free point in each cup . thus , if the partial diagram has @xmath204 cups with only one free point inside and a total of @xmath502 edges then the remaining @xmath663 free points can be added to the diagram in @xmath664 different ways in such a way that the resulting diagram has only intersected edges . moreover , the number of diagrams on @xmath665 points with @xmath204 empty cups and a total of @xmath502 edges is again @xmath666 . this is because the condition of no cups in the connection of the remaining @xmath667 edges is similar to the condition of composing a graph with @xmath204 free points and @xmath667 intersected edges . putting everything together we get a new recurrence formula @xmath668 reducing the problem to the computation of @xmath669 . next , we want to find a recurrence relation for @xmath670 by looking at the connectivity of the first point in the partial diagrams on @xmath671 points with @xmath83 intersected edges . the leftmost vertex in the partial diagram has to be connected to some other vertex at position @xmath501 . the connectivity of the @xmath672 points to the left of the point at position 1 is equivalent to that in a partial diagram with @xmath673 intersected edges and a free point except for the case where @xmath513 . therefore we have that @xmath674 now , eq .   yields @xmath675 . using again eq .   for @xmath673 we finally get that @xmath676 the solution of the recurrence eq .   with the initial conditions @xmath677 and @xmath678 is @xmath679 and coincides with the absolute value of bessel polynomials @xmath680 @xmath681 evaluated at @xmath682 . besides the careful definition of the spin model and its sectors , the main point of this first paper is the algebraic set up necessary to analyze its symmetries . this is a non trivial task since we are dealing with non semi - simple algebras , and that the action of @xmath0 and @xmath3 are meshed through a complex structure of indecomposable representations . the main results are the decomposition formulas  ( [ eq : rez1],[eq : rez2 ] ) for @xmath481 viewed as a @xmath349 and a @xmath3 module . the decomposition in eq . has been computed in two essentially different ways : first , by decomposing tensor products between @xmath349 representations and @xmath124 without knowing anything about the brauer algebra and , second , starting from eq .   with the assumption that the representations of @xmath349 and @xmath3 on @xmath2 generate the full centralizers of each other ( schur duality ) . the fact that we arrive at the same result using both methods highly suggests that our assumption about the schur duality between @xmath349 and @xmath3 on @xmath2 is correct .    when the question of decomposing @xmath2 is addressed in sec . [ heart ] , the notion of block appears to be a particularly useful concept for organizing indecomposable representations . these results will be applied to educated conjectures about the conformal field theory in the next paper . although there are many things left unclear about the representation theory of @xmath683 , it is very tempting to speculate the form of the decomposition of @xmath684 . before making the guess , observe that as a @xmath349 module @xmath685 , where @xmath31 is a direct sum of projectives organized in blocks , while @xmath158 is a direct sum of simples indexed by the same young tableau ( in the partition notation for dominant weights ) as the irreps of @xmath39 . more than that , they appear with the same multiplicities as their partners in @xmath686 . that the same phenomenon occurs for @xmath687 . in this case @xmath158 is the trivial representation . ] therefore , @xmath158 and @xmath688 are similar in all but the internal structure of their simple summands . the similarity between the two modules has to be understood in terms of their centralizers , because these are precisely the objects that do not `` see '' the internal structure of simples . is a simple module for the algebra @xmath689 then @xmath690 . ] in conclusion , one should have @xmath691 , which is quite natural once there is a schur duality between @xmath349 and @xmath3 on @xmath684 . it is tantalizing to speculate that as a @xmath0 module @xmath692 , with @xmath31 projective and @xmath693 . thus , the problem of the decomposition of @xmath684 as a @xmath0 module is reduced to understanding the projective representations of the supergroup , i.e. to finding the quiver diagram for each block . it has been suggested in @xcite that the quiver diagram of blocks does not depend on @xmath29 provided the degree of atypicality @xmath501 and the action of the outer automorphism @xmath302 are fixed . can act in two ways : either leave invariant all the weights in the block or pairwise transform some of them . ] the discussion of sec . [ sec : osp42 ] suggests that the two types of quivers for a block of @xmath694 and a fixed @xmath501 will give rise to the same quiver for the induced blocks in @xmath0 . we also succeeded in computing the multiplicity of temperley lieb representations in a standard @xmath3-module @xmath66 . finally , we gave a combinatorial description of @xmath42 blocks as the set of minimal partitions dressed by balanced removable border strips and have shown that there is a similar description for @xmath40 blocks . * acknowledgments : * this work was supported by the agence national pour la recherche under a programme blanc 2006 int - ads / cft . we would like to thank j. germoni for fruitful correspondence , and j. jacobsen and p. martin for useful discussions . cc i. bena , j. polchinski and r. roiban , phys . d 69 ( 2004 ) 046002 , hep - th/0305116 . f. wegner , nucl . phys . b316 ( 1989 ) 663 . m. zirnbauer,``conformal field theory of the integer quantum hall plateau transition '' , hep - th/9905054 . n. read and h. saleur , nucl . b613 ( 2001 ) 409 . n. mann and j. polchinski , phys . d72 ( 2005 ) 086002 and n. mann and j. polchinski ; hep - th/0508232 . n. read and h. saleur , nucl . b 777 , 263 ( 2007 ) . n. read and h. saleur , nucl . b 777 , 316 ( 2007 ) . b. nienhuis , e. k. riedel and m. schick , phys . b27 , 5625 . e. domany , d. mukamel , b. nienhuis and a. schwimmer , nucl . b190 ( 1981 ) 279 . b. nienhuis , j. stat . phys . 34 ( 1984 ) 731 . j. l. jacobsen , n. read and h. saleur , phys . ( 2003 ) 090601 ; cond - mat/0205033 . g. parisi and n. sourlas , j. physique lett . ( paris ) 41 ( 1980 ) l403 . a. ram , pac . j. math . 169 ( 1995 ) 173 . w. f. doran , d. b. wales and p. j. hanlon , j. algebra 211 ( 1999 ) 647 . g. benkart , c. l. shader and a. ram , j. pure appl . algebra 130 ( 1998 ) 1 . j. birman and h. wenzl , trans . am . ( 1989 ) 249 . j. murakami , osaka j. math . 24 ( 1987 ) 745 . m. wadati , t. deguchi and y. akutsu , phys . 180 ( 1989 ) 249 . h. weyl , `` the classical groups '' , princeton university press ( 1946 ) . f. gavarini , `` on the radical of brauer algebras '' , math.rt/0606501 ( 2006 ) . f. goodman , p. de la harpe and v. jones , `` coxeter graphs and towers of algebras '' , msri publications 14 , springer , berlin 1989 . p.p . martin , `` potts models and related problems in statistical mechanics '' , world scientific , singapore ( 1991 ) . a. cox , m. de visscher and p. martin , `` the blocks of the brauer algebra in characteristic zero '' , math.rt/0601387 . a. cox , m. de visscher and p. martin , `` a geometric characterization of the blocks of the brauer algebra '' , math.rt/0612584 . a. balantekin and i. bars , j. math . ( 1981 ) 1149 . c. j. cummins and r. c. king , j. phys . a 20 ( 1987 ) 3103 . r. green , jktr 7 ( 1998 ) 155 . d. levy , int . phys . a6 ( 1991 ) 5127 . j. germoni , boletin de la academia nacional de ciencias 65 ( 2000 ) 147 ` http://hal.archives-ouvertes.fr/hal-00002750/fr/ ` . c. hamer , j. phys . a 19 ( 1986 ) 3335 . b. sutherland , phys . b 12 ( 1975 ) 3795 . h. saleur , nucl.phys . b578 ( 2000 ) 552 . p. hanlon and d. wales , math . comp . 54 ( 1990 ) 771 . v. serganova proceedings of the international congress of mathematicians , 1998 . v. kac , adv . ( 1977 ) 8 . b. morel , a. sciarrino , p. sorba , j. phys . a : math . gen . 18 ( 1985 ) 1597 . j. p. hurni , j. phys . a 20 ( 1987 ) 5755 . r. farmer , p. jarvis , j. phys . a : math . 17 ( 1984 ) 2365 . v. kac , lecture notes in math . 676 ( 1978 ) 597 . v. kac , commun . algebra 5 ( 1977 ) 889 . j. van der jeugt , j. math . phys . 26 ( 1985 ) 913 . i. macdonald , `` symmetric functions and hall polynomials '' , clarendon press , oxford ( 1979 ) , 10 on the left and @xmath695 on the right . ]    in this section we recall standard facts about the @xmath40 lie superalgebra mainly following the pioneering work of kac @xcite . for more details on @xmath40 young supertableaux see @xcite . let @xmath124 be a vector space with an additive @xmath696 grading @xmath129 , that is @xmath697 and @xmath698 . let @xmath699 , @xmath700 and @xmath701 $ ] . choose in @xmath124 a basis @xmath702 with @xmath703 and @xmath704 . we take the vector @xmath705 in brackets because it appears for odd @xmath12 only . the grading of @xmath124 induces a grading on @xmath706 , that is @xmath707 preserves the degree of @xmath708 and @xmath709 changes it . define the supertranspose of a matrix @xmath710 by @xmath711 let @xmath15 denote the matrix with the only nonzero components @xmath712 the lie superalgebra @xmath40 is realized as a subset of @xmath706 with elements @xmath158 satisfying @xmath713    in terms of elementary matrices @xmath714 the generators of @xmath40 read @xmath715 the generators @xmath716 span the cartan subalgebra @xmath717 . denote by @xmath718 the basis in @xmath719 dual to @xmath720 . it can be easily checked that generators in eq . correspond to roots of the type @xmath721 , generators in eq . correspond to roots of the type @xmath722 and generators in eq . correspond to roots of the type @xmath723 . the bilinear invariant form @xmath724 induces a scalar product on @xmath719 . the standard basis is recovered by putting @xmath725 for @xmath726 and @xmath727 for @xmath728 . elementary weights @xmath729 are orthogonal in @xmath719 and @xmath730 . the first @xmath731 simple roots are chosen to be @xmath732 , @xmath733 , @xmath734 for @xmath728 and @xmath735 . the last simple root is @xmath736 for odd @xmath12 and @xmath737 . the roots @xmath738 are called odd and the rest  even .    the component of a weight @xmath739 along the hidden simple @xmath740 root @xmath741 is @xmath742 according to @xcite , an @xmath40 highest weight is dominant iff it has integer dinkyn labels @xmath743 and integer @xmath744 satisfying the following consistency conditions @xmath745 all irreducible finite dimensional representations are indexed by dominant weights @xmath739 . given a dominant weight @xmath746 in the standard basis , the first @xmath731 dynkin labels are @xmath747 for @xmath748 , @xmath749 for @xmath750 . the last dynkin label is @xmath751 for @xmath12 odd and @xmath752 for @xmath12 even . from eq . we also get @xmath753 . the set of numbers @xmath754 define a partition , shown in fig . [ f : hook_shape ] , provided that consistency conditions plus some additional constraints depending on @xmath12 are satisfied . these additional constraints require @xmath755 and @xmath756 to be even if @xmath12 is even , and @xmath757 to be to be even if @xmath12 is odd .     representation to which belongs the highest weight state of a @xmath40 representation . ] the last two conditions define tensorial weights .    partitions @xmath51 such that @xmath758 are called hook shape . let @xmath302 denote the outer automorphism induced by the symmetry of the @xmath695 dynkin diagram under the exchange of the last two roots in fig . [ f : dynkin_diag ] . this automorphism is extremely important in understanding the difference between the representation theory of the supergroup @xmath41 and its lie superalgebra . note that @xmath302 can be explicitly realized through the discrete transformation @xmath759 exchanging the last two basis vectors in @xmath760 . indeed , @xmath761 for @xmath735 and @xmath762 because @xmath763 are the duals of @xmath764 . therefore @xmath765 .    in the case of @xmath12 odd , there is a bijective correspondence between hook shape partitions @xmath51 and dominant weights @xmath739 . the same holds for @xmath12 even , except for @xmath51 with @xmath766 when @xmath51 represents both @xmath739 and @xmath767 .    if there is a pair @xmath768 such that at least one of the conditions below are satisfied @xmath769 the weight @xmath51 is called _ atypical_. the highest weight @xmath739 is always atypical . ] see @xcite for the origin of these conditions and note ref . @xcite , where these have been presented in the form ( [ e:1_atyp],[e:2_atyp ] ) . if none of these conditions is satisfied , the weight is called _ typical _ and , according to @xcite , the associated kac module @xmath770 ( which is a finite dimensional quotient of the corresponding highest weight module ) is irreducible , its ( super)character is given by the weyl - kac formula @xcite and , in particular , its superdimension is zero . let @xmath771 be a grassman algebra . the supergroup @xmath41 may be realized as a subset of even supermatrices @xmath772 with entries in @xmath689 and @xmath58 belonging to @xmath773 , and entries in @xmath256 and @xmath105 belonging to @xmath774 , which satisfies @xmath775 equivalently , @xmath41 can be seen as the set of linear transformations leaving invariant the graded symmetric form @xmath776 where @xmath777 are arbitrary points in a superspace parametrized by coordinates @xmath778 and @xmath779 and @xmath780 . representing @xmath781 with infinitesimal @xmath782 and expanding eq . one gets the definition of the superalgebra @xmath40 . thus , the subgroup of @xmath41 connected to identity is an exponential of @xmath40 . the representation theory of both is the same as long as we restrict to tensor representations which are the only ones appearing in the tensor space @xmath2 .    from the definition   any matrix @xmath783 has superdeterminant @xmath784 . the supergroup has two disconnected parts @xmath785 , which correspond to the value of the superdeterminant of its elements , that is @xmath786 .    to see this , one can repeat the same reasoning typical of @xmath10 groups . elementary transformations susceptible to change the sign of the superdeterminant belong to the discrete symmetry group @xmath787 of the @xmath41 invariant form . the generators of @xmath787 are read out from eq .   to be `` reflections '' @xmath788 and @xmath789 , and permutations @xmath790 and @xmath791 . for odd @xmath12 there is also the reflection @xmath792 . the subgroup @xmath787 is in fact the weyl group of the root system of @xmath793 . denote by @xmath794 the set of elements of @xmath787 embedded in @xmath785 . it is easy to see that all elements of @xmath795 are conjugate in @xmath796 to a single reflection @xmath759 , which one can take @xmath797 if @xmath12 is even and @xmath798 if @xmath12 is odd . therefore , we see that indeed @xmath799 . let @xmath800 be a vector of weight @xmath801 . then , as seen in sec . [ sec : superalg ] , there is an action of @xmath759 on @xmath802 provided by @xmath803 . in the case of @xmath475 the outer automorphism @xmath302 exchanges @xmath804 with @xmath805 . the representations induced from @xmath40 to @xmath41 are of the form @xmath806 there are two possible cases now : i ) either @xmath807 and then obviously @xmath808 with @xmath809 and @xmath810 or ii ) @xmath811 and the induced module in eq . is irreducible . two representations @xmath812 of @xmath696 are called _ associate_. the modules @xmath813 and @xmath814 are also called associate . in contrast , @xmath815 is isomorphic to its associate because there is an equivalence transformation between @xmath816 and @xmath817 through the change of sing of basis vectors in the subspace @xmath818 . therefore @xmath819 is called selfassociate . a direct implication following from the definitions of ( self ) associate modules is @xmath820 , where @xmath821 are ( self)associate weights of @xmath41 . for a selfassociate weight @xmath198 this equality implies @xmath822 if @xmath823 . note that the centralizer of @xmath42 on @xmath2 is the direct product algebra @xmath824 rather then @xmath40 . this algebra has the same tensor irreducible representations as the supergroup @xmath41 . this is a compact resum of the results presented in @xcite plus some additional remarks on the representation theory of @xmath349 . the superalgebra @xmath475 has minor differences with respect to the general context of @xmath40 superalgebras , because of the isomorphism @xmath825 . the even part of the superalgebra is @xmath826 . the odd part is a representation of the even part of dimension @xmath827 . the standard basis vectors @xmath828 of @xmath719 are normalized as @xmath829 . the even and the odd positive root systems are @xmath830 and @xmath831 . the simple roots are traditionally chosen as @xmath832 , @xmath833 , @xmath834 . the hidden root will then be @xmath835 . consistency conditions for a dominant weight @xmath836 , require @xmath837 and @xmath838 . we associate to @xmath739 a hook shape partition @xmath51 with symplectic part @xmath839 and orthogonal part @xmath840 . to make the correspondence @xmath841 bijective we mark the partition @xmath51 by @xmath842 when @xmath843 . irreps can be written as a couple @xmath844 of @xmath845 irreps . the sign attached to @xmath51 distinguishes between @xmath846 and @xmath847 when @xmath848 . ] atypicality conditions  ( [ e:1_atyp],[e:2_atyp ] ) take the form @xmath849 the solutions can be parametrized by two integers @xmath501 and @xmath502 . for @xmath390 these are @xmath850 , while for @xmath851 , @xmath852 and @xmath853 .     and @xmath854 for the blocks of @xmath475 . ]    denote by @xmath855 the @xmath475 simple modules . for @xmath51 typical @xmath856 and @xmath857 can be computed by decomposing @xmath858 into ( at most 16 ) representations of @xmath859 @xmath860 the dimensions of @xmath861 can be computed with the help of character formulas given in @xcite .    for @xmath390 we get @xmath862 , @xmath863 and @xmath864 , where @xmath865 .    for @xmath851 we get @xmath866 and @xmath867 , and @xmath868 . the set of weights @xmath534 with @xmath501 fixed belong to the same block of @xmath475 . to ( at least partially ) see this one has to check that the second order casimir invariant takes the same value @xmath869 on the whole block @xmath501 . define the _ central character _ of @xmath855 . if two weights @xmath870 are in the same block then @xmath871 have the same central characters . this is a consequence of the extension of the schur lemma ( in the form known to physicists ) to indecomposable representations . ] the actual construction of the set of indecomposable modules providing the equivalence relation of sec . [ sec : tmb ] between the weights of a block is done in @xcite . the quiver diagram representing the structure of @xmath475 projective modules in a block is represented in fig . [ f : diag_ind ] . the projective covers @xmath872 of the modules @xmath873 in the block @xmath390 have the submodule structure @xmath874 while in the block @xmath851 their submodule structure is @xmath875    the dimensions of projective modules in the block @xmath390 are @xmath876 , while in the block @xmath851 there are @xmath877 . the superdimension of projective ( including typical ) modules vanishes . let us apply the general discussion of sec . [ sec : supergr ] to the supergroup @xmath349 . the outer automorphism @xmath302 acts on @xmath719 by exchanging @xmath804 with @xmath805 . consequently , @xmath878 for @xmath390 or @xmath879 and @xmath880 otherwise . we claim that the quiver diagram of type @xmath881 for the block @xmath390 of @xmath475 will give rise to two quiver diagrams of type @xmath881 , as shown in fig . [ fig : cool ] , and , consequently , to two associate blocks for the algebra @xmath570 , which we call @xmath882 . as we shall see bellow , the weights in the block @xmath390 are @xmath883 , while the weights in the block @xmath505 are @xmath884 . we also claim that the quiver diagram of type @xmath854 for the block @xmath885 of @xmath475 will give rise to a single quiver diagram of type @xmath881 , as shown in fig . [ fig : cool ] , and a selfassociate block for the algebra @xmath570 , which we label also by @xmath501 . as we shall sea bellow , the weights in the block @xmath886 are @xmath887 .     and @xmath854 ( grey dots and dotted lines ) for @xmath475 blocks , represented in fig . [ f : diag_ind ] , into quiver diagrams of type @xmath881 for @xmath570 blocks . white and black dots represent weights of the type @xmath888 and @xmath305 respectively . double circles represent selfassociate weights @xmath889 . ] our claim follows from the analysis of shift operators @xmath890 introduced in @xcite . these operators are very practical for decomposing kac modules @xmath891 into @xmath859 irreps . of arrows in the quiver diagram of a block is irrelevant for the following . one can take instead of @xmath891 the standard modules @xmath892 as well , which are defined by cohomological induction in @xcite . ] to be more specific , let @xmath893 be a vector of weight @xmath198 maximal for the algebra @xmath859 . then , @xmath894 is again a maximal vector of weight @xmath895 for @xmath859 . to identify @xmath475 irreducible components in @xmath891 one has to search for @xmath859 maximal vectors with dominant weights @xmath198 in the same block as @xmath51 . consider first the block @xmath390 of @xmath475 . then @xmath896 has a @xmath859 maximal vector @xmath897 and one can check that all positive odd generators annihilate it . therefore @xmath897 is a maximal vector for @xmath475 and @xmath898 contains at least @xmath899 and @xmath900 . in fact , these are the only two irreducible factors of @xmath901 because , using the weyl - kac formula for characters and the results of sec . [ sec : osp42 ] , one can check that @xmath902 . in the case @xmath903 one has that @xmath904 and , therefore , @xmath905 contains also the trivial representation .    in order to see how the four weights @xmath906 split into two different blocks of @xmath907 one has to check out how @xmath908 transforms under the action of @xmath759 . from the explicit expression of shift operators in @xcite it follows that @xmath909 and , consequently , @xmath910 are in the same block and @xmath911 are in an other same block of @xmath570 . the module @xmath912 has a maximal vector @xmath913 , of weight zero , corresponding to the trivial representation @xmath914 . with the help of relations in appendix @xcite for the shift operator products of type @xmath915 , one can show that @xmath916 and , therefore , @xmath914 belongs to the block @xmath390 of @xmath570 as claimed . consider now the block @xmath851 of @xmath475 . then @xmath917 will have a single @xmath475 maximal vector ( besides @xmath918 ) corresponding to the irrep @xmath919 given by @xmath920 if @xmath921 , @xmath922 if @xmath923 and @xmath924 if @xmath925 . the induced module @xmath926 will be the sum of @xmath917 glued together by the action of @xmath759 . finally , the induced module @xmath927 has two irreducible components @xmath928 and @xmath929 with @xmath570 maximal vectors @xmath930 . in this section we explain carefully the notion of block appearing in the representation theory of nonsemisimple algebras . we also look in details at the similarity between the blocks of @xmath40 and @xmath42 . in the representation theory of non semisimple algebras the block is an essential notion . the _ blocks _ are conjugacy classes of irreps with respect to the equivalence relation @xmath931 defined as follows . let @xmath932 be the category of indecomposable modules of the algebra . write @xmath933 if there is an indecomposable module in @xmath932 with simple summands @xmath934 . extend the relation @xmath931 by transitivity in order to get an equivalence . in a semisimple algebra the notion of block is irrelevant because indecomposable representation are irreducible and the congruence @xmath931 becomes an equality . a relevant example is the temperley lieb algebra , with fugacity for loops @xmath85 in its adjoint / diagrammatic representation . for generic values of @xmath85 the algebra is semisimple and , thus , has only completely reducible representation . restricting to subsets of planar diagrams , with the number of vertical lines fixed to @xmath185 , and treating all the other diagrams as zero , we get all irreps , which are parametrized by @xmath185 . however , at special points @xmath935 with coprime integers @xmath936 , the algebra becomes nonsemisimple , irreps labeled by @xmath185 become reducible and @xmath185 becomes a label for a whole block of the algebra , see @xcite . the irreducible components @xmath577 of indecomposable modules @xmath937 when the brauer algebra @xmath42 is nonsemisimple , where first studied by mathematicians hanlon _ et al _ in @xcite . recently martin _ et al _ gave a complete description for the blocks of the brauer algebra in @xcite . we introduce the same notation as in @xcite to formulate their block result for @xmath42 . if the box @xmath938 is in the row @xmath82 and column @xmath83 of the young tableau of a partition @xmath198 , then its content is @xmath939 . two boxes @xmath940 are called balanced if @xmath941 . for two partitions @xmath301 , the skew partition @xmath942 is called balanced if it is composed of balanced pairs of boxes . the necessary condition for @xmath66 to contain @xmath577 is : i ) @xmath301 and @xmath942 is balanced ; ii ) if @xmath85 is even and the boxes of content @xmath943 in @xmath942 are configured as shown in case @xmath944 fig . [ fig:2conf ] , then the number of columns in this configuration is even . the given necessary criterion has the structure of a partial ordering . if @xmath945 satisfy i ) and ii ) we write @xmath946 . the splitting of the set of weights @xmath336 into posets with respect to @xmath947 gives the blocks of @xmath42 . as shown in @xcite , there is a unique minimal partition in a block , which can serve as a label . a sufficient criterion for the module @xmath66 to contain @xmath577 was derived in @xcite and requires @xmath51 to be the least weight @xmath948 .    . ]    we want to give a combinatorial description of the weights in a block . consider the young tableau of a partition @xmath51 in the block of the minimal partition @xmath198 . let @xmath949 ( @xmath950 ) be the box with the highest ( lowest ) content in the skew partition @xmath942 . let @xmath804 ( @xmath951 ) denote the box bellow ( on the left of ) @xmath949 ( @xmath950 ) , if there is one , and the box on the left of ( above ) @xmath949 ( @xmath950 ) otherwise . define by recurrence the balanced pairs @xmath952 until @xmath953 if @xmath85 is even or @xmath954 if @xmath85 is odd . by construction , the set of boxes @xmath955 belongs to a balanced removable border strip of width one or , simply , a _ balanced strip_. one can repeat the same reasoning with the young tableau of @xmath956 ( which is not necessarily in the same block as @xmath198 because of ii ) ) . thus , we clearly see that partitions @xmath51 in the same block can be constructed by dressing up with balanced strips a certain partition @xmath198 with no removable balanced strips . denote by @xmath957 the balanced strip of smallest length addable to @xmath198 . if @xmath85 is odd denote by @xmath958 the minimal partition @xmath959 . if @xmath85 is even denote by @xmath958 the minimal partition @xmath959 only if the two boxes with content @xmath960 in @xmath957 are disposed horizontally and @xmath961 otherwise . partitions which are of the form @xmath198 dressed up with an even ( odd ) number of balanced strips are in the same block as @xmath962 . note that it is irrelevant in what order the strips are dressed on @xmath198 . also , there can not be two balanced strips of the same length . thus , a partition @xmath51 in the block @xmath963 is unambiguously specified by the length of balanced strips in the skew partition @xmath942 . we claim now and show bellow that a block of @xmath40 is composed , in the partition notation of sec . [ sec : superalg ] , of hook shaped partitions built up by dressing with balanced strips an atypical partition with no removable balanced strips . weights . in particular the minimal partitions @xmath964 discussed above are in the same block of @xmath40 if there are hook shaped and atypical . condition ii ) is clearly related to the @xmath696 we neglect by looking at the representation theory of @xmath40 instead of @xmath41 . ] for that we need to reformulate the original block result @xcite for @xmath40 . let the _ degree of atypicality _ @xmath501 of a dominant weight @xmath739 , be the dimension of the subspace @xmath965 of the root lattice orthogonal to @xmath966 , where @xmath759 is the weyl vector of @xmath40 . each atypicality condition in eq . ( [ e:1_atyp],[e:2_atyp ] ) is , in fact , an orthogonality condition between an odd root @xmath967 and @xmath968 . therefore , @xmath501 is the number of odd roots orthogonal to each other and to @xmath968 or , equivalently , the number of atypicality conditions labeled by couples @xmath768 with distinct @xmath82 and @xmath83 . from the definition of the highest weight module @xmath969 it is clear that irreducible finite dimensional components of @xmath969 must have dominant weights of the form @xmath970 , where the sum is over all odd positive roots @xmath971 spanning @xmath965 . consider a @xmath40 weight @xmath51 , which , in the notation of app . [ sec : superalg ] , has symplectic part @xmath759 and orthogonal part @xmath207 . suppose @xmath972 are the first columns of @xmath51 satisfying @xmath973 and @xmath974 . then , one can find rows @xmath975 , such that @xmath976 and the pairs @xmath977 satisfy the atypicality condition   for @xmath978 if @xmath12 is odd and @xmath979 if @xmath12 is even and the atypicality condition   for @xmath980 . indeed , from eq . with @xmath981 the condition @xmath982 implies @xmath983 and thus @xmath984 , while @xmath985 implies @xmath986 if @xmath12 is even and @xmath987 is @xmath12 is odd . from eq .   with @xmath981 the condition @xmath980 implies @xmath985 while @xmath988 follows directly from @xmath989 . conversely , if @xmath990 satisfies an atypicality condition with @xmath981 , then @xmath991 . as shown in fig . [ fig : imagine ] , @xmath992 is the width of the foot of the narrowest hook with arm width @xmath993 in which the young tableau of @xmath51 can be drawn in . two atypicality conditions @xmath768 and @xmath994 are called independent if @xmath995 and @xmath996 . clearly , conditions @xmath977 are pairwise independent for @xmath997 and for @xmath980 . let us show that an atypicality condition @xmath998 is independent of conditions @xmath999 iff there is a row shorter then @xmath29 such that the box @xmath938 at the end this row and the box @xmath1000 at the end of column @xmath83 are balanced .    in order to do that it is useful to imagine the partition @xmath51 drawn on an infinite square lattice , as in fig . [ fig : imagine ] , with each square having its content written inside . the following cases are possible    * suppose first that there is no box with at the end of the row @xmath83 . from @xmath987 follows @xmath1001 . observe that the column @xmath1002 is also empty . therefore @xmath1003 and the atypicality conditions @xmath977 and @xmath1004 are not independent . * let @xmath1005 and let @xmath763 be the box at the end of column @xmath83 . no suppose that @xmath51 has a rightmost box @xmath1006 with content @xmath1007 and let @xmath1008 be the column of that box . condition @xmath1009 gives @xmath1010 . the equality sign can not hold because otherwise @xmath1011 which contradicts @xmath987 . thus @xmath1012 and therefore @xmath1013 . if @xmath1014 then there is a box @xmath938 bellow @xmath1006 , which is balanced with @xmath1000 and has no box to the right , thus it is the end of a row shorter then @xmath29 . if @xmath1015 then comparing the @xmath82 s from the two atypicality conditions we get @xmath1016 and the two atypicality conditions are not independent * finally , if there is no box with content @xmath235 then the column @xmath1017 gives an atypicality condition @xmath1004 with @xmath1016 . next , by the definition of @xmath185 , a column @xmath1018 can satisfy an atypicality condition only with a row @xmath1019 . after inserting @xmath1020 in eqs . ( [ e:1_atyp],[e:2_atyp ] ) we get @xmath1021 the lhs in eq .   is the hook length of the box in the row @xmath82 and column @xmath83 of @xmath51 , thus , always positive . on the other hand , eq . requires the box @xmath1000 at the foot of column @xmath83 be balanced with the box @xmath242 at the end of row @xmath82 .    in the end , we see that there are two sources for _ independent _ atypicality conditions satisfied by a weight @xmath51 . first , if @xmath992 is the width of the foot of the narrowest hook with arm width @xmath993 , in which the young tableau of @xmath51 can be drawn in , then there are @xmath1022 atypicality conditions satisfied by the weight and we call them of type 1 . second , to each balanced pair of boxes @xmath1023 , such that @xmath938 is a box at the end a row and @xmath938 is a box at the end of a column , corresponds an atypicality condition of type 2 . if @xmath51 satisfies @xmath1024 atypicality condition of type 2 then the degree of atypicality of the weight is @xmath1025 . let @xmath1026 be the set of balanced pairs satisfying atypicality conditions of type 2 . applying the iterative construction explained above to the boxes @xmath1027 one can see that there is a removable balanced strip @xmath1028 in @xmath1029 , with its ends in @xmath1030 . clearly , by the same reasoning , one can identify a new balanced strip @xmath1031 removable in @xmath1032 . the end @xmath1033 of this iterative procedure has no more removable balanced strips . note that @xmath1033 satisfies @xmath501 atypicality conditions all of type @xmath1034 and the sequence of weights @xmath1035 has the same degree of atypicality @xmath501 .    in order to complete the proof of the claim it remains to notice two things first , if @xmath1036 is an odd root generating an atypicality condition of type 1 , then @xmath1037 is not dominant . second , if @xmath1038 is an odd root generating an atypicality condition of type 2 , then @xmath739 has a removable strip with its ends in the last box @xmath242 of row @xmath82 and @xmath1039 of column @xmath83 and @xmath1040 is dominant and can be represented by a partition of the form @xmath1041 , where @xmath51 is the young tableau of @xmath739 .     drawn on a hook shaped square lattice and fitting exactly inside a hook with foot width @xmath992 and arm width @xmath993 . the black boxes represent the diagonal of squares with content @xmath1042 . ] the explicit form of the characters of classical groups is easier derived in the limit of infinite rank of the corresponding lie algebra . the inverse limit exists and is given by the _ modification rules _ for characters . the concepts of infinite rank and inverse limit are rigorously defined for the case of schur symmetric functions , connected to the irreducible characters of @xmath1043 , in @xcite . let us clarify this point . the characters of classical groups , evaluated on a group element , are polynomials in the eigenvalues of that element in the defining representation for the group . the infinite rank limit corresponds to considering polynomials depending on an infinite number of such variables . irreducible characters are polynomials with a very specific symmetry , which is not obscured by the restriction of finite number of variables in the infinite rank limit . these objects are known as _ symmetric functions_. when the number of variables is set finite most symmetric functions become functionally dependent . once an algebraically independent subset of symmetric functions is chosen , which is the actual set of characters in the case of classical groups , the modification rules `` for characters '' represent arbitrary symmetric functions along this basis . one can introduce generalized symmetric functions @xmath332 for the supergroup @xmath41 according to eqs . ( [ eq : elchar ] , [ eq : onchar ] ) , see @xcite , @xcite . the major difference with respect to classical groups is that functionally independent generalized symmetric functions are no longer irreducible characters of the supergroup . however , modification rules for @xmath332 exist and have been derived in @xcite . we bring them bellow in the form of eq .   with the notations of our paper . suppose that @xmath51 is a typical @xmath40 weight . then , according to @xcite , only @xmath332 with @xmath198 of the form @xmath51 dressed by balanced strips @xmath1044 modify to @xmath73 @xmath1045 here @xmath1046 is the superdeterminant representation . we have also introduced the weight function @xmath1047 defined on skew partitions composed of balanced strips and @xmath1048 is the number of columns in @xmath1049 . consider now , the @xmath40 block labeled by the weight @xmath1050 with degree of atypicality @xmath501 and no removable balanced strips . then , any weight @xmath51 in the block @xmath1051 of @xmath198 is of the form @xmath1050 dressed up by @xmath1052 balanced strips . then , according to @xcite , @xmath332 with @xmath198 of the form @xmath1050 dressed up by @xmath1053 balanced strips modifies to @xmath1054    for a typical weight @xmath51 we put @xmath320 equal to @xmath51 dressed up by the balanced strip of minimal length @xmath1055 if @xmath1056 . in order to prove that @xmath1057 one has to show that @xmath1055 runs over an odd number of columns . indeed , each box in @xmath1055 belongs either to a horizontal or a vertical part of the strip , except for the boxes at the corners of @xmath1055 , which belong to both . we say the balanced pair @xmath1060 has an allowable configuration if both boxes belong either to horizontal or vertical parts of @xmath1055 , otherwise @xmath1061 has a non allowable configuration . as discussed in app . [ sec : tmb ] , to every non allowable configurations of @xmath1061 with content @xmath1062 corresponds a removable balanced strip in @xmath51 with its ends in the border boxes with content @xmath1063 or @xmath1064 depending on weather @xmath938 is on a horizontal or a vertical part of @xmath957 . because @xmath51 is a @xmath40 typical weight and , thus , has no removable balanced strips , there are only allowable configuration of balanced pairs in @xmath1055 . thus , a balanced pair in @xmath1055 , which is not in the same column , indexes either two different columns or none . there is at most one column containing the whole balanced pair and it appears always if @xmath85 is odd and only for @xmath1056 if @xmath85 is even , as shown in fig . [ fig : corner ] . thus , @xmath1058 is always odd for @xmath12 odd and even only if @xmath1059 for @xmath12 even . let @xmath1049 denote the @xmath82th lowest length strip addable to @xmath51 and @xmath1048 the number of columns in it . one can prove by the same method that @xmath1065 has an even number of columns , one of which is already in @xmath1049 and , consequently , @xmath1066 is odd .    again by the same method it is possible to prove that @xmath1067 has an even number of columns . this is because @xmath1068 contains a substrip @xmath1069 which can be obtained by moving down along the diagonal the strip @xmath1055 . applying what was said above about @xmath1065 to @xmath1070 we see that @xmath1071 . if @xmath51 is typical and @xmath1072 then @xmath1073 is the next partition in the block of @xmath51 , while if @xmath1074 then @xmath1075 is the next partition in the bloc of @xmath51 . therefore we have just shown , as claimed in sec . [ heart ] that @xmath1076 .
we define and study a lattice model which we argue is in the universality class of the @xmath0 supercoset sigma model for a large range of values of the coupling constant @xmath1 . in this first paper 
 , we analyze in details the symmetries of this lattice model , in particular the decomposition of the space of the quantum spin chain @xmath2 as a bimodule over @xmath0 and its commutant , the brauer algebra @xmath3 . 
 it turns out that @xmath2 is a nonsemisimple module for both @xmath0 and @xmath3 . 
 the results are used in the companion paper to elucidate the structure of the ( boundary ) conformal field theory .    # 
 1#2#3#4    # 1#2#3#4    constantin candu@xmath4 and hubert saleur@xmath5    _ service de physique thorique , cea saclay , + gif sur yvette , 91191 , france@xmath4 + _ _ department of physics and astronomy , university of southern california + los angeles , ca 90089 , usa@xmath6 + _