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in the 1980 s , belavin and drinfeld classified solutions @xmath0 of the classical yang - baxter equation ( cybe ) for simple lie algebras @xmath3 satisfying @xmath4 @xcite . they proved that all such solutions fall into finitely many continuous families and introduced combinatorial objects to label these families , belavin - drinfeld triples ( see section [ bd ] ) . in 1993 , gerstenhaber , giaquinto , and schack attempted to quantize such solutions for lie algebras @xmath5 as a result , they formulated a conjecture stating that certain explicitly given elements @xmath6 satisfy the quantum yang - baxter equation ( qybe ) and the hecke relation @xcite . specifically , the conjecture assigns a family of such elements to any belavin - drinfeld triple of type @xmath7 . this conjecture is stated in section [ ggsss ] . recently , etingof , schiffmann , and the author found an explicit quantization of all @xmath0-matrices from the belavin - drinfeld list . they did so by twisting the coproduct in the drinfeld - jimbo quantum group @xmath8 . for @xmath9 , one can evaluate the universal @xmath10-matrix of the twisted @xmath8 in the vector representation of @xmath8 . this gives an element @xmath11 of @xmath12 which satisfies the qybe and the hecke relation . this element is presented in section [ ptp ] .    in this paper i show that the elements @xmath11 and @xmath13 from @xcite and @xcite coincide . this proves the ggs conjecture . this is done by explicitly expanding the formula for @xmath11 using combinatorial techniques involving belavin - drinfeld triples . the proof occupies all of section 2 . note that the ggs conjecture was proved in some special cases ( the cremmer - gervais and orthogonal disjoint cases ) by hodges in @xcite and @xcite . the ggs conjecture was proved in some additional cases ( orthogonal generalized disjoint ) by the author in @xcite . also , the disjoint case was completed in @xcite by pavel etingof and the author . the author actually found the matrix @xmath11 with the help of a computer before the general twist given in @xcite was found . the matrix was constructed to coincide with the ggs matrix in many cases , and motivated in part the general construction given in @xcite . see @xcite for details . also , many steps of this proof were motivated by and checked with computer programs . let @xmath14 be the standard basis for @xmath15 . let @xmath16 be the set of simple roots of @xmath17 . we will use the notation @xmath18 . let @xmath19 denote the inner product on @xmath15 having @xmath20 as an orthonormal basis . @xcite a _ belavin - drinfeld triple of type @xmath7 _ is a triple@xmath21 where @xmath22 and @xmath23 is a bijection , satisfying two relations :    \(a ) @xmath24 preserves the inner product : @xmath25 , @xmath26 . \(b ) @xmath24 is nilpotent : @xmath27 such that @xmath28 . let @xmath29 be the lie algebra of complex @xmath30 matrices . let @xmath31 be the subspace of diagonal matrices . elements of @xmath15 define linear functions on @xmath32 by @xmath33 . let @xmath34 be the casimir element inverse to the standard form on @xmath3 . it is easy to see that @xmath35 , for any @xmath36 . let @xmath37 be the projection of @xmath38 to @xmath39 .    for any belavin - drinfeld triple , consider the following equations for @xmath40 : @xmath41 s = \frac{1}{2 } \bigl[(\alpha + t \alpha ) \otimes 1\bigr ] p^0.\end{gathered}\ ] ] belavin and drinfeld showed that solutions @xmath42 of the cybe satisfying @xmath43 , up to isomorphism , are given by a discrete datum ( the belavin - drinfeld triple ) and a continuous datum ( a solution @xmath44 of ) . we now describe this classification .    for @xmath45 , set @xmath46 . define @xmath47 . for any @xmath48 , set @xmath49 ( the set of positive roots of the subalgebra of @xmath17 having @xmath50 as the set of simple roots ) . in particular we will often use the notation @xmath51 . we extend @xmath24 additively to a map @xmath52 , i.e. @xmath53 . whenever @xmath54 for @xmath55 , we say @xmath56 . clearly @xmath57 is a partial ordering on @xmath58 . we will also use @xmath59 to denote @xmath56 or @xmath60 . suppose @xmath61 for @xmath45 and @xmath62 . then there are two possibilities on how @xmath63 sends @xmath64 to @xmath65 , since @xmath24 is an automorphism of the dynkin diagram . namely , either @xmath66 and @xmath67 , or @xmath68 and @xmath69 . in the former case , call @xmath63 _ orientation - preserving on @xmath64 _ , and in the latter , _ orientation - reversing on @xmath64_. let @xmath70 now we define @xmath71 ( @xmath72 is the standard solution of the cybe satisfying @xmath73 . ) the element @xmath0 is the solution of the cybe corresponding to the data @xmath74 . it follows from @xcite that any solution @xmath75 is equivalent to such a solution @xmath0 under an automorphism of @xmath76 . the ggs conjecture suggests a quantization of the matrix @xmath0 given in , given by a matrix @xmath77 conjectured to satisfy the quantum yang - baxter equation ( qybe ) , @xmath78 , and the hecke relation , @xmath79 . this may be formulated and justified as follows ( which is more or less the original motivation ) .    if we write @xmath80 , where @xmath81 , then we can consider the constraints imposed by the qybe and the hecke relation modulo @xmath82 . one may easily check that the qybe becomes the cybe for @xmath0 , while the hecke relation becomes the condition @xmath83 . since it is not difficult to see that @xmath84 is symmetric , the unique symmetric choice for @xmath85 is @xmath86 where @xmath87    [ ggs1 ] there exist unique polynomials @xmath88 of the form + @xmath89 such that @xmath90 . the proof is easy . define @xmath91 , with the @xmath88 uniquely determined by proposition [ ggs1 ] . the matrix @xmath13 is called the ggs @xmath10-matrix . we will use the notation @xmath92 for elements @xmath93 . define the following matrices :    @xmath94    where @xmath95 is the standard drinfeld - jimbo solution to the qybe , which is a quantization of @xmath96 . the matrix @xmath13 equals @xmath97 . this is a straightforward computation . we see that @xmath98 , although @xmath99 in general .    * `` the ggs conjecture '' @xcite * [ ggs ] the matrix @xmath13 satisfies the qybe and the hecke relation . [ or0 ] it is sufficient to check the qybe for one value of @xmath100 since the space of solutions to the homogeneous equation corresponding to is exactly the space @xmath101 where @xmath102 is the space of all @xmath103 such that @xmath104 for any @xmath105 . indeed , it is easy to see that @xmath106 implies @xmath107 = 0 $ ] , and it follows that @xmath108 satisfies the qybe iff @xmath13 does , for any @xmath109 . our formulation is from @xcite , correcting misprints . the original formulation in @xcite is somewhat different . we will write @xmath110 to denote the matrix @xmath103 with @xmath111 substituted for @xmath112 . define @xmath113 where @xmath114 is the transpose of @xmath103 , for @xmath115 . then , the original form of @xmath13 can be written as follows : @xmath116 we have @xmath117 . thus , @xmath13 satisfies the hecke relation iff @xmath10 satisfies the hecke relation . in this case , we have @xmath118 , so @xmath119 , and thus @xmath10 satisfies the qybe iff @xmath13 does . thus , the two formulations are equivalent .      in this section we give a combinatorial formula for @xmath120 . first let us introduce some definitions , which will be used in this formula as well as in the proof of the main theorem . a _ positive @xmath24-pair _ is a pair @xmath121 for @xmath122 . we define the _ order _ to be @xmath123 . the set of positive @xmath24-pairs is denoted @xmath124 .    in section 2 , we will also define negative @xmath24-pairs , but so far we do nt need them . if @xmath125 satisfies the property that @xmath126 for some @xmath122 , we say its _ right - passing order _ is @xmath127 , and denote it by @xmath128 . if there is no such @xmath127 , say @xmath129 . similarly define _ left - passing order _ @xmath130 . a positive @xmath24-pair @xmath131 is _ right - passed _ if@xmath132 and @xmath133 . we say that @xmath134 is _ half right - passed _ if @xmath135 . similarly define the left versions , and we denote the sets of all such @xmath24-pairs by @xmath136 , and @xmath137 ( right - passed , half right - passed , left - passed , half left - passed , respectively ) . the @xmath138 subscripts indicate positive @xmath24-pairs .    for a positive @xmath24-pair @xmath131 , we define @xmath139 to be @xmath140 if @xmath141 , @xmath142 if @xmath143 , and otherwise @xmath144 . similarly define @xmath145 . it will be useful to define _ symmetric _ and _ anti - symmetric _ versions : @xmath146 , @xmath147 . this allows us to state a simple combinatorial formula for @xmath120 :    [ fep ] we may rewrite @xmath120 as follows : @xmath148 ( e_\beta \o e_{-\alpha } + e_{-\alpha } \o e_{\beta}).\end{gathered}\ ] ]    this is proved in section [ es ] and also follows from the proof of the main theorem , theorem [ mt ] ( see remark [ efr ] for details ) . [ frg ] @xmath149 is given as follows : @xmath150 + r_{st}\end{gathered}\ ] ]    clear . [ gcge ] for a given @xmath151 , there are exactly @xmath152 triples ( @xmath153 is the euler @xmath153-function ) in which @xmath154 @xcite . these are called _ generalized cremmer - gervais _ triples . these are indexed by @xmath155 , where @xmath156 , and given by @xmath157 , @xmath158 , and @xmath159 , where @xmath160 gives the residue modulo @xmath151 in @xmath161 . for these triples , there is a unique @xmath100 with first component having trace 0 , which is given by @xmath162 , and @xmath163 for @xmath164 ( this is easy to verify directly and is also given in @xcite ) . with this @xmath100 , @xmath13 has a very nice combinatorial formula , which was conjectured by giaquinto and checked in some cases . we now state and prove this formula . [ gp ] @xmath13 is given as follows for generalized cremmer - gervais triples : @xmath165.\end{gathered}\ ] ]    see appendix a.      in @xcite , an explicit quantization is given for any classical @xmath0-matrix described in section [ bd ] . this is given by a twist of the standard coproduct on the quantum universal enveloping algebra @xmath166 . in particular , in the @xmath151-dimensional representation , this gives an element @xmath167 so that @xmath168 satisfies the qybe and the hecke relation . in fact , @xmath169 is triangular , i.e. @xmath170 and @xmath171 . suppose we are given @xmath24 so that @xmath172 . define the following matrices :    @xmath173    [ esst ] @xcite the element @xmath11 satisfies the qybe and the hecke relation . now , we state the main theorem of this paper :    [ mt ] for any given @xmath24 and @xmath100 , @xmath174 . this theorem clearly implies the ggs conjecture . in this section we will introduce combinatorial objects and arguments which are sufficient to prove the combinatorial formula for @xmath120 . to do this , we wish to directly expand . this involves expanding a quadratic expression in terms of the form @xmath175 and @xmath176 for @xmath177 . most of the monomials in the expansion are zero . thus , the first step is to restrict our attention to those that are not . in this vein , we define _ compatible @xmath24-quadruples _ ( definition [ ctd ] ) . to further simplify the formula for @xmath178 , we will have to show that most of these monomials cancel pairwise . this is accomplished with bijections @xmath179 and @xmath180 ( definitions [ phd]-[pssd ] ) between the corresponding @xmath24-quadruples . for all of section 2 we will not need to refer to the dimension of the representation ( formerly @xmath151 ) , so we will reuse @xmath151 for other purposes .    for any subset @xmath48 , define @xmath181 . let @xmath182 . these will be called _ negative @xmath24-pairs_. we define @xmath183 , and @xmath184 for @xmath185 .    for negative @xmath24-pairs , we allow the order to be zero , but not for positive ones !    for @xmath186 , define @xmath187 if @xmath188 and @xmath189 if @xmath190 . we define @xmath191 and @xmath192 just as in the positive case , but with components in @xmath24-pairs permuted . let @xmath193 and @xmath194 . note that all passed elements must have positive order , so there is nothing new in the case of negative passed @xmath24-pairs .    for @xmath195 , set @xmath196 and @xmath197 ( _ lower , higher , _ respectively . ) for @xmath198 , similarly set @xmath199 and @xmath200 . an element @xmath186 is uniquely given by @xmath201 , and the sign of @xmath103 . this is obvious .    for convenience , we will often give elements @xmath186 in terms of @xmath201 , and the sign of @xmath103 . for convenience , we will say that @xmath202 and @xmath203 for @xmath24-pairs @xmath103 .    for any pair @xmath204 , define @xmath205 to be @xmath206 . we will , however , only consider cases where @xmath207 and @xmath208 . [ ql ] suppose @xmath209 . then , if @xmath210 , then there is a unique @xmath211 such that @xmath212 . namely , this is given by componentwise addition of all @xmath213 . this follows from nilpotency . it is easy to see that we need only show that @xmath214 for any @xmath215 and @xmath216 . equivalently , we have to show that the componentwise addition of all @xmath213 is not @xmath217 . first , we generalize @xmath218 and @xmath219 . clearly , every @xmath220 can be written as @xmath221 for some @xmath222 . we say that @xmath223 for @xmath224 if @xmath221 and @xmath225 for nonnegative integers @xmath226 . in this case , we define @xmath227 . note that , when @xmath186 , @xmath228 . now , we set @xmath229 and @xmath230 . let @xmath231 and let @xmath232 . we see that componentwise addition of two elements of @xmath233 , if in @xmath234 , yields another element of @xmath233 , where the orders are summed . in particular , this means that , when @xmath235 , for @xmath236 , then @xmath237 . since @xmath238 , it follows that any componentwise sum of elements of @xmath239 , if in @xmath240 , yields a nonzero element of @xmath241 . this is all we need . define @xmath242 as follows . when @xmath243 , @xmath244 , for @xmath245 , @xmath246 . otherwise , @xmath247    we now apply these general definitions to the task of setting aside the important subsets of @xmath248 . these subsets parameterize nonzero terms that arise upon expansion of @xmath178 . [ ctd ] define @xmath249 and @xmath250 ( _ compatible @xmath24-quadruples of same / opposite sign _ ) by @xmath251    define @xmath252 ( _ increasing , constant , _ and _ decreasing _ in * order * ) by @xmath253    for any @xmath254 , @xmath255 . if @xmath256 , then it is clear from @xmath254 that @xmath257 , negating in each component . but then @xmath258 , which is impossible . define @xmath259 and @xmath260 ( _ increasing _ and _ decreasing _ in * size * ) by @xmath261    @xmath250 and @xmath262 are partitioned into increasing and decreasing quadruples by different criteria ( size and order , respectively ) ! it turns out that these criteria are the useful ones . now we define _ good _ and _ bad _ subsets of the sets we have defined .    @xmath263    and @xmath264 moreover , set @xmath265 and similarly for @xmath266 . the definitions above are symmetric in the two components of @xmath24-quadruples because @xmath267 iff @xmath268 iff @xmath269 iff @xmath270    in the following definitions , particularly in the orientation - reversing cases , it may not be completely obvious that the maps have the images indicated ( i.e. are well - defined ) . see proposition [ qc ] for well - definition . [ phd ] we define @xmath271 . we first consider the increasing case . take @xmath272 and suppose @xmath273 , @xmath274 , @xmath275 , and @xmath276 . suppose that @xmath277 for @xmath278 . if @xmath279 ( i.e. @xmath280 is good ) , then set @xmath281 where @xmath282 and @xmath283 have the same sign as @xmath284 and are given by @xmath285 , @xmath286 , @xmath287 , and @xmath288 . in other words , @xmath282 and @xmath283 are the unique same - sign @xmath24-pairs such that @xmath289 and @xmath290 . in the case that @xmath280 is bad , we simply set @xmath291 . in the good decreasing case , we take @xmath292 , set @xmath112 and @xmath293 as above , and again let @xmath294 where @xmath295 are the unique @xmath24-pairs with the same sign as @xmath284 such that @xmath289 and @xmath296 . in the bad decreasing case , we again set @xmath297 . we define @xmath298 as follows . take @xmath299 . if @xmath300 is good , i.e. @xmath301 does not divide @xmath302 , then write @xmath303 + q$ ] for @xmath304 . in the case @xmath300 is bad , i.e. @xmath301 divides @xmath302 , we again write @xmath303 + q$ ] , this time choosing @xmath305 when @xmath306 and @xmath307 when @xmath308 . then , we define @xmath309 where @xmath284 has the same sign as @xmath282 and @xmath283 , while @xmath103 has the opposite sign , and @xmath310 , @xmath311 } + \ldots + t^{p(\o(v)-\o(u))})(u_l+v_l)+t^{(p+1)(\o(v)-\o(u ) ) } v_l$ ] , @xmath312 , and @xmath313 .    in the case of decreasing quadruples , we begin with @xmath314 and set @xmath315 , with @xmath316 and @xmath112 all defined as above . [ pssd ] we similarly define @xmath317 and @xmath318 . suppose @xmath319 . if @xmath283 is good , i.e. @xmath320 does not divide @xmath321 , we write @xmath322 for @xmath323 . if @xmath283 is bad , i.e. @xmath320 divides @xmath321 , we set @xmath305 when @xmath324 and @xmath325 when @xmath326 , and again write @xmath327 . now , we define @xmath328 where @xmath284 has the same sign as @xmath283 , @xmath103 has the opposite sign , and @xmath329 and @xmath330 . for @xmath331 , we define @xmath332 , with @xmath316 and @xmath112 all defined as above . we had to define two separate maps @xmath180 and @xmath333 because sometimes @xmath334 and @xmath335 intersect , and @xmath333 and @xmath180 do not agree . we say that a pair @xmath195 reverses orientation if @xmath336 , and in this case , we set @xmath337 . otherwise @xmath103 preserves orientation and @xmath338 . for negative pairs @xmath339 , we say @xmath103 preserves orientation and @xmath340 if @xmath60 ; otherwise , we set @xmath341 and say that @xmath103 preserves / reverses orientation iff @xmath131 does . define @xmath342 $ ] as follows : @xmath343 and @xmath344 . [ nc ] if @xmath345 then @xmath282 preserves orientation . in the event @xmath283 reverses orientation , then @xmath346 . if @xmath272 , then @xmath284 preserves orientation . in the event @xmath103 reverses orientation , @xmath347 . the same results hold considering @xmath348 and @xmath349 , respectively . in the first case , if @xmath282 reversed orientation , then @xmath350 , thus @xmath351 by nilpotency  this is a contradiction . on the other hand , when @xmath283 reverses orientation , @xmath352 must reverse orientation , and thus by nilpotency can not be defined on all of @xmath353 . thus @xmath354 , hence the desired result . the second case follows easily from nilpotency . [ qc ] the maps @xmath355 and @xmath356 are inverse to each other . the maps @xmath357 and @xmath358 are inverse to each other . finally , the maps @xmath359 and @xmath360 are inverse to each other . all maps @xmath361 and @xmath333 preserve @xmath362 and @xmath363 . the fact that the maps are well - defined , inverse to each other , and preserve @xmath362 is easy to see from construction when orientations are preserved ( it helps to draw a picture ) . also , when orientations are preserved , @xmath363 is trivially preserved . so it remains to consider orientation - reversing cases . given @xmath364 , if orientation is reversed in @xmath103 or @xmath284 , it can only be reversed in @xmath103 , and @xmath347 by lemma [ nc ] . in the case @xmath365 , it follows that @xmath366 , where @xmath282 preserves orientation , @xmath283 reverses orientation , @xmath367 , and @xmath368 . so @xmath363 and @xmath362 are preserved , and @xmath309 . if , instead , @xmath369 , then @xmath370 , where latexmath:[$|v| =    reverses orientation . again , @xmath363 and @xmath362 are preserved and @xmath328 . now consider an element @xmath372 in which either @xmath282 or @xmath283 reverses orientation . by lemma [ nc ] , only @xmath283 reverses orientation and @xmath373 . so , we get @xmath374 where @xmath375 . in this case , @xmath103 reverses orientation and @xmath284 does not , so @xmath362 and @xmath363 are preserved , and clearly @xmath281 . finally , suppose @xmath376 reverses orientation . by definition , this means that @xmath377 reverses orientation on @xmath283 . by nilpotency , @xmath378 , and it follows that @xmath379 . we then have @xmath380 and @xmath381 . it follows that @xmath103 reverses orientation , and so @xmath363 and @xmath362 are preserved , and @xmath370 . the decreasing and left cases follow in exactly the same way as the increasing and right cases . [ psq ] suppose @xmath272 . if @xmath281 and not all of @xmath382 preserve orientation , then @xmath383 reverse orientation , @xmath384 preserve orientation , and @xmath385 . if @xmath370 and not all of @xmath386 preserve orientation , then @xmath383 reverse orientation , @xmath284 preserves orientation , and @xmath385 . the same facts hold under the assumptions @xmath387 with @xmath294 or @xmath283 . this follows directly from the argument above . @xmath388    fix a choice of sign @xmath389 for this proof . clearly , whenever @xmath390 , @xmath391 , and @xmath392 , then @xmath393 . in this case , @xmath394 , as @xmath395 also has reversed orientation and the same order as @xmath103 and @xmath284 . it remains only to see that , for any @xmath396 with @xmath397 , there are @xmath398 ways of writing @xmath399 for @xmath400 , and they all are of this form . the formula follows immediately . * direct proof of proposition [ fep ] . * set @xmath401 and + @xmath402 , so that @xmath403 . then , using @xmath153 and @xmath404 , @xmath405    equation follows immediately . [ efr ] the combinatorial formula [ fe ] for @xmath178 also follows from the proof of theorem [ mt ] . namely , in the proof we actually show that @xmath11 has the form of @xmath13 but we use the combinatorial formula for @xmath178 instead of the original one . on the other hand , since the combinatorial formula is symmetric , and because @xmath11 satisfies the qybe and the hecke relation , @xmath11 must be the unique element satisfying the hypotheses of proposition [ ggs1 ] by the discussion in section [ ggsss ] . the proof of proposition [ fep ] above is , however , given for pedagogical reasons and because the results used will be needed later .      by _ passing properties _ of a @xmath24-pair we mean information about its left- and right - passing order . in particular , usually we will be concerned with whether a pair is ( half ) right- or ( half ) left - passed .    in this section , we will list all possible passing properties of compatible increasing and decreasing quadruples , in connection with those properties of their images under @xmath153 or @xmath404 . these results , lemmas [ pq ] and [ ppp ] , are essential in order to consider quadratic terms which arise in the hecke condition for @xmath13 , which are similar to those in the formula for @xmath178 but include powers of @xmath112 which depend on the passing properties . as a consequence of these results , one can prove the hecke condition for @xmath13 directly ( see @xcite ) .    in order to prove these results , we will first need to develop some more powerful combinatorial tools and notation regarding belavin - drinfeld triples . the combinatorics can best be pictured on the dynkin diagram for @xmath17 . we picture this diagram as the line segment @xmath406 $ ] with integer vertices . we then picture the positive root @xmath407 , for @xmath408 , as the line segment @xmath409 $ ] . in this context , @xmath410 and @xmath411 can be thought of as subsets of the graph consisting of the union of all the length-1 segments which make them up , and maps @xmath24 are nilpotent graph isomorphisms @xmath412 . assume @xmath413 and @xmath414 . we say @xmath415 if @xmath416 ( @xmath407 is to the left of @xmath417 ) . as subcases of this , we say that @xmath418 for @xmath419 , @xmath420 if @xmath421 , and @xmath422 for @xmath423 . similarly define @xmath424 by @xmath425 whenever @xmath426 , and the same for @xmath427 and @xmath428 . if @xmath429 or @xmath430 , then we say that @xmath431 ( meaning there are orthogonal subsets @xmath432 so that @xmath433 and @xmath434 ) .    here and in the sequel we will make frequent use of the following key combinatorial lemma :    take @xmath435 . let @xmath436 be the smallest positive integer such that @xmath437 . let @xmath438 be the smallest positive integer less than @xmath436 such that @xmath439 , if such an integer exists . otherwise set @xmath440 . if @xmath438 is finite and there is a positive integer @xmath441 which is not a multiple of @xmath438 satisfying @xmath442 , then let @xmath443 be the smallest such @xmath444 . otherwise set @xmath445 . [ c1 ] suppose @xmath435 and @xmath446 . ( i ) for any positive integer @xmath447 such that @xmath442 , either @xmath444 is a multiple of @xmath438 , or @xmath448 . ( ii ) @xmath449 and @xmath450    take some @xmath444 such that @xmath442 . we show in the following paragraph that either @xmath438 divides @xmath444 , or @xmath451 and @xmath452 . this proves the lemma  all that remains is to see that , in the latter case , @xmath448 . if , instead , @xmath453 , then applying the above result also to @xmath443 , we find that both @xmath454 and @xmath455 are perpendicular to @xmath456 . by space concerns on the diagram , it follows that @xmath457 , but then @xmath458 , which would show that @xmath438 divides @xmath459 , in contradiction to @xmath460 . so , take any @xmath444 such that @xmath442 , and assume that @xmath444 is not a multiple of @xmath438 . define @xmath461 by @xmath462 if @xmath463 preserves orientation on @xmath464 , and @xmath465 otherwise . define @xmath466 as follows . for any @xmath467 ( @xmath151 is the length of the dynkin diagram ) , let @xmath112 be given by @xmath468 . then we define @xmath469 . define @xmath470 . clearly , @xmath471 is defined so that if @xmath472 is a connected segment of the diagram , then @xmath471 is constant on @xmath473 , for each fixed @xmath293 , @xmath474 . since @xmath475 for @xmath476 , it follows that @xmath471 is @xmath438-periodic in the first component . for the same reason , @xmath471 is @xmath444-periodic in the first component . if @xmath477 , @xmath471 must be @xmath478-periodic in the first component . this follows since @xmath479 , @xmath480 implies @xmath481 whenever @xmath482 , @xmath483 . by minimality , @xmath484 , which is impossible . hence , @xmath485 . furthermore , @xmath486 , because otherwise @xmath487 , which would imply that @xmath488 by the above results applied to @xmath489 , which is clearly contradictory . [ c3 ] suppose that @xmath490 for @xmath491 where @xmath492 and @xmath493 . then for any @xmath494 , @xmath495 , where @xmath496 and @xmath497 , then @xmath498 , and @xmath56 implies that @xmath499 . by reversing all directions , given @xmath500 , @xmath501 , with @xmath502 , then for any @xmath503 , @xmath504 , where @xmath496 and @xmath505 , then @xmath498 , and @xmath56 implies @xmath506 .    first , it is clear that @xmath0 divides @xmath507 iff @xmath508 for @xmath122 an integer . in this case , the theorem is satisfied ; so suppose not . by applying @xmath509 some number of times to @xmath64 or @xmath65 , it suffices to assume @xmath510 . now , if @xmath511 is defined on @xmath64 , then @xmath512 together with the lemma gives the desired result . so assume @xmath511 is not defined on @xmath64 , and hence it is not defined on @xmath65 either . suppose @xmath513 for some positive integer @xmath514 . write @xmath515 . now , by applying @xmath516 some number of times to each @xmath517 , we can obtain @xmath518 for some @xmath519 , showing that @xmath511 is defined on @xmath65 , which is a contradiction . but then @xmath516 is defined the direction - reversed case is the same .    for convenience , let @xmath520 for any or no superscript @xmath521 . when @xmath63 acts on some segment of the dynkin diagram ( i.e. some subset of @xmath522 ) by sending @xmath464 to @xmath523 , we say it acts by _ shifting to the right by @xmath127 _ when @xmath127 is positive , and by _ shifting to the left by @xmath127 _ when @xmath127 is negative . in particular , on each segment , @xmath63 acts by _ shifting _ iff orientation is preserved . the next two lemmas summarize all of the possible passing properties of an opposite - sign quadruple and its image under @xmath153 . [ pq ] take @xmath524 and @xmath281 . then exactly one of the following must hold :    @xmath525a@xmath526 @xmath527 + @xmath525b@xmath526 @xmath528    similarly , exactly one of the following must hold :    @xmath525c@xmath526 @xmath529 + @xmath525d@xmath526 @xmath530 .    these results , after interchanging superscripts of @xmath531 with @xmath0 , also hold when one considers @xmath292 and @xmath294 , instead of the original hypothesis . let @xmath532 and @xmath533 for some @xmath127 ( which exists by construction ) . suppose that @xmath534 . we will analyze all possible cases by considering the value of @xmath535 . write @xmath536 + q$ ] where @xmath537 . first , i claim that @xmath538 . suppose instead that @xmath539 . in this case , @xmath540 , which immediately implies from the proof of proposition [ qc ] that @xmath541 preserves orientation on @xmath353 . so @xmath283 preserves orientation , and by definition @xmath542 preserves orientation on @xmath283 . now , if we set @xmath543 , we find that @xmath544 because @xmath545 . now , @xmath540 shows that @xmath546 . finally , @xmath547 , because @xmath539 . these facts , however , contradict lemma [ c1 ] . so , it must be that @xmath548 . we divide into the two cases , ( 1 ) @xmath549 and ( 2 ) @xmath550 . first consider the case @xmath551 . set @xmath552 - po^l(v)$ ] . then we have three cases : ( i ) @xmath553 , ( ii ) @xmath554 , and ( iii ) @xmath555 . first consider ( i ) . now , @xmath556 , so @xmath557 . conversely , whenever @xmath557 , clearly @xmath558 with @xmath559 - po^r(y)$ ] ( we use that @xmath284 always preserves orientation ) . in this case , @xmath553 and @xmath549 , as desired . this situation , characterized by @xmath560 , falls into ( a ) and we will call it ( a1 ) . next , take ( ii ) . in this case , @xmath561 , so @xmath562 and @xmath563 . conversely , it is clear that @xmath563 iff @xmath562 from the construction of @xmath153 , and in this case , @xmath558 with @xmath564 . thus , @xmath565 and @xmath549 , as desired . this is a different case of ( a ) , so let us call it ( a2 ) .    finally , consider ( iii ) . in this case , @xmath566 . set @xmath567 $ ] . then it follows that @xmath568 , so that @xmath569 with @xmath570 . conversely , if @xmath571 , then @xmath572 with @xmath573 + po^l(x)$ ] , as desired ( this can be checked separately when @xmath103 reverses orientation  here @xmath574 so there is no difficulty . ) hence , @xmath575 and @xmath549 . this is the final case of ( a ) , so let us call it ( a3 ) .    next , consider the case @xmath576 . set @xmath577 $ ] . because @xmath576 , it follows that @xmath578 , hence @xmath579 with @xmath580 . since @xmath581 iff @xmath582 , we have @xmath583 . conversely , whenever @xmath579 , then @xmath534 with @xmath584 $ ] . hence , @xmath550 and @xmath583 . this accounts for case ( b ) . we have proved the first part of the lemma , because we have considered all possible nonzero values of @xmath585 and @xmath586 , and grouped them into the cases ( a1 ) , ( a2 ) , ( a3 ) , and ( b ) . we have shown that each of these is associated with different values of @xmath535 , which justifies the zero values of @xmath585 and @xmath586 in each case . next , we apply the same analysis used in the first part to show that exactly one of ( c),(d ) holds . let @xmath587 and @xmath588 . we suppose that @xmath589 and divide into the cases @xmath590 and @xmath591 . first consider @xmath590 . then it is clear that @xmath592 , with equality iff @xmath593 . now , @xmath594 is defined on @xmath595 , and @xmath596 , so @xmath597 preserves orientation on @xmath595 . this shows that @xmath598 , so that @xmath599 with @xmath600 . that is , @xmath601 . conversely , whenever @xmath599 , we know from the fact that @xmath282 preserves orientation that @xmath602 preserves orientation on @xmath603 , and hence that @xmath604 with @xmath605 . it is then clear that @xmath606 . this accounts for case ( d ) . now , suppose that @xmath589 with @xmath591 . we divide into the cases @xmath540 and @xmath574 . first suppose @xmath540 . in particular , this implies that @xmath377 is defined on @xmath595 , and by nilpotency , it must preserve orientation . also , @xmath541 must preserve orientation on @xmath595 . so , we see that @xmath607 , thus @xmath608 with @xmath609 . conversely , if @xmath540 and @xmath610 , we see from @xmath611 and lemma [ c1 ] that @xmath612 . it follows that @xmath613 with @xmath614 , and @xmath615 , as desired . this accounts for one situation of ( c ) ; call this ( c1 ) .    next , suppose @xmath574 and @xmath591 . we further divide into the three cases ( i ) @xmath616 , ( ii ) @xmath617 , and ( iii ) @xmath618 .    in case ( i ) , we use that @xmath284 preserves orientation to see that @xmath619 , so that @xmath620 with @xmath621 . conversely , whenever @xmath620 , it follows that @xmath574 using lemma [ c1 ] : otherwise we would have @xmath622 with @xmath623 while + @xmath624 . hence , @xmath625 with @xmath626 . call this situation ( c3 ) .    in case ( ii ) , we have that @xmath627 . conversely , whenever @xmath628 , we must have @xmath574 by lemma [ c1 ] , and then @xmath629 and @xmath630 , as desired . if @xmath631 , then it also follows from lemma [ c1 ] considering @xmath595 that @xmath574 , and then @xmath632 . call this situation ( c2 ) . finally , we consider case ( iii ) . now , @xmath618 shows that @xmath633 . by nilpotency , @xmath594 must preserve orientation on @xmath634 , and it follows that @xmath635 with @xmath636 . conversely , if @xmath635 with @xmath574 , then it follows that @xmath637 , so@xmath638 . hence , @xmath625 with @xmath639 , and @xmath640 . this has the same passing properties as ( c1 ) , so call this situation ( c1 ) . we have finished the second half of the lemma , since we have accounted for all possible values of @xmath641 , and @xmath642 in cases ( d ) , ( c1 ) , ( c1 ) , ( c2 ) , and ( c3 ) . each of these are associated with distinct values of @xmath643 with rexspect to the @xmath644 , and @xmath321 , once @xmath127 is fixed . to obtain the result for decreasing quadruples , simply reverse all directions and permute the components of all @xmath24-quadruples ( not pairs ! ) in this proof . [ ppp ] take @xmath645 and @xmath370 . then @xmath646 , and exactly one of the following hold :    @xmath525a@xmath526 @xmath527 . + @xmath525b@xmath526 @xmath583 , @xmath647 . under the hypotheses @xmath648 and @xmath649 , these results still hold upon interchanging superscripts of @xmath531 and @xmath0 . clearly @xmath593 iff @xmath632 . suppose @xmath650 . hence @xmath651 . suppose for a contradiction that @xmath620 . then , @xmath652 . however , @xmath653 and lemma [ c1 ] imply that @xmath654 , a contradiction . so , the identity @xmath655 easily follows ( since @xmath656 ) . the rest of the proof is almost exactly the same as the proof of the first part of lemma [ pq ] , getting rid of @xmath282 . again , the results follow with simple modifications in the decreasing case .      in this section , we consider the longer monomial terms that arise in the expansion of the formula for @xmath11 . as in the previous cases , most terms cancel ; we therefore explicitly give the groupings which cancel ( corollary [ top ] ) or almost cancel , and show that what remains is simply @xmath13 , proving theorem [ mt ] . in order to do this , we need to define the notion of a @xmath24-chain , which generalizes @xmath24-quadruples to the objects needed to handle the longer monomials , and prove some more combinatorial results regarding these . @xmath657 @xmath239-chain _ is a chain @xmath658 , so that @xmath210 . let @xmath659 denote the set of @xmath657 tp - chains . let @xmath660 denote the set of all @xmath239-chains , i.e. @xmath661 .    a chain @xmath662 is said to be _ outer _ if@xmath663 and @xmath664 . let @xmath665 denote the set of such @xmath657-chains , and let @xmath666 denote the set of all outer chains for any @xmath667 . lemmas [ ail]-[onp ] below have obvious analogues obtained by changing the sign of all @xmath24-pairs , reversing the order of @xmath24-chains , and replacing @xmath659 with @xmath668 in all forms . these results are not stated but will be referred to in the same manner as the actual results stated . [ ail ] @xmath525i@xmath526 suppose @xmath669 . suppose @xmath670 and @xmath671 where @xmath672 . then @xmath673 iff @xmath674 , @xmath675 iff @xmath676 , and @xmath677 iff @xmath678 . @xmath525ii@xmath526 suppose @xmath679 . suppose @xmath680 and @xmath681 where @xmath682 and @xmath683 . then @xmath684 iff @xmath685 for @xmath686 any order relation @xmath687 or @xmath424 . the proof is the same for both ( i ) and ( ii ) , does not mention @xmath688 , and is given in the following paragraphs . first , we note that @xmath675 iff @xmath689 and @xmath690 reverse orientation , which is true iff @xmath691 and @xmath692 reverse orientation , which is true iff @xmath676 . now , we show that @xmath693 implies @xmath694 . to reach a contradiction , suppose that @xmath693 and @xmath695 . by nilpotency , @xmath689 and @xmath692 preserve orientation . thus @xmath691 and @xmath690 also preserve orientation . write @xmath696 . first suppose @xmath697 . then @xmath698 , so that @xmath699 is defined on @xmath700 and therefore on @xmath701 . however , this implies that @xmath702 , while @xmath703 . this contradicts lemma [ c1 ] . so @xmath704 . then @xmath705 and @xmath706 . since @xmath693 , it must be that @xmath707 , which implies that @xmath708 . so @xmath709 \overline { < } ( z_2)_l + ( y_3)_l$ ] . now , write @xmath710 + m$ ] , for @xmath711 . since @xmath712 , it follows that @xmath713 ( @xmath714 ) or @xmath715 ( @xmath716 ) . by lemma [ c1 ] , it follows that @xmath717 divides both @xmath514 and @xmath718 . but then @xmath719 divides @xmath720 and @xmath721 , which shows that @xmath722 , contrary to assumption .    next , suppose @xmath722 . we show @xmath678 . find @xmath293 and @xmath112 such that @xmath723 and @xmath724 . in this case , @xmath725 . hence , @xmath726 . by assumption , @xmath727 . since @xmath728 , this in particular implies that @xmath678 , as desired . [ ail2 ] @xmath525i@xmath526 suppose @xmath729 with @xmath730 and + @xmath731 . let @xmath732 and suppose @xmath733 . . then @xmath694 . @xmath525ii@xmath526 similarly , suppose @xmath735 with @xmath736 and @xmath737 . then set @xmath738 and suppose @xmath733 . setting @xmath739 , it follows that @xmath694 . again , parts ( i ) and ( ii ) have nearly the same proof , which follows . suppose , on the contrary , that @xmath740 . clearly @xmath741 , else @xmath692 would have reversed orientation , which is not possible by lemma [ nc ] . so @xmath678 . now , @xmath742 is defined on @xmath701 since , in case ( i ) , @xmath742 is defined on @xmath743 and @xmath744 , which follows from @xmath745 , and in case ( ii ) , @xmath742 is defined on @xmath743 . in particular , @xmath746 is defined on @xmath701 . but now , @xmath747 but @xmath748 , contradicting lemma [ c1 ] .    [ pmol ] suppose @xmath749 with @xmath750 and @xmath751 . then @xmath752 iff @xmath753 and @xmath754 iff @xmath755 . first we note that , if @xmath752 , then @xmath756 , for the following reason . suppose @xmath752 and write @xmath757 or @xmath758 , depending on whether @xmath759 is bad or good . then @xmath760 , so lemma [ ail2 ] applies and shows that @xmath761 , as desired . by symmetry , @xmath754 implies that @xmath755 . so , it suffices to show that @xmath752 if @xmath762 and @xmath754 if @xmath763 . by the symmetry of the situation , we need only prove the first .    assume , for sake of contradiction , that @xmath754 and @xmath764 . first suppose that @xmath765 . then we note that @xmath766 is defined on @xmath767 because it is defined on @xmath768 and @xmath769 . next we note that @xmath770 because @xmath771 . also , @xmath772 , and @xmath773 . this contradicts lemma [ c1 ] .    on the other hand , it is impossible that @xmath774 shifted @xmath775 to the right by @xmath776 , while @xmath777 . this would contradict corollary [ c3 ] . [ bps ] suppose @xmath735 and @xmath778 . then @xmath779 . clearly @xmath780 would imply that @xmath781 and @xmath689 reverse orientation , which is not possible since latexmath:[$|x_1| >    that @xmath742 , and hence @xmath783 , is defined on @xmath743 , and hence @xmath784 . since @xmath785 , we find that @xmath786 \overline { < } t^{\o(x_2 ) } [ ( x_2)_l + ( y_2)_l ] \gtrdot t^{\o(y_2 ) } [ ( x_2)_l + ( y_2)_l]$ ] , contradicting lemma [ c1 ] . so @xmath779 . [ onp ] suppose @xmath787 with @xmath788 . then @xmath789 and @xmath790 . first , note that @xmath766 is defined on @xmath768 since it is defined on @xmath791 , and the former is a subset of the latter on the diagram . if @xmath792 , then set @xmath793 . if @xmath794 and @xmath795 , set @xmath796 . in either case , @xmath797 is defined on @xmath798 . indeed , @xmath799 is defined on @xmath798 , and @xmath800 . note that , by lemma [ nc ] , @xmath688 and @xmath690 must preserve orientation . hence , @xmath801 and @xmath799 must preserve orientation on @xmath802 . set @xmath803 . we have @xmath804 , contradicting lemma [ c1 ] . now , we are ready to define maps @xmath805 , and @xmath806 which parameterize terms which cancel in the expansion of @xmath807 ( given in corollary [ top ] ) .    for any map @xmath808 taking two arguments , let @xmath809 be the map @xmath808 applied to the @xmath215 and @xmath216-th components of some larger @xmath127-tuple . similarly , for any map @xmath810 taking only one argument , define @xmath811 to be the map @xmath810 applied to the @xmath215-th component of a larger @xmath127-tuple . define @xmath812 for @xmath813 as follows . for @xmath814 , let @xmath815 . define @xmath816 for @xmath817 by @xmath818 . for @xmath819 , define @xmath820 by @xmath821    note that , unlike in the case of the map @xmath404 ( i.e. the case of @xmath822 chains ) , we can have chains on which both maps @xmath823 and @xmath824 are defined . we see , however , that in many cases ( namely , the hypotheses of lemma [ pmol ] ) , only one map will yield an outer chain . when @xmath825 and @xmath826 , @xmath103 is said to be _ left - passed _ and we write @xmath827 . if @xmath828 and @xmath829 or @xmath830 , then @xmath103 is said to be _ right - passed _ or _ half - passed _ , and @xmath831 or @xmath832 , respectively . note that , given @xmath828 with @xmath833 , @xmath834 iff @xmath835 , so in these cases the half - passed chains not only have a half right - passed pair but also a half left - passed pair . define @xmath836 by @xmath837 @xmath838 similarly , define @xmath839 by @xmath840 .    as in the case of @xmath841 , we have the following :    [ ppi ] let @xmath842 . then @xmath843 iff @xmath844 for some sign @xmath389 or @xmath845 for some superscript @xmath846 . this follows easily from proposition [ qc ] .    a chain @xmath828 is said to be _ negatively special _ if @xmath847 and @xmath848 . the set of such chains is denoted @xmath849 . similarly , @xmath850 is _ positively special _ ( or in @xmath851 ) if @xmath852 and latexmath:[$|x_{m}| >    general term will be simply _ special_.    a chain @xmath828 is said to be _ negatively reversed _ if @xmath854 and @xmath855 and @xmath856 both reverse orientation . the set of such chains is @xmath857 . a chain @xmath828 is _ positively reversed _ ( or in @xmath858 ) if @xmath859 and @xmath860 and @xmath861 both reverse orientation . as before , @xmath862 is the set of _ reversed _ chains . [ rcl ] if @xmath863 , then @xmath864 or @xmath865 . if @xmath866 , then @xmath867 for any @xmath868 . in fact , if @xmath869 , then @xmath824 is not defined on @xmath103 , and @xmath870 satisfies @xmath871 . additionally , @xmath872 or @xmath873 where @xmath874 .    similarly , if @xmath875 , then @xmath823 is not defined on @xmath103 , and @xmath876 satisfies @xmath877 . additionally , @xmath878 or @xmath879 where @xmath880 . this easily follows from lemmas [ nc ] and [ psq ] . [ rs ] no outer chain is both reversed and special . we define maps @xmath881 as follows . if @xmath882 where @xmath883 , set @xmath884 . if @xmath885 , set @xmath886 , and if @xmath887 , set @xmath888 . similarly , if @xmath889 where @xmath890 , set @xmath891 . if @xmath892 , set @xmath893 , and if @xmath894 , then set @xmath895 .    [ pg1 ] ( i ) take @xmath896 for some fixed sign @xmath389 . then @xmath897 or @xmath898 , but not both . in the former case , @xmath899 as well . \(ii ) if @xmath900 , for @xmath901 , then either ( a ) @xmath902 , ( b ) @xmath903 , or ( c ) neither . in case ( c ) , @xmath904 . in case ( a ) , @xmath905 , and in case ( b ) , @xmath906 . \(iii ) if @xmath907 $ ] , then @xmath908 . additionally , @xmath909 if @xmath910 and @xmath911 if @xmath912 $ ] . \(i ) this follows immediately from lemma [ ail ] in the cases that @xmath913 and @xmath914 . ( note that it is impossible to have @xmath904 by corollary [ rs ] . ) in the case that @xmath915 and @xmath916 or in the case that @xmath917 and @xmath918 , it is clear that @xmath898 . \(ii ) it follows that only one of the @xmath919 can be in @xmath666 from lemma [ pmol ] . if @xmath920 , then no @xmath921 is outer by lemma [ rcl ] . by lemma [ pmol ] , if @xmath922 , there is nothing to prove . if @xmath923 and @xmath924 , then lemma [ ail2 ] gives the desired result . for @xmath925 the result is easy . \(iii ) the two statements follow from lemmas [ onp ] , [ bps ] , respectively . we define the following subsets of @xmath666 : + ( a ) @xmath926 + ( a ) @xmath927 + ( b ) @xmath928 \mid \psi_{\pm}(x ) \in tpc^o \}$ ] + furthermore , let @xmath929 , @xmath930 , and @xmath931 .    [ ds ] the subsets @xmath932 and @xmath933 are disjoint and their union is all of @xmath666 .    that @xmath934 , and @xmath935 are disjoint from @xmath936 follows from corollary [ rs ] , lemma [ rcl ] , and the definition , respectively . the other facts regarding disjointedness are obvious . to check that the union is all of @xmath666 , we apply lemma [ pg1].ii , which shows that @xmath937 $ ] . it is clear , though , that @xmath938 . this proves the desired result . [ ctc ] @xmath525i@xmath526 for each choice of sign , @xmath939 maps @xmath940 injectively to @xmath941 . + @xmath525ii@xmath526 @xmath942 maps @xmath943 \cap [ rpc \cup hpc]$ ] injectively to @xmath944 $ ] and @xmath945 maps @xmath946 \cap lpc$ ] injectively to @xmath947 . + @xmath525iii@xmath526 @xmath948 = \emptyset$ ] . + @xmath525iv@xmath526 @xmath949 $ ] and @xmath950 $ ] are both disjoint from @xmath951 . + @xmath525v@xmath526 @xmath952 is disjoint from @xmath666 for each choice of sign . \(i ) this follows immediately from lemmas [ pg1].i and [ ppi ] . \(ii ) this follows immediately from lemmas [ pg1].ii , [ ppi ] , and [ ppp ] . \(iii ) this follows immediately from lemma [ onp ] . \(iv ) this follows immediately from the definition of @xmath953 . \(v ) this follows from lemma [ pg1].i , since we know @xmath951 is disjoint from @xmath936 . [ top ] @xmath666 is partitioned by all sets of the form : + @xmath525a@xmath526 for @xmath954 , @xmath955 @xmath525a@xmath526 those of the form , replacing @xmath138 with @xmath956 and @xmath531 with @xmath0 . ( call these also @xmath957 , distinguished from the above by the `` sign '' of @xmath103 . ) + @xmath525b@xmath526 for a fixed @xmath958 , @xmath959 @xmath525b@xmath526 those of form , replacing @xmath138 with @xmath956 , @xmath531 with @xmath0 , and @xmath960 with @xmath933 . call these @xmath957 as well .    here it is meant that , if @xmath961 is not defined on @xmath103 , then @xmath962 for any @xmath284 .    for every @xmath954 , we see from the lemma , part ( i ) , that @xmath963 . furthermore , @xmath964 and @xmath965 where these are defined ( lemma , part ii ) . now take any @xmath966 . first note that either @xmath967 or @xmath968 because @xmath969 . note that @xmath970 because @xmath971 and @xmath972 are disjoint from @xmath666 ( lemma [ rcl ] ) . now , if @xmath973 , then using lemma [ ds ] , @xmath974 ( of type a or b , depending on whether @xmath975 or @xmath968 , respectively ) . if @xmath976 , then clearly @xmath966 . in this case , @xmath977 by the lemma , parts ( ii ) , ( iv ) , and ( v ) , using lemma [ ppi ] . furthermore , in this case @xmath978 , again from the lemma , parts ( iv ) and ( v ) . then , @xmath979 . thus , we see that all elements @xmath966 fit into at least one of the sets ( a ) or ( b ) . to see that all elements in @xmath980 fit into at most one , note that the way in which @xmath284 appears is uniquely determined by which of the following holds : ( a ) @xmath973 , @xmath981 ; ( b ) @xmath973 , @xmath982 ; ( c ) @xmath976 , @xmath983 . now , it remains to consider those elements of @xmath936 . for any @xmath984 , we know that @xmath985 by lemma [ rcl ] . the former two are subsets of @xmath935 , so we know that @xmath986 for any @xmath984 , and clearly @xmath987 is unique . the negative case is almost identical to this one .    for @xmath988 , set @xmath989 if @xmath990 or @xmath991 and @xmath144 otherwise . now , given @xmath992 , set @xmath993 . [ jrjl ]    the terms in the expansion of @xmath807 are @xmath995 for each @xmath996 , as well as the terms @xmath997 for @xmath164 and @xmath998 for each @xmath215 .    [ cop ] @xmath525i@xmath526 if @xmath999 , then @xmath1000 . @xmath525ii@xmath526 if @xmath1001 , then @xmath1002 . \(i ) let @xmath954 . the negative case is similar ( see comments at the end of the proof of this part ) . let @xmath828 and let @xmath1003 be the part of the formula for @xmath1004 which will not change upon applying @xmath823 , @xmath945 , and @xmath1005 ( where applicable ) . let @xmath1006 if @xmath827 and @xmath1007 otherwise . set @xmath1008 . first we show that @xmath1009 first , by lemma [ pq ] , @xmath1010 hence , @xmath1011 if @xmath1012 , then @xmath1007 and @xmath1013 . in this case , follows from the definition of @xmath1014 . if @xmath827 , then @xmath1015 , and implies @xmath1016 . then lemma [ ppp ] shows that @xmath1017 so that the left - hand side of is @xmath1018 . by , this is the same as the right - hand side . next , we show that @xmath1019 f.\ ] ] naturally , we may assume that @xmath860 reverses orientation . since , in this case , @xmath1020 also reverses orientation , it must be that @xmath1021 . suppose now that @xmath1022 and @xmath1023 . now , for every @xmath293 such that @xmath1024 , set @xmath1025 , @xmath1026 , and @xmath1027 . then @xmath1028 are exactly those reversed chains such that @xmath1029 . now , we consider the possible passing properties of the @xmath24-pairs involved . note first that , since @xmath1030 , @xmath1031 for all @xmath293 . for the same reason , @xmath1032 for all @xmath293 . next , note that @xmath1033 for all @xmath293otherwise , applying @xmath1034 to @xmath689 would contradict nilpotency . for the same reason , @xmath1035 for all @xmath293 . next , note that @xmath1036 for all @xmath293 . this follows from the fact that @xmath1037 for all @xmath293 , making use of lemma [ pq ] . now , follows readily . let @xmath1038 when @xmath1039 and @xmath1040 otherwise . finally , we show that @xmath1041 the proof is similar to the proof two paragraphs back . if @xmath1042 , then @xmath1040 . by lemma [ pq ] , @xmath1043 . thus @xmath1044 , proving in this case ( note that @xmath1045 and recall the definition of @xmath1014 . ) if @xmath1039 , then @xmath1046 and by lemma [ pq ] ( or the equation above ) , @xmath1047 and @xmath1048 . if we set @xmath1049 , then it follows from lemma [ ppp ] that neither @xmath1050 nor @xmath1051 is passed , so @xmath1052 . hence , @xmath1053 , so the left - hand side of is @xmath1054 , which is equal to the right - hand side by corollary [ psq ] and the above analysis .    now , putting , , and together , we get @xmath1055 as desired . the negative case is almost the same as the above , except that the `` error term '' @xmath1014 over - corrects , but this is counteracted by the fact that now @xmath942 is defined on @xmath832 as well as @xmath1056 . ( the details are omitted . ) \(ii ) this is almost the same as the proof above . first we note that the result is clear when @xmath1057 , as @xmath1058 and @xmath1004 gives the desired formula ( bearing in mind the definition of @xmath1059 ) . suppose @xmath1060 . let @xmath1006 and let @xmath1061 if @xmath827 and @xmath1062 , and @xmath1040 otherwise . let @xmath1063 f.\ ] ]    lemma [ ppp ] shows that @xmath1064 and @xmath1065 first suppose @xmath1040 . in this case , implies easily . if @xmath1066 , then @xmath1067 , and @xmath1068 . then , @xmath1069 . furthermore , @xmath1070 . from this follows . next , we claim that @xmath1071 f.\ ] ] this follows by exactly the same arguments as used in the second paragraph of the proof of part ( i ) . but , @xmath1064 , so , putting and together , the result easily follows . the negative/@xmath933 case is almost the same as this , bearing in mind the final comments in the proof of the previous part . _ proof of the main theorem , [ mt ] . _ this follows immediately from lemmas [ jrjl ] and [ cop ] and corollary [ top ] , using for @xmath13 . i would like to thank pavel etingof for introducing me to this problem and advising me . i would also like to thank the harvard college research program for their support . finally , i am indebted to gerstenhaber , giaquinto , and hodges for valuable discussions and for sharing some unpublished results . in this section we explicitly compute @xmath13 for generalized cremmer - gervais triples , the only triples satisfying @xmath1072 . recall from example [ gcge ] the results given in @xcite : these triples are indexed by @xmath1073 where @xmath1074 is the dimension , @xmath514 is relatively prime to @xmath151 , @xmath1075 , @xmath1076 , and @xmath1077 , where @xmath160 give the residue mod @xmath151 in @xmath161 . moreover when @xmath100 is taken to have trace zero in the first component , it is uniquely given by @xmath1078 and @xmath1079 where @xmath1080 . then , the only difficulty is in computing @xmath1081 , so here we use to prove . clearly we have @xmath1082 where @xmath1083 , for @xmath1084 , since @xmath338 for all @xmath103 . it suffices , then , to show @xmath1085 for all @xmath103 . take @xmath1086 . below we use the notation @xmath1087 = 1 $ ] if `` statement '' is true and 0 if it is false . one sees that @xmath1088 - \frac{1}{2}[i = k ] - \\ \frac{1}{n}\text{res}\bigl(\frac{i+k-2j}{m}\bigr ) - \frac{1}{n}\text{res } \bigl(\frac{k - i}{m}\bigr ) \\ = 1 - \frac{1}{2}[2j = i+k ] - \frac{1}{2}[i = k ] - \frac{2}{n } \text{res}\bigl(\frac{k - j}{m}\bigr ) + m_{i , j , k},\end{gathered}\ ] ] where @xmath1089[\text{res}\bigl(\frac{k - j}{m}\bigr ) > \text{res}\bigl(\frac{i+k-2j}{m } ) ] \\ - [ i \neq k][\text{res}\bigl(\frac{k - j}{m}\bigr ) < \text{res}\bigl(\frac{k - i}{m}\bigr)].\end{gathered}\ ] ] thus , since @xmath1090 , it suffices to show @xmath1091 + \frac{1}{2}[i = k ] + p^r(x ) + p^l(x ) = 1 + m_{i , j , k}$ ] . note that @xmath1092 iff @xmath632 and @xmath1093 iff @xmath562 , and in these cases @xmath1094 , so it suffices to consider @xmath1095 . in this case we need to show @xmath1096 . now , @xmath1097 + [ \text{res}\bigl(\frac{k - j}{m}\bigr ) > \text{res}\bigl(\frac{k - i}{m}\bigr)],\end{gathered}\ ] ] and it is not difficult to see that @xmath1098 = p^l(x)$ ] while @xmath1099 = p^r(x)$ ] . this finishes the proof . m. gerstenhaber and a. giaquinto , and s. d. schack , _ construction of quantum groups from belavin - drinfeld infinitesimals _ , israel math . * 7 * ( 1993 ) , 4564 . a. giaquinto and t. j. hodges , _ nonstandard solutions of the yang - baxter equation _ * 44 * ( 1998 ) , 6775 .
we prove the ggs conjecture @xcite ( 1993 ) , which gives a particularly simple explicit quantization of classical @xmath0-matrices for lie algebras @xmath1 , in terms of a matrix @xmath2 which satisfies the quantum yang - baxter equation ( qybe ) and the hecke condition , whose quasiclassical limit is @xmath0 . 
 the @xmath0-matrices were classified by belavin and drinfeld in the 1980 s in terms of combinatorial objects known as belavin - drinfeld triples . 
 we prove this conjecture by showing that the ggs matrix coincides with another quantization from @xcite , which is a more general construction . 
 we do this by explicitly expanding the product from @xcite using detailed combinatorial analysis in terms of belavin - drinfeld triples .