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we assume basic familiarity with hopf algebras , as found in the opening chapters of the book  @xcite . let , be a pair of dual hopf algebras over a field @xmath3 . this means that there is a nondegenerate pairing @xmath4 for which the structure of @xmath5 is dual to that of @xmath6 and vice - versa . for example , @xmath6 could be finite - dimensional and @xmath5 its linear dual , or @xmath6 could be graded with each component finite - dimensional and @xmath5 its graded dual . these algebras naturally act on each other  @xcite : suppose that @xmath7 and @xmath8 and set @xmath9 ( we use sweedler notation for the coproduct , @xmath10 . ) these left actions are the adjoints of right multiplication : for @xmath11 and @xmath12 , @xmath13 this shows that @xmath5 is a left @xmath6-module under the action in . in fact , @xmath5 is a left @xmath6module algebra , meaning that for @xmath12 and @xmath7 , @xmath14 recall that the @xmath15 and @xmath16 satisfy @xmath17 and @xmath18 for all @xmath7 . [ lem : hopf ] for @xmath11 and @xmath8 , we have @xmath19    let @xmath20 . we prove first the formula @xmath21 ( this is essentially @xmath22 in the proof of lemma 2.1.4 in  @xcite . ) expanding the sum using   and coassociativity , @xmath23 , gives @xmath24 here , follows as @xmath5 is an @xmath6-module and   from the antipode and counit conditions . note that @xmath25 . using this becomes @xmath26 which proves the lemma , as this holds for all @xmath20 . this proof is identical to the argument in the appendix to  @xcite , where @xmath27 was a complete homogeneous symmetric function in the hopf algebra @xmath6 of symmetric functions . we apply lemma [ lem : hopf ] to produce skew littlewood - richardson rules for several hopf algebras in algebraic combinatorics . we isolate the common features of those arguments .    in the notation of section [ sec : harpoon ] , let @xmath28 and @xmath29 be dual bases indexed by some set @xmath30 , so @xmath31 for @xmath32 . define structure constants for @xmath6 and @xmath5 via @xmath33{0pt}{16pt}\end{aligned}\ ] ] the @xmath34 and @xmath35 defined above co - multiply according to @xmath36 ( note that the structure of @xmath5 can be recovered from the structure of @xmath6 . thus , we may suppress the analogs of and the second formula in in the coming sections . )    finally , suppose that the antipode acts on @xmath6 in the @xmath37-basis according to the formula @xmath38 for some functions @xmath39 and @xmath40 . then lemma [ lem : hopf ] takes the following form . [ th : algebraic rule ] for any @xmath41 , we have @xmath42 swapping @xmath43 and @xmath44 in yields the analog for the skew elements @xmath45 in @xmath5 .    the actions in   together with the second formulas for the coproducts in   and show that @xmath46 and @xmath47 . now use   and   to obtain @xmath48 this equals the right hand side of  , since @xmath49 . the condition is highly restrictive . it implies that the antipode @xmath50 , as a linear map , is conjugate to a signed permutation matrix . nevertheless , it holds for the hopf algebras we consider . more generally , it holds if either @xmath6 or @xmath5 is commutative , for then @xmath50 is an involution ( * ? ? ? the commutative hopf algebra @xmath51 of symmetric functions is graded and self - dual under the hall inner product @xmath52 . a systematic study of @xmath53 from a hopf algebra perspective appears in @xcite . we follow the definitions and notation in chapter i of  @xcite . the schur basis of @xmath53 ( indexed by partitions ) is self - dual , so and become @xmath54 where the @xmath55 are the and the @xmath56 are the ( * ? ? ? combinatorial expressions for the @xmath57 and inner products @xmath58 are derived using the hopf algebraic structure of @xmath53 in @xcite . the coefficients occur in the triple product @xmath59 , @xmath60 write @xmath61 for the conjugate ( matrix - transpose ) of @xmath62 . then the action of the antipode is @xmath63 which is just a twisted form of the fundamental involution @xmath64 that sends @xmath65 to @xmath66 . indeed , the formula @xmath67 shows that   holds on the generators @xmath68 of @xmath53 . the validity of   follows as both @xmath50 and @xmath64 are algebra maps .    since @xmath69 unless @xmath70 , we may write as @xmath71    we next formulate a combinatorial version of  . given partitions @xmath62 and @xmath72 , form the skew shape by placing @xmath62 southwest of @xmath72 . thus , @xmath73 similarly , if @xmath74 is a tableau of shape @xmath62 and @xmath50 a tableau of shape @xmath72 , then @xmath75 is the skew tableau of shape @xmath76 obtained by placing @xmath74 southwest of @xmath50 . fix a tableau @xmath77 of shape @xmath78 . the littlewood - richardson coefficient @xmath79 is the number of pairs @xmath80 of tableaux of respective shapes @xmath62 and @xmath72 with @xmath75 knuth - equivalent to @xmath77 . see  ( * ? ? ? * ch . 5 , cor . 2(v ) ) . similarly , @xmath81 is the number of triples @xmath82 of tableaux of respective shapes @xmath83 , @xmath62 , and @xmath72 with @xmath84 knuth - equivalent to @xmath77 . write for the shape of a tableau @xmath50 and @xmath85 if @xmath50 is knuth - equivalent to @xmath77 . [ l : delta(skew ) ] let @xmath86 be partitions and fix a tableau @xmath77 of shape @xmath78 . then @xmath87 the sum taken over triples @xmath88 of tableaux with @xmath89 and @xmath90 . note that @xmath91 and the operation @xmath92 makes sense for skew tableaux . if @xmath50 is a tableau of skew shape @xmath93 , put @xmath94 . [ th : two ] let @xmath95 be partitions and fix a tableau @xmath77 of shape @xmath78 . then @xmath96 the sum taken over triples @xmath97 of skew tableaux of respective shapes @xmath98 , @xmath99 , and @xmath72 such that @xmath100 . if @xmath77 is the unique of shape @xmath78 whose @xmath101th row contains only the letter @xmath101 , then this is conjecture 6.1 in  @xcite . indeed , in this case @xmath50 is yamanouchi of shape @xmath72 , so the sum is really over pairs of tableaux , and this explains the @xmath72-yamanouchi condition in @xcite . the difference lies in the tableau @xmath102 and the reading word condition in  @xcite . it is an exercise in tableaux combinatorics that there is a bijection between the indices @xmath103 of theorem  [ th : two ] and the corresponding indices of conjecture 6.1 in  @xcite . we reinterpret   in terms of tableaux . let @xmath88 be a triple of tableaux of partition shape with @xmath89 and @xmath104 . if @xmath105 , then by  ( * ? ? ? * ch . 5 , cor . 2(i ) ) , @xmath106 counts skew tableaux @xmath102 of shape @xmath98 with @xmath107 . likewise , if @xmath108 , then @xmath109 counts skew tableaux @xmath110 of shape @xmath99 with @xmath111 . now   may be written as @xmath112 summing over skew tableaux @xmath113 with @xmath114 of partition shape , @xmath89 , @xmath90 , @xmath115 , @xmath116 , and @xmath117 . finally , note that @xmath114 is the unique tableau of partition shape knuth - equivalent to @xmath118 . since @xmath119 is knuth - equivalent to @xmath77 ( by transitivity of @xmath120 ) , we omit the unnecessary tableaux @xmath114 from the indices of summation and reach the statement of the theorem . the self - dual hopf algebra of symmetric functions has a natural self - dual subalgebra . this has dual bases the schur @xmath0- and @xmath1-functions  ( * ? ? ? * iii.8 ) , which are indexed by @xmath121 . write for the length of the partition @xmath122 . as in section [ sec : schur s ] , the constants and skew functions in the structure equations @xmath123 have combinatorial interpretations ( see below ) . also , each basis @xmath124 and @xmath125 is _ almost _ self - dual in that @xmath126 and @xmath127 . the algebra @xmath128 is generated by the special @xmath1-functions @xmath129 ( * ? ? ? * iii , ( 8.1 ) ) . this implies that @xmath130 , from which we deduce that @xmath131 as @xmath132 unless @xmath133 , we may write the algebraic rule   as @xmath134 with a similar identity holding for @xmath135 ( swapping @xmath136 and @xmath137 ) . we formulate two combinatorial versions of  . strict partitions @xmath138 are written as ( where row @xmath101 begins in column @xmath101 ) . skew shifted shapes @xmath139 are defined in the obvious manner : @xmath140 in what follows , tableaux means _ semi - standard ( skew ) shifted tableaux on a marked alphabet _ * iii.8 ) . we use shifted versions of the jeu - de - taquin and plactic equivalence from  @xcite and  @xcite , denoting the corresponding relations by and , respectively . given tableaux @xmath141 we write @xmath142 when representative words @xmath143 ( built via `` @xmath144 ''  @xcite ) of the corresponding shifted plactic classes satisfy @xmath145 .    stembridge notes ( following ( * ? ? ? 8.2 ) ) that for a fixed tableau @xmath146 of shape @xmath147 , @xmath148 serrano has a similar description of these coefficients in terms of @xmath149 . fixing a tableau @xmath77 of shape @xmath78 , it follows from ( * ? ? ? 1.15 ) that the coefficient @xmath150 in @xmath151 is given by @xmath152    if @xmath77 is a tableau of shape @xmath122 , write @xmath153 for @xmath154 . the formula relating the @xmath155 s and @xmath156 s combines with and to give our next result . [ th : schur - q ] let @xmath95 be shifted partitions and fix a tableau @xmath77 of shape @xmath78 . then @xmath157 the sum taken over quintuples @xmath113 with @xmath114 of partition shape , @xmath89 , @xmath158 , @xmath115 , @xmath159 , and @xmath160 . serrano conjectures an elegant combinatorial description ( * ? ? ? * conj . 2.12 and cor . 2.13 ) of the structure constants @xmath161 in : for any tableau @xmath146 of shape @xmath147 , he conjectured @xmath162 ( note that if @xmath163 are tableaux , then @xmath164 does not necessarily imply that @xmath165 . ) this leads to a conjectural reformulation of theorem [ th : schur - q ] in the spirit of theorem [ th : two ] . [ th : conj schur - q ] let @xmath95 be partitions and fix a tableau @xmath77 of shape @xmath78 . then @xmath166 the sum taken over triples @xmath97 of skew tableaux of respective shapes @xmath167 , @xmath99 , and @xmath72 such that @xmath168 .    there is a unique shifted tableau @xmath74 in any shifted plactic class  ( * ? ? so the conditions @xmath169 and @xmath170 in and may be replaced with the single condition @xmath168 . the hopf algebra of noncommutative symmetric functions was introduced , independently , in @xcite as the ( graded ) dual to the commutative hopf algebra of quasisymmetric functions . we consider the dual bases ( indexed by compositions ) @xmath171 of gessel s quasisymmetric functions and @xmath172 of noncommutative ribbon schur functions . the structure constants in @xmath173 may be given combinatorial meaning via the descent map @xmath174 from permutations to compositions and a section of it @xmath175 . these maps are linked via , edge - connected skew young diagrams ( written in the french style ) , with no @xmath176 subdiagram present . by way of example , @xmath177 ( in the intermediate step for @xmath178 , new rows in the ribbon begin at descents of @xmath179 . in the intermediate step for @xmath180 , the boxes in the ribbon are filled left - to - right , bottom - to - top . ) a ribbon @xmath181 may be extended by a ribbon @xmath182 in two ways : affixing @xmath182 to the rightmost edge or bottommost edge of @xmath181 ( written @xmath183 and @xmath184 , respectively ) : @xmath185 if a ribbon @xmath186 is formed from @xmath181 and @xmath182 in either of these two ways , we write @xmath187 . the coefficient @xmath188 is @xmath189 , if @xmath187 , and @xmath190 otherwise . if @xmath191 is the shifted shuffle product on permutations ( see ( 3.4 ) in @xcite ) , then the coefficient @xmath192 is the number of words @xmath179 in @xmath193 such that @xmath194 . the coefficient @xmath195 has the analogous description . antipode formulas for the distinquished bases were found , independently , in @xcite : @xmath196 where @xmath197 is the conjugate of @xmath181 ( in the sense of french style skew partitions ) . for example , @xmath198 . the descriptions of the antipode and structure constants in and give a formula for the product of two skew ribbon schur functions . [ th : schur - r ] let @xmath199 be compositions . then @xmath200 the sum taken over factorizations @xmath201 , extensions @xmath202 , and words @xmath179 in the shuffle product @xmath203 such that @xmath204 . [ rem : f skew ] the nonzero skew ribbon schur functions @xmath205 do not correspond to skew ribbon shapes in a simple way . for example , @xmath206 is not a ( connected ) sub - ribbon of @xmath207 , yet @xmath208 . contrast this with the skew functions @xmath209 , where @xmath210 if and only if @xmath211 for some ribbon @xmath64 . that is , @xmath212 . thus we may view @xmath181 as the last @xmath213 boxes of the ribbon @xmath182 and @xmath214 as the complementary ribbon @xmath64 . interpreting @xmath215 alternately as a product of ordinary functions or skew functions yields the curious identity @xmath216 fix an integer @xmath217 . let @xmath218 denote the hopf subalgebra of the hopf algebra of symmetric functions generated by the homogeneous symmetric functions @xmath219 . let @xmath220 denote the hopf - dual quotient hopf algebra of symmetric functions . we consider the dual bases @xmath221 and @xmath222 of @xmath2-schur functions and dual @xmath2-schur functions of @xcite , also called strong schur functions and weak schur functions in @xcite . the @xmath2-schur functions were first introduced by lapointe , lascoux , and morse in the context of macdonald polynomials , and were later shown by lam to represent schubert classes in the affine grassmannian of @xmath223 . we refer the reader to the references in @xcite . here @xmath122 varies over all @xmath2-bounded partitions , that is , those partitions satisfying @xmath224 . there is an involution @xmath225 on @xmath2-bounded partitions called . we have @xmath226    if @xmath227 is a one - part partition , then @xmath228 is a homogeneous symmetric function . we have the @xmath2-pieri and dual @xmath2-pieri rules @xcite ( called weak and strong pieri rules in @xcite ) @xmath229 for @xmath230 . here @xmath231 denotes an connecting @xmath122 and @xmath147present if and only if both @xmath93 and @xmath232 are horizontal @xmath233-strips . the notation @xmath234 denotes an as introduced in @xcite , which we will not define here . the terminology comes from the relationship with the weak and strong ( bruhat ) orders of the affine symmetric group . we remark that there may be distinct strong strips @xmath234 and @xmath235 which start and end at the same partition , so that the second pieri rule of may have multiplicities . ( strictly speaking , the strong strips in @xcite are built on @xmath236-cores , and our @xmath234 denotes the strips obtained after applying a bijection between @xmath236-cores and @xmath2-bounded partitions . ) we define skew functions @xmath237 and @xmath238 using and . there is an explicit combinatorial description of @xmath238 in terms of the weak tableaux of @xcite , but only a conjectured combinatorial description of @xmath237 ( * ? ? ? 4.18(3 ) ) . [ thm : skewkpieri ] for @xmath2-bounded partitions @xmath239 , and @xmath230 , @xmath240    ( \lambda^-)^{\omega_k } { { \,\rightsquigarrow_{j}\ , } } \lambda^{\omega_k } } }    s_{\mu^+/\lambda^-}^{(k)}\ ] ]    in theorem [ th : algebraic rule ] , take @xmath241 . for @xmath81 , use the formula @xmath242 , and for @xmath243 and @xmath244 , use . [ thm : skewdualkpieri ] for @xmath2-bounded partitions @xmath245 , and @xmath230 , @xmath246    ( \mu^-)^{\omega_k } { { \,\rightarrow_{j}\ , } } \mu^{\omega_k } } } f_{\lambda^+/\mu^-}^{(k)}\ ] ]    identical to the proof of theorem [ thm : skewkpieri ] .    as an example , let @xmath247 , @xmath248 , @xmath249 , and @xmath250 . then theorem [ thm : skewkpieri ] states that @xmath251 which one can verify using and the expansions @xmath252 and @xmath253 . theorem [ thm : skewdualkpieri ] states that @xmath254 one can verify that both sides are equal to @xmath255 . fix @xmath256 . the hopf algebra @xmath128 of section [ sec : schur pq ] contains a hopf subalgebra @xmath257 generated by the schur @xmath0-functions @xmath258 . in @xcite , it was shown that @xmath257 is isomorphic to the homology ring @xmath259 of the affine grassmannian of the symplectic group @xmath260 . a distinguished basis @xmath261 , representing the schubert basis , was studied there . the symmetric functions @xmath262 are shifted versions of the @xmath2-schur functions of section [ sec : kschur ] . the indexing set for the basis @xmath263 is the set @xmath264 of affine grassmannian type @xmath265 permutations : they are the minimal length coset representatives of @xmath266 in @xmath267 . a lower bruhat order ideal @xmath268 of the affine type @xmath265 weyl group is defined in @xcite . let @xmath269 denote those @xmath270 with length @xmath271 . for each @xmath272 , there is a nonnegative integer @xmath273 , called the of @xmath274 . we note that @xmath275 . with this notation , for each @xmath276 , we have the pieri rule ( * ? ? ? 1.3 and 1.4 ) @xmath277 where the sum is over all @xmath278 such that @xmath279 , and @xmath280 .        in theorem [ th : algebraic rule ] , take @xmath292 and use . for the constants @xmath195 , use the formula @xmath293 if @xmath294 , the product @xmath295 in becomes @xmath296 . if @xmath297 ( resp . , @xmath298 ) , it becomes @xmath299 ( resp . , @xmath300 ) . susan montgomery , _ hopf algebras and their actions on rings _ , cbms regional conference series in mathematics , vol .  82 , published for the conference board of the mathematical sciences , washington , dc , 1993 .
we use hopf algebras to prove a version of the littlewood - richardson rule for skew schur functions , which implies a conjecture of assaf and mcnamara . 
 we also establish skew littlewood - richardson rules for schur @xmath0- and @xmath1-functions and noncommutative ribbon schur functions , as well as skew pieri rules for @xmath2-schur functions , dual @xmath2-schur functions , and for the homology of the affine grassmannian of the symplectic group . 
 assaf and mcnamara  @xcite recently used combinatorics to give an elegant and surprising formula for the product of a skew schur function and a complete homogeneous symmetric function . 
 their paper included a conjectural skew version of the littlewood - richardson rule , and also an appendix by one of us ( lam ) with a simple algebraic proof of their formula . 
 we show how these formulas and much more are special cases of a simple formula that holds for any pair of dual hopf algebras . 
 we first establish this hopf - algebraic formula , and then apply it to obtain formulas in some well - known hopf algebras in combinatorics .