A RELATION BETWEEN SCHUR P AND S. S. Leidwanger. Universite de Caen, CAEN. cedex FRANCE. March 24, 1997

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1 A RELATION BETWEEN SCHUR P AND S FUNCTIONS S. Leidwanger Departement de Mathematiques, Universite de Caen, 0 CAEN cedex FRANCE March, 997 Abstract We dene a dierential operator of innite order which sends Schur S- functions to Schur P -functions. Using the properties of we deduce algebraic identities satised by the cardinalities of certain sets of tableaux. Resume Nous denissons un operateur dierentiel d'ordre inni qui envoie les fonctions S de Schur sur les fonctions P de Schur. A l'aide des proprietes de, nous deduisons des identites algebriques satisfaites par les cardinalites de certains ensembles de tableaux. Introduction The subject of this paper is to describe a dierential operator which sends a Schur S-function s to a Schur P -function P c() where c() is a composition associated to the partition by an easy combinatorial method. The denition of was motivated by the results of [5]. Using the properties of, we can then deduce algebraic identities satised by the cardinalities of certain sets of tableaux (Yamanouchi tableaux, Yamanouchi domino tableaux, \Stembridge" tableaux). It would be interesting to obtain bijective proofs of these identities. The operator has a natural interpretation in the representation theory of ane Lie algebras [6]. Indeed it is well-known that S-functions form a linear basis of the algebra of symmetric functions Sym and P -functions form a linear basis of T () the subalgebra of Sym generated by odd power sums p i. Furthermore there are natural actions on Sym and T () of the innite dimensional Lie algebra b (see [] ; we follow Kac's notation for ane Lie algebras). It can be shown that is the unique linear map from Sym to T () sending to and commuting with the lowering operators of b. This viewpoint will not be developed here but see [7].

2 Symmetric functions We review the necessary background in the theory of symmetric functions (plethysms, dierential operators) and recall some denitions concerning partitions (-quotient, - core and -sign). We also introduce a linear involution on Sym, and we associate to each partition a composition c() and another partition J(). Our notations for symmetric functions are as in [8]. A partition = ( ; : : : ; n ) is a weakly decreasing sequence of nonnegative integers. We denote by 0 the conjugate partition of, jj the weight of and l() the length of. To a partition = ( : : : l j : : : l ) in Frobenius' notation, we associate the composition c() = ( ; ; ; ; : : : ; l ; l ): An example is shown in Figure. Figure : = (6; ; ; ; ) = (5; j; ); c() = (6; ; ; ) Products of power sums, Schur S-functions, Schur P -functions are respectively denoted by p, s, P. The Schur P -functions are indexed by strict partitions i.e partitions with distinct parts. We denote by Sym(A) the algebra of symmetric functions in an innite set of variables A = fa ; a ; : : :g with coecients in C. When there is no danger of confusion we shall omit A and simply write Sym. Sym can also be regarded as the polynomial ring Sym = C [p i ; i N ]. Let T () be the subalgebra of Sym generated by odd power sums T () = C [p i ; i N]. The Schur S-functions give a linear basis of Sym and the Schur P -functions give a linear basis of T (). Let ( ; ) be the scalar product of Sym dened by (s ; s ) = where and are partitions and is Kronecker's symbol. We denote by D f the adjoint of the multiplication by f, that is, (D f g; h) = (g; fh); ( f ; g ; h Sym) : D f is also called a dierential operator since we have for f = f(p ; p ; : : :) D f @p ; : : : n ; : : :) : In particular we have D s (s ) = s =. We denote by the ring homomorphism of Sym which sends p i to p i. ( (f) is the plethysm of p by f introduced by Littlewood.) The -core () of a partition is dened as follows (see []). If has no -hooks = () otherwise remove as many -hooks as possible from the diagram of ; the partition obtained in this way is (). We recall that () is a staircase partition k =

3 (k? ; : : : ; ; 0). We can also compute the -core using the algorithm that computes the -quotient and the -sign as follows. Make into a partition of even length n (where n is the length of ) by adding if necessary a zero part. Add to the staircase partition n. Reduce modulo from right to left the successive parts of = n without using two times the same representative. This gives a sequence, which is put in decreasing order by a permutation. We can now compute the -core of by subtracting the staircase n from (). The -sign is () = sign(). Finally, subtract from the even parts of the corresponding residues in and divide by to obtain 0. The same procedure applied to the odd parts gives the second partition. There exists a one-to-one correspondence between a partition and its -core and -quotient []. We write = ( () ; ( 0 ; )). We denote by Sym ; the subspace spanned by Schur S-functions indexed by partition without -core. We can now dene J() and : J() = (;; (; ;)); = (;; (( 0 ) 0 ; )): At last we introduce the linear involution of Sym ; dened by (s ) = (?) j0 j () ( )s : This denition becomes more understandable after the statement of Lemma 6 (ii). Tableaux In this section we describe the combinatorial objects used in the sequel: Yamanouchi domino tableaux, \Stembridge" tableaux. A domino diagram of shape is a diagram of this shape whose cells are or rectangles called dominoes. We can see an example on Figure. x y Figure : A domino diagram of shape (5; ; ; ; ; ) Semistandard domino tableaux are dened as semistandard (ordinary) tableaux, namely, they are obtained by numbering all the dominoes of a domino diagram with nonnegative integers weakly increasing along the rows (from left to right), strictly down the columns (see []). To an ordinary tableau T we can associate a word called column reading obtained by reading the successive columns of T from bottom to top and left

4 to right. The column reading of a domino tableau is obtained in the same way except that horizontal dominoes which belong to two successive columns i and i are read only when reading column i. A Yamanouchi word is a word w = w : : : w n such that each right factor w i w i : : : w n contains for every j at least as many letters j than j. For example the word w = is a Yamanouchi word while w 0 = is not. A Yamanouchi tableau (resp. domino tableau) is a tableau (resp. domino tableau) whose column reading is a Yamanouchi word. A marked tableau T of shape is a diagram of this shape whose cells are numbered with a special alphabet = f 0 ; ; 0 ; ; : : :g (with 0 < < 0 < : : :) and satisfying the following three rules:. T (i; j) T (i ; j), T (i; j) T (i; j );. Each column has at most one k (k = ; ; : : :),. Each row has at most one k 0 (k 0 = 0 ; 0 ; : : :). The row reading of a marked tableau T is obtained by reading the successive rows of T from bottom to top and left to right. Given any word w = w w : : : w n over we dene m i (j), the multiplicity of i among w n?j : : : w n (0 < j n), m i (n j) = m i (n) the multiplicity of i 0 among w w : : : w j (0 < j n). The word w is said to satisfy the \lattice property" if, whenever m i (j) = m i? (j) we have w n?j 6= i; i 0 if (0 j n) w j?n 6= i? ; i 0 if (n j n). The word w is a \Stembridge" word if it satises the \lattice property" and if the leftmost i in jwj is unmarked in w (where jwj denotes the word obtained by erasing the marks of w). A \Stembridge" tableau is a tableau whose row reading is a Stembridge word. The content of a \Stembridge" tableau is the content of the corresponding tableau unmarked (see [9]). On Figure, example isn't a \Stembridge" tableau because in w = the leftmost occurrence of is marked. Example isn't a \Stembridge" tableau because w = doesn't verify lattice property. Indeed m (8) = = m (8) and w 8?7 = w = 0 = i 0. Example is a \Stembridge" tableau.

5 The operator In [5], a dierential operator has been introduced. It sends Schur S-functions on Schur P -functions but with restrictions on the length of the partitions. To improve these results we have considered a new operator, dened in terms of the partition J() introduced in Section. Denition Let be the dierential operator on Sym of innite order : = X (?) jj (J( 0 ))s J( 0 ) D (s ): Our main result is that: Theorem is a projector from Sym to T () and (s ) = P c() : The proof of this theorem is in two steps. We rst establish the following characterization of : Proposition is the unique linear operator satisfying (i) (fg) = f(g); (f T () ; g Sym); (ii) ( (g)) = ( (g)); ( g Sym): We then prove: Proposition Let D be the linear map dened by: D : Sym! Sym s! P c() : D satises (i) and (ii) of Proposition. For the proofs of Propositions and we need to introduce new denitions and objects. Let A 0 ; A be two innite sets of independent variables. We shall denote the symmetric functions of A 0 by Sym(A 0 ), and the symmetric functions of A by Sym(A ). By Sym(A 0 ; A ) we shall mean the functions which are separately symmetric in A 0 and A. Finally Sym(A 0 A ) will denote the symmetric functions of the whole set A 0 [ A. We will use the algebra automorphism w of Sym(A 0 ) which sends p k (A 0 ) to p k (?A 0 ) =?p k (A 0 ). In general, w(f(a 0 )) will be denoted by f(?a 0 ) (-ring notation). We recall that s (?A 0 ) = (?) jj s 0(A 0 ); s (A 0 A ) = X s = (A 0 )s (A ): Clearly, w can be extended to Sym(A 0 ; A ) by setting w(f(a 0 )g(a )) = f(?a 0 )g(a ); f; g Sym: We shall write w(sym(a 0 A )) = Sym(?A 0 A ). We denote by S () the subalgebra (Sym). We now introduce the main tool of the proof.

6 Denition 5 Let be the isomorphism of C -vector spaces: : Sym ; (A)! Sym(A 0 ; A ) s (A)! ()s 0(A 0 )s (A ): We remark that can be generalized at Sym and in this case it is the lifting at the level of symmetric functions of the combinatorial bijection which sends to ( () ; ( 0 ; )). Lemma 6 (i) (Sym) = Sym(A 0 A ) More precisely, for f Sym; ( (f)) = f(a 0 A ) Lemma 7 ( (Sym)) T () : (ii) =? w Using these results we can prove the two propositions above. We remark on the one hand that (i) of Proposition is obvious from Denition since involves only derivations with respect to the even power sums p i. On the other hand (ii) of Proposition is proved using the form? w of and the algebras Sym(A 0 A ), Sym(?A 0 A ). We then prove that is a projector using (i), (ii) of Proposition and Lemma 7. Proposition is proved. We now give a sequence of lemmas to prove Proposition. Lemma 8 D (fg) = fd (g); f T () ; g Sym, and so D is a projector. Lemma 9 D (s ) = D (s ) for s Sym ; : The proof is totally combinatorial and uses domino diagrams. Corollary 0 D ( (s )) = ( (s )) for s Sym: It comes from Lemmas 8, 7, and 9 and the fact that (Sym) Sym ;. This completes the proof of Proposition and of Theorem. We deduce immediately from Theorem the following expression of P -functions in terms of S-functions. Corollary Let = ( ; : : : ; k ) be a strict partition and be a partition given by (? ;? ; : : : j ; : : :) in Frobenius' notation. Then X P = (?) jj (J( 0 ))s J( 0 ) D (s )s

7 Examples:. We compute the Schur P-functions indexed by partitions of weight 6. For this it is sucient to compute in terms indexed with partitions of weight less or equal to. = s (;) (D s()? D s(;) ) s (;;;) (D s()? D s(;) D s(;) )?s (;) (D s(;)? D s(;;) D s(;;;) ) s (;;;;;) (D s(6)? D s(5;) D s(;)? D s(;) )?s (;;;) (D s(;)? D s(;;)? D s(;) D s(;;)? D s(;;;) D s(;;;) ) s (5;) (D s(;;)? D s(;;;) D s(;;;;)? D s(;;;;;) ) ::: For = (6), = (5j0) = (6), we have P (6) = (s (6) ) = s (6) : s (;) s () s (;;;) s ()? 0 s (;;;;;) : 0 0 For = (5; ), = (5; ), we have P (5;) = (s (5;) ) = s (5;) : s (;) s (;) s (;;;) s (;)? 0 s (;;;;) :(?) For = (; ), = (j) = (; ; ), we have P (;) = (s (;;) ) = s (;;) :s (;) (?s () s (;;) )s (;;;) (?s () )?s (;) (?s () )0?s (;;;) (?)0:. For l() k we have : D (s )s k = X c s k where c is the Littlewood-Richardson coecient. We then have P c(k ) = X c (?) jj (J( 0 ))s J( 0 ) s k 5 Applications We now use this operator to prove that various sets of tableaux have the same cardinality. First we have to recall some formulas on Schur S-functions and P -functions. We have P = X g s ()

8 where g is the number of \Stembridge" tableaux of unshifted shape and content, s s = X c s () where c is the Littlewood-Richardson coecient i.e. the number of Yamanouchi skew-tableaux of shape = and weight, X D (s )s = (=)d s () ;jj=jj?jj where d is the number of Yamanouchi domino tableaux of shape = and weight, and nally X (s ) = ()d s () ;jj=jj where d is the number of Yamanouchi domino tableaux of shape and weight. The rst formula is proved in [9], the second one is well-known, and the two last formulas are given in []. For a strict partition, we shall denote by () ; () ; : : : ; (k) all the partitions such that c( (i) ) = w i () for some permutation w i. Theorem kx (w i )g (i) = i= if = 0 else The proof is obtained by applying (which is linear and a projection) to equation (). This corollary shows that the number of marked tableaux with (w i ) > 0 and the number of those with (w i ) < 0 are equal. For example, the set and the set on Figure have the same cardinality. This corresponds to the case = (8; ; ; ) and = (6; 5; ; ; ). Theorem g c() = X ; (?) jj (J( 0 )) (=)d c J( 0 ) This follows from Denition and relations (), (). On Figure 5 we have computed g (8;;)(6;;;;;), drawing all \Stembridge" tableaux of unshifted shape (6,,,,) and weight (8,,). On Figure 6 we have computed d (8;;;;) (where (8,,,,) is the partition such that c() = (8; ; )) and c (6;;;;) J( 0 ) with the sign of the pairs of tableaux. We see that adding the number of pairs of tableaux (being careful of the sign) of Figure 6 we obtain the same number as that of Stembridge tableaux of Figure 5.

9 6 Generalizations The operator can be generalised in two ways. We can dene with a formula similar to that of Denition a family of operators k, where we put the staircase k in place of the empty -core in J(). These operators satisfy properties analogous to those of. In particular we obtain again the main formula of [5] and give a dierent proof of it. We can also dene a dierential operator n which sends Sym to the subalgebra T (n) = C [p i ; i 6 0(n)]. For this we have just to put n-core and n-quotient in place of -core and -quotient. References [] C. Carre, B. Leclerc, Splitting the square of a Schur function into its symmetric and antisymmetric parts, J. Algebraic Combinatorics, (995), 0-. [] G. D. James, A. Kerber, The representation theory of the symmetrics groups,(98) Addison-Wesley, Readings, Massachusetts. [] M. Jimbo, T. Miwa, Solitons and innite dimensional Lie algebras, Publ. RIMS, Kyoto Univ. 9 (98), [] V. Kac, Innite dimensional Lie algebras, rd edition, Cambridge 990. [5] A. Lascoux, B. Leclerc, J.-Y. Thibon, Une nouvelle expression des fonctions P de Schur, C. R. Acad. Sci. Paris 6 (99), -. [6] B. Leclerc, S. Leidwanger, Fonctions P de Schur et representations d'algebres de Lie anes, C. R. Acad. Sci. Paris, (997), 7-. [7] B. Leclerc, S. Leidwanger, Schur functions and ane Lie algebras preprint. [8] I. G. Macdonald, Symmetric functions and Hall polynomials, nd edition, Oxford 995. [9] J. R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Ad. Math. 7 (989) 87-.

10 Figure : Figure :

11 Figure 5: 5 5 Figure 6: