Schur Functions and Affine Lie Algebras

210, 103]144 Ž1998. JA987453

JOURNAL OF ALGEBRA ARTICLE NO.

Schur Functions and Affine Lie Algebras Bernard Leclerc and Severine Leidwanger ´ Departement de Mathematiques, Uni¨ ersite´ de Caen, 14032 Caen cedex, France ´ ´ Communicated by Peter Littelmann Received April 15, 1997

We make use of $the representation theory of the infinite-dimensional Lie algebras a` , b` , and sl2 to derive explicit formulas relating Schur’s P-functions to Schur’s S-functions. Q 1998 Academic Press

1. INTRODUCTION This article is the result of our attempt to obtain a better understanding of the relationship between two families of symmetric functions known as Schur’s S-functions and P-functions. The use of S-functions in the character theory of the symmetric groups S n is well known w24x, and Schur introduced the P-functions to play a similar role for the spin characters of the double covering groups of S n w33, 25x. Since that time S-functions and P-functions have appeared in a number of topics: t-functions of the KP and BKP hierarchies of soliton equations w11, 39x; irreducible characters of the Lie algebras gl n and the Lie superalgebras QŽ n. w34x; cohomology of grassmannians and isotropic grassmannians w31, 13x; polynomials universally supported on general and symmetric degeneracy loci w30x; untwisted and twisted boson]fermion correspondences and vertex operators w12x; zonal spherical functions of Gelfand pairs Ž S 2 n , H n . twisted by a sign character, H n being a subgroup of S 2 n isomorphic to the hyperoctaedral group w35x. As discovered by the Kyoto school ŽSato, Date, Jimbo, Kashiwara, Miwa., the interpretation in terms of hierarchies of partial differential equations can be reformulated in the framework of infinite-dimensional $ Lie$algebras w11x. Indeed, the basic representations of a` s gl` and b` s go` can be realized in the vector spaces spanned by S-functions and 103 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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LECLERC AND LEIDWANGER

P-functions, respectively, and in both cases Schur functions form the canonical basis of weight vectors Žall weights have multiplicity one.. This fact has already been used by You w39x to derive the formula Pl2 s 2y l Ž l .h Ž sl . ,

Ž 1.

where l denotes the partition obtained from l by a certain doubling procedure Žsee Section 5., and h is the map sending even power sums on zero and multiplying odd power sums by a factor 2. So, roughly speaking, P-functions are square roots of S-functions indexed by double partitions. A generalization of Ž1. to skew Schur functions was given by Jozefiak and ´ Pragacz w14x. Their proof was based on the determinantal expressions of these functions. In w40x, You showed that the formula of Jozefiak and ´ Pragacz can also be derived by using the techniques of w39x. In fact, the S-functions can be used to construct all of the fundamental representations LŽ L k . of a` . Thus, they can be regarded as skew-symmetric tensors of infinite rank. On the other hand, the basic b` -module is an infinite analogue of the spin representation of so 2 nq1 , and therefore the P-functions should be seen as infinite-dimensional spinors. This means that one way of understanding the problem of the relationship between S-functions and P-functions is to view it as an infinite analogue of the classical problem considered by Brauer, Weyl, and Cartan of expressing the tensor product of two spinors in terms of skew-symmetric tensors. In the finite-dimensional case, it was proved by Brauer, Weyl w2x, and Cartan w4x that the tensor square of the spin representation LŽ L 0 . of so 2 nq1 decomposes under the action of so 2 nq1 as LŽ L 0 .

m2

, L Ž 2L 0 . [ L Ž L 1 . [ ??? [ L Ž L ny1 . [ L Ž 0 . .

Ž 2.

Here, for 1 F i F n y 1, LŽ L i . is the Ž n y i .th exterior power of the vector representation C 2 nq1 , LŽ2L 0 . is the nth exterior power of C 2 nq1 , and LŽ0. is its 0th power, that is, the trivial representation. As is customary, we have denoted by L i the fundamental weights for the root system Bn , but we have adopted an unusual indexation to agree with the natural labeling for b` . The analogue of Ž2. for b` reads LŽ L 0 .

m2

, L Ž 2L 0 . [

ž[ kG1

LŽ L k . .

/

Ž 3.

This will be our first tool for understanding how to pass from S-functions to P-functions. Indeed, since all irreducible components in the right-hand side of Ž3. have multiplicity one, there is Žup to normalization of the highest weight vectors. a unique homomorphism Hk of b` -modules, from

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

105

LŽ L k . to LŽ L 0 .m2 if k G 1, and from LŽ2L 0 . to LŽ L 0 .m2 if k s 0. Now, via the natural embedding b` ; a` and the restriction rules w11x, L Ž L k . a` x , L Ž L k . b`

Ž k G 1. ,

L Ž L 0 . a` x , L Ž 2L 0 . b` ,

Ž 4. Ž 5.

the S-functions form a basis of each irreducible summand of the right-hand side of Ž3.. Our first result is an explicit formula for the expansion of Hk Ž sl . on the basis  Pa m Pb 4 . From a combinatorial point of view, this involves a construction on partitions closely related to Frobenius symbols and their generalizations w26, 28x. This is a further instance of the phenomenon already observed in w1, 27, 17, 19, 20x that combinatorial objects introduced in the representation theory of the symmetric groups and their spin groups arise naturally in the context of affine Lie algebras. Using the description of Hk , one can obtain two different types of relations between P-functions and S-functions. Indeed, there are two natural ways of mapping LŽ L 0 .m2 to LŽ L 0 .. The first one is simply the multiplication m : Pa m Pb ¬ Pa Pb . We show that m ( Hk s 0 for k G 1 and m ( H0 s h. This provides a new proof of formula Ž1., and more generally this gives for any partition l a simple expression for h Ž sl .. Our argument is based on the fact that m ( Hk , which of course cannot be a homomorphism of b` -modules, is nevertheless a homomorphism of s-modules, where s denotes the principal Heisenberg subalgebra of b` . The second way of mapping LŽ L 0 .m2 to LŽ L 0 . is inspired by Dynkin’s construction of subordinate modules w6x. It relies on the decomposition L Ž L 0 . s C ¨L 0 [ L low ,

Ž 6.

where ¨L 0 is a highest weight vector, and L low is the sum of all weight spaces except the highest one. Hence any tensor T g LŽ L 0 .m2 has a unique expansion, T s G Ž T . m ¨L 0 q T X ,

Ž 7.

where GŽT . g LŽ L 0 . and T X g LŽ L 0 . m L low . This defines, for each k G 0, a map D k s G( Hk , which is a homomorphism of b`y-modules, that is, which commutes with the action of the lowering operators of b` . In particular, D 0 is a b`y-map from LŽ2L 0 . to LŽ L 0 . sending the highest weight vector ¨ 2 L 0 to ¨L 0 , and this shows that these two representations are subordinate in the sense of Dynkin w6x. It follows from our description of Hk that the maps D k send each S-function to a single P-function Žup to sign. or 0. More precisely, D k Ž sl . s Pc k Ž l . ,

Ž 8.

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where c k Ž l. is a sequence of integers computed from l in a simple way Žsee Section 4.. There is a quite different method for relating P-functions to S-func$ tions, based this time on the affine Lie algebra AŽ1. 1 s sl2 . This method does not seem to have any finite-dimensional counterpart. Recall that the space spanned by S-functions is the polynomial ring Sym in the power sum symmetric functions pk s Ý i x ik , whereas the P-functions span the subalgebra T generated by the odd power sums p 2 kq1 w24x. Using the embedding $ sl2 ; a` and restricting the representation Sym , LŽ L 0 .a` , one gets an $ action of sl2 on Sym with highest irreducible component T , LŽ L 0 . $ w11x. $

sl2

In this realization, the principal Heisenberg subalgebra s of sl2 Žwhich coincides with the principal subalgebra s of b` when both algebras are embedded in a` . acts by ˆ p 2 kq1 Žmultiplication by p 2 kq1 . and ­r­ p 2 kq1. Thus one obtains $in this way the principal realization of r P of the basic representation of sl2 , first constructed by Lepowsky and Wilson w22x. Using the combinatorics of 2-cores and 2-quotients of partitions w9x, which is known to be strongly connected with the basic representation $ of sl2 w1, 17x, we introduce a linear involution n of Sym such that

nŽT. s

[s kG0

rkc 2

Ž Sym. ,

Ž 9.

where r k denotes the staircase partition Ž k, k y 1, . . . , 1., and c 2 is the ring homomorphism of Sym sending pj to p 2 j . If we now define linear operators on n Ž T . by putting

rH Ž x . s n ( rP Ž x . (n

$

Ž x g sl2 . ,

Ž 10 .

it turns out that the generators hŽ j. of the homogeneous Heisenberg $ subalgebra of sl2 are represented by

r H Ž hŽ j. . s "2 j

­ ­ p2 j

,

r H Ž hŽyj. . s "p ˆ2 j

Ž j ) 0. ,

Ž 11 .

where the sign " is equal to Žy1. ky 1 on the kth sector sr k c 2 ŽSym. of n Ž T .. Thus, r H is a version of the Frenkel]Kac homogeneous realization of LŽ L 0 . $ w7x, and the involution n intertwines the principal and the sl2 homogeneous pictures. At this point, because of Eq. Ž9., one can define natural maps DX k from Sym to T by requiring that, for f g T and g g Sym, DX k Ž fg . s fDX k Ž g . ,

DX k Ž c 2 Ž g . . s n Ž sr k c 2 Ž g . . .

Ž 12 .

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

107

One of our main results is that DX k s D k . This is rather unexpected, since $ the P-functions seem to play no role in the representation theory of sl2 . On the other hand, this result allows us to describe D k as a differential operator of infinite order: Dk s

< <

Ý Ž y1. l e 2 Ž Jk Ž lX . . s J Ž l . Dc Ž s . . l

k

X

2

l

Ž 13 .

The notation is explained in detail below. We merely point out here that the partition Jk Ž lX . and the sign e 2 Ž Jk Ž lX .. are defined by means of the notions of 2-core and 2-quotient of a partition, whose relation with b` and P-functions is unclear. As a consequence of Ž8. and Ž13., we obtain for any P-function several quadratic expressions in terms of S-functions. For example, one has PŽ4, 2. s D 0 Ž sŽ4 , 1, 1. . s sŽ4, 1, 1. y sŽ1 , 1. sŽ4. q sŽ1 , 1. sŽ2 , 1, 1. y sŽ1 , 1, 1, 1. sŽ2. q sŽ3 , 1. sŽ2. q sŽ3 , 1, 1, 1. , but also, PŽ4, 2. s D 1 Ž sŽ3 , 2. . s sŽ1. sŽ3 , 2. q sŽ1 , 1, 1. sŽ3. q sŽ2 , 2, 1. sŽ1. , or PŽ4, 2. s D 2 Ž sŽ2 , 1. . s sŽ2 , 1. sŽ2 , 1. . These formulas extend those of w16x, which suffered from a limitation on the length of the partitions. The arguments of w16x were also different, being based on the calculus of divided difference operators and on an induction with respect to the number of variables. The main motivation of this work has been to find a representation-theoretical understanding of the formulas of w16x. The article is organized as follows. In Section 2, we recall the definition of a` and describe its fundamental representations using S-functions. Section 3 is concerned with the algebra b` and the realization of its fundamental representations using S-functions and P-functions. In Section 4, we consider the tensor square of the spin representation, and we describe the intertwining operators Hk . This is applied in Section 5 to get formula Ž1. and its generalization, and in Section 6 to obtain our first description of the linear operators D k . The second part of the paper begins with some background material on cores and quotients of partitions, and certain related symmetric function ŽSection 7.. In Section 8, we review the basic representation of identities $ sl2 and we introduce the involution n . Then in Section 9, we prove that the

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operators D k are characterized by Ž12., and we derive expression Ž13.. Finally, in Section 10 we compare our formulas with those of w16x, and we discuss some generalizations. Most of the results of this paper were announced in w18x. A notable exception is the generalization of formula Ž1. that we discovered afterward. During the preparation of this manuscript, we realized from a correspondence with P. Pragacz that he had found independently the same generalization from a different approach Žsee w32x..

2. THE FUNDAMENTAL REPRESENTATIONS OF a` We recall, following w11, 15x, the construction of the level 1 representations of the infinite-dimensional Lie algebra a` . Let a` denote the Lie algebra of complex Z = Z-matrices A s Ž a i j . such that a i j s 0 for < i y j < 4 0. In other words, the elements of a` have only a finite number of nonzero diagonals, and matrix multiplication makes sense, which allows us to define the Lie bracket as the commutator AB y BA. The completed infinite rank affine algebra a` is the central extension a` s a` [ Cc with Lie bracket

w A q l c, B q m c x s Ž AB y BA . q c Ž A, B . c

Ž A, B g a` , l , m g C . , Ž 14.

where c is the skew-symmetric bilinear form defined on the matrix units Ei j by

c Ž Ei j , Eji . s 1 s yc Ž Eji , Ei j . ,

Ž 15 .

if i F 0, j G 1, and c Ž Ei j , Em n . s 0 otherwise. The Chevalley generators of a` , denoted by e`i , f i` , h`i Ž i g Z., are expressed in terms of the matrices Ei j by e`i s Ei , iq1 ,

Ž 16 .

f i`

Ž 17 .

s Eiq1, i ,

h`i s e`i , f i` s Ei , i y Eiq1, iq1 q d i , 0 c.

Ž 18 .

We denote by L k Ž k g Z. the fundamental weights, that is, L k Ž h`i . s d i k . The irreducible highest weight representations of a` with highest weight L k are called the fundamental representations and are denoted by LŽ L k .a` Žor simply by LŽ L k . when there is no risk of confusion.. The infinite wedge construction of LŽ L k . by Kac and Peterson Žsee w15x. shows clearly that LŽ L k . is a natural infinite analogue of the representation of gl n in Hk C n.

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

109

In particular, we can see that, as in the finite-dimensional case, all weights have multiplicity one, which implies that there is up to scaling a unique basis of weight vectors. There is, however, an alternative construction of LŽ L k . by means of differential operators acting on a polynomial algebra, which was found by the Kyoto school in relation to the investigation of the KP hierarchy w11x. This is the realization we shall summarize now. Let Bk s Ý i g Z Ei, iqk Ž k g ZU .. It is easily checked, using Ž14., that

w Bk , Bl x s k d k , yl c,

Ž 19 .

which means that the Bk generate an infinite-dimensional Heisenberg subalgebra s a of a` . The natural representation of s a on the polynomial ring Cw pi , i g NU x is given by Bk f s k

­f ­ pk

,

Byk f s pk f ,

cf s f

Ž f g C w pi x , k g NU . . Ž 20.

This is the so-called canonical commutation relations representation. From now on, we shall identify Cw pi x with the algebra SymŽ X . s Sym of symmetric functions in a countable set of variables X s  x 1 , x 2 , . . . 4 by setting pk s Ý i x ik , the power sum symmetric function w24x. In particular, we have deg pk s k. The action of s a on Sym can be extended to an action of the whole algebra a` by using differential operators of infinite order called ¨ ertex operators. It turns out that the canonical basis of weight vectors is given in this setting by Schur’s S-functions, sl s

Ý xl Ž m . pmrzm ,

Ž 21 .

m

where for l and m s Ž1m 1 ??? r m r ., two partitions of m, zm s 1m 1 m1! . . . r m r m r !, and xlŽ m . denotes the irreducible character xl of S m evaluated on the conjugacy class of cycle-type m w24x. The action of the Chevalley generators on sl is described as follows. Let Ž i, j . be the node of the Young diagram of the partition l situated on row i and column j. The content d of Ž i, j . is j y i. The node Ž i, j . is removable from l if the diagram obtained when we remove it is still the diagram of a partition m. In this case Ž i, j . is also an addable node for m , and we use the notation lrm s d . For example, as illustrated in Fig. 1, the partition l s Ž5, 2, 1. has three removable nodes with content 4, 0, y2, and four addable nodes with content 5, 1, y1, y3. We can now state the following.

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LECLERC AND LEIDWANGER

FIG. 1. The Young diagram of l s Ž5, 2, 1..

THEOREM 1 w11x. by

Let k g Z. The Che¨ alley generators of a` act on Sym

¡ if lrm s i y k ¢0 otherwise ¡s if nrl s i y k ~ f s s ¢0 otherwise ¡ys if l has a remo¨ able Ž i y k . -node ~ h s s s l has an addable Ž i y k . -node ¢0 ifotherwise. e`i sl s

~s

` i l

m

Ž 22 .

n

Ž 23 .

l

` i l

l

Ž 24 .

This extends by linearity to an irreducible highest weight representation of a` isomorphic to LŽ L k ., with highest weight ¨ ector sŽ0. s 1. In this representation the Heisenberg subalgebra s a acts according to Eq. Ž20.. 3. THE FUNDAMENTAL REPRESENTATIONS OF b` Our main references for this section are w5, 11, 15, 40x. Let b` denote the Lie subalgebra of a` consisting of the matrices A s Ž a i j . satisfying a i j s Ž y1 .

iqjq1

ayj, yi

Ž i , j g Z. .

Ž 25 .

The completed infinite rank affine algebra b` is the central extension b` s b` [ Cc, with Lie bracket given by Ž14., Ž15.. Thus b` is a subalgebra of a` . We shall take as Chevalley generators the elements Ei`, Fi`, Hi` Ž i g N., given by ` Ei` s e`i q eyiy1 s Ei , iq1 q Eyiy1 , yi

E0`

s 2Ž

e`0

q

` ey1

. s 2 Ž E0, 1 q Ey1, 0 .

` Fi` s f i` q fyiy1 s Eiq1, i q Eyi , yiy1

Hi` s Ei` , Fi`

Ž i ) 0.

Ž i G 0. .

Ž 26 . Ž 27 .

Ž i G 0.

Ž 28 . Ž 29 .

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

111

The canonical central element is K s H0` q 2 Ý iG 1 Hi` s 2 c. Again we denote by L k Ž k g N. the fundamental weights: L k Ž Hi` . s d i k .

Ž 30 .

The corresponding fundamental representations will be denoted by LŽ L k . b` Žor simply LŽ L k . when there is no possible ambiguity.. Note that K acts on LŽ L k . by multiplication by 2 if k G 1, that is, LŽ L k . has level 2 for k / 0 and LŽ L 0 . has level 1. More generally, for a dominant integral weight L Ži.e., a sum of fundamental weights. the irreducible highest weight representation with highest weight L will be denoted by LŽ L . b` . One should think of b` as an infinite-dimensional analogue of the Lie algebra so 2 nq1 associated with the symmetric bilinear form i Ž u i , u j . s Ž y1. d i , yj

Ž 31 .

n on C 2 nq1 s [isyn Cu i . ŽThis choice of the form Ž?, ? . has the advantage that the diagonal matrices of so 2 nq1 form a Cartan subalgebra, and thus the triangular decomposition of sl 2 nq1 induces a triangular decomposition of so 2 nq1.. Recall that the fundamental representations of so 2 nq1 are all obtained by restriction from the fundamental representations of sl 2 nq1 , with the exception of the spin representation that requires a separate construction Žsee w8x, for instance .. The infinite-dimensional situation is similar. First we have the following branching rules.

THEOREM 2 w11x. The restrictions to b` of the fundamental representations of a` are irreducible, and we ha¨ e the isomorphisms L Ž L i . a` x , L Ž L i . b` L Ž L i . a` x , L Ž 2L 0 . b` L Ž L i . a` x , L Ž Lyiy1 . b`

Ž i G 1. ,

Ž 32 .

Ž i s 0, y1. ,

Ž 33 .

Ž i F y2. .

Ž 34 .

It follows that the level 2 representations LŽ2L 0 . b` and LŽ L k . b` Ž k G 1. can all be realized in the polynomial algebra Sym. Let us denote by p k the representation of b` in Sym regarded as the b` -module LŽ L k .a` x Ž k g Z.. The action of the endomorphisms p k Ž Ei` ., p k Ž Fi` . on the basis of S-functions is deduced from Theorem 1 and Eqs. Ž26. ] Ž29.. To describe it combinatorially, one simply has to draw the Young diagram of l and write in each box its content qk. Let us denote by Yk Ž l. this Young diagram with shifted labels. Then p k Ž Fi` . sl is obtained by adding to Yk Ž l. all possible nodes with label i or yi y 1 Žif any., and similarly, p k Ž Ei` . sl is obtained by erasing the removable nodes

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LECLERC AND LEIDWANGER

of Yk Ž l. with label i or yi y 1, and multiplying by 2 if i s 0. For instance, one has

p 0 Ž F0` . sŽ6 , 4, 2, 1, 1. s sŽ6 , 4, 3, 1, 1. ,

p 0 Ž F1` . sŽ6 , 4, 2, 1, 1. s sŽ6 , 4, 2, 2, 1. ,

p 0 Ž F2` . sŽ6 , 4, 2, 1, 1. s 0, as is easily seen from the left diagram of Fig. 2, whereas the right diagram shows that

p 2 Ž F0` . sŽ6 , 4, 2, 1, 1. s sŽ6 , 4, 2, 2, 1. ,

p 2 Ž F1` . sŽ6 , 4, 2, 1, 1. s 0,

p 2 Ž F2` . sŽ6 , 4, 2, 1, 1. s sŽ6 , 4, 3, 1, 1. q sŽ6 , 4, 2, 1, 1, 1. . We note the following obvious symmetry:

p k Ž Ei` . sl s pyky1 Ž Ei` . slX ,

p k Ž Fi` . sl s pyky1 Ž Fi` . slX ,

Ž 35 .

where lX denotes the partition conjugate to l Žsee w24x.. To construct the spin representation LŽ L 0 . b` , we introduce the principal Heisenberg subalgebra s of b` : s s s b s s a l b` s

ž[ kgZ

C B2 kq1 [ C K .

/

Ž 36 .

By Ž19., one has the commutation rule

w B2 kq1 , B2 lq1 x s Ž k q 12 . d 2 kq1, y2 ly1 K .

Ž 37 .

Hence the natural action of s on T s Cw p 2 kq1 , k g Nx is given by B2 kq1 f s k q

ž

1

/

­f

2 ­ p 2 kq1

,

By2 ky1 f s p 2 kq1 f ,

Kf s f

Ž f g T , k g N . . Ž 38 .

FIG. 2. The Young diagrams Y0 Ž6, 4, 2, 1, 1. and Y2 Ž6, 4, 2, 1, 1..

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

113

Using vertex operators, one can extend to b` the action of s on T. Again, one gets a representation with all weight multiplicities equal to 1, the canonical basis of weight vectors Žup to scaling. being given this time by Schur’s P-functions. These symmetric functions were introduced by Schur as a spin analogue of the S-functions, in the sense that their expansion on the basis of power sums yields the value of the spin characters of the double covering groups ˆ m of the symmetric groups S m . Namely, for l a bar partition of m, i.e., S a partition whose parts are pairwise distinct, Pl s

Ý 2 uŽ l Ž m .y l Ž l..r2 vjl Ž m . pmrzm ,

Ž 39 .

m

where l Ž l. is the length of l Ži.e., its number of parts., u x v stands for the smallest k g Z such that k G x, and jlŽ m . denotes the value of the spin ˆ m labeled by m s Ž1m 1 3 m 3 ??? Ž2 r character jl on the conjugacy class of S m 2 rq 1 . w25x. q 1. It is convenient to define Pa for all sequences a s Ž a 1 , . . . , a r . of positive integers by requiring that PwŽ a . s sgn wPa

Ž w g Sr. .

Ž 40 .

In particular, Pa s 0 if a i s a j for some pair of indices. Using this convention, we can state the following spin analogue of Theorem 1. THEOREM 3 w5, 10x. The Che¨ alley generators of b` act on T by Fi`Pl s Pn ,

Ei`Pl s Pm ,

Ž 41 .

where n Ž resp. m . is obtained from l by replacing its part i by i q 1 Ž resp. its part i q 1 by i ., the result being 0 if i Ž resp. i q 1. is not a part of l. Ž For con¨ enience, we assume here that a part equal to 0 has been added at the end of l.. This extends by linearity to an irreducible highest weight representation of b` isomorphic to LŽ L 0 ., with highest weight ¨ ector PŽ0. s 1. In this representation the Heisenberg subalgebra s acts according to Eq. Ž38.. For example, F0`PŽ3 , 2. s PŽ3 , 2, 1. , E1`PŽ3 , 2. s PŽ3 , 1. ,

F3`PŽ3 , 2. s PŽ4 , 2. , E2` PŽ3 , 2. s PŽ2 , 2. s 0.

It is an easy consequence of Ž41. that the weight of Pl as a weight vector of LŽ L 0 . is given by wt Pl s

Ý ni L i , iG0

Ž 42 .

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LECLERC AND LEIDWANGER

where n i s 1 if i is a part of l and i q 1 is not, n i s y1 if i q 1 is a part of l and i is not, and n i s 0 otherwise. For example, wt PŽ5, 4, 2. s L 5 y L 3 q L 2 y L1 q L 0 . Note that the polynomial realizations of the b` -modules that we have described are graded vector spaces. This gradation is compatible with the action of b` , in the sense that deg Fi` ¨ s deg ¨ q 1,

deg Ei` ¨ s deg ¨ y 1

Ž i g N . . Ž 43 .

Therefore, the usual gradation on Sym and T coincides Žup to sign. with the so-called principal gradation. This yields immediately the principally specialized characters ch L Ž L k . b` s ch L Ž 2L 0 . b` s ch L Ž L 0 . b` s

Ł Ž1 y q 2 ny1 .

Ł Ž 1 y q n . y1

Ž k G 1. ,

Ž 44 .

nG1 y1

nG1

s

Ł Ž1 q qn. .

Ž 45 .

nG1

4. THE TENSOR SQUARE OF THE SPIN REPRESENTATION OF b` In this section, we give the infinite-dimensional analogue of the theorem of Brauer and Weyl cited in the Introduction. In addition, we perform the explicit calculation of the intertwining operators Hk , which is crucial for the applications of the next two sections. THEOREM 4. The tensor square of the spin representation of b` is isomorphic, as a b` -module, to the direct sum of all le¨ el 2 irreducible highest weight representations: L Ž L 0 . m L Ž L 0 . , L Ž 2L 0 . [ The proof follows from two lemmas. subsequence m s Ž m 1 , . . . , m r . of r k , r k , that is, the subsequence of r k instance, if k s 5 and m s Ž3, 1. then LEMMA 1.

ž[ k)0

LŽ L k . .

/

Ž 46 .

Set r k s Ž k, k y 1, . . . , 1.. Given a we denote by m its complement in obtained by deleting the m i . For m s Ž5, 4, 2..

For k g N, let ¨ k g T m2 , LŽ L 0 .m2 be defined by ¨k s

<

<

Ý Ž y1. m Pm m Pm , m

Ž 47 .

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SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

where the sum is o¨ er all subsequences m of r k . Then ¨ 0 s 1 m 1 is a highest weight ¨ ector of weight 2L 0 , and for k ) 0, ¨ k is a highest weight ¨ ector of weight L k . For example, ¨ 2 s PŽ2 , 1. m 1 y PŽ2. m PŽ1. q PŽ1. m PŽ2. y 1 m PŽ2 , 1.

is a highest weight vector of weight L 2 and ¨ 3 s PŽ3 , 2, 1. m 1 y PŽ3 , 2. m PŽ1. q PŽ3 , 1. m PŽ2. y PŽ2 , 1. m PŽ3.

y PŽ3. m PŽ2 , 1. q PŽ2. m PŽ3 , 1. y PŽ1. m PŽ3 , 2. q 1 m PŽ3 , 2, 1. is a highest weight vector of weight L 3 . Proof. The case k s 0 is clear. Let us assume k ) 0. It follows from Eqs. Ž40., Ž41. that Ei`Ž Pm m Pm . s 0 unless either i is a part of m and i q 1 a part of m , or i q 1 is a part of m and i a part of m. Such terms Pm m Pm go in pairs obtained one from the other by exchanging i and i q 1. Since these terms have opposite sign, their sum is annihilated by Ei`. Finally, wt ¨ k s L k follows from Ž42. and the fact that wtŽ Pl m Pm . s wt Pl q wt Pm . Let Vk [ UŽ b` . ¨ k , where UŽ b` . is the enveloping algebra of b` . Then by Lemma 1, Vk is an irreducible component of LŽ L 0 .m2 isomorphic to LŽ L k . for k ) 0, and to LŽ2L 0 . for k s 0. Consider the direct sum W s [k Vk ; LŽ L 0 .m2 . To prove Theorem 4, one has to establish that this inclusion is actually an equality. This will be a consequence of LEMMA 2.

One has ch L Ž L 0 .

m2

s ch W.

Ž 48 .

Proof. Using Eq. Ž45., we have 2

Ž ch L Ž L 0 . . s

1

žŁ

1 y q 2 ny1

nG1

2

/

s

1 y q2n

1

Ł

nG1

1 y qn

Ł

nG1

1 y q 2 ny1

.

On the other hand, since the degree of ¨ k is equal to k Ž k q 1.r2, there holds by Ž44. ch W s ch L Ž 2L 0 . q

Ý q kŽ kq1.r2 ch L Ž L k . s Ý q kŽ kq1.r2 Ł kG1

nG1

kG0

1 1 y qn

Thus, we are reduced to proving that

Ý kG0

q kŽ kq1.r2 s

1 y q2n

Ł

nG1

1 y q 2 ny1

.

But this is a classical identity of Gauss Žsee w24, p. 13x, for example..

.

116

LECLERC AND LEIDWANGER

Since the irreducible components of LŽ L 0 .m2 are pairwise nonisomorphic by Theorem 4, there exists for k g N a unique isomorphism Hk of b` -modules Žup to a multiplicative constant. from LŽ L k .a` x to LŽ L 0 .m2 . Using the polynomial realizations of Section 3, we can regard Hk as a map from Sym to T m T , that we normalize by requiring that Hk Ž 1 . s ¨ k .

Ž 49 .

The remainder of this section is devoted to the description of Hk . First we need some combinatorial definitions. Let F0 Ž l. s Ž a 1 , . . . , a r N b 1 , . . . , br . denote the Frobenius symbol of the partition l Žsee w24x.. This is readily obtained from the Young diagram Y0 Ž l.. Namely, a 1 , . . . , a r are the rightmost numbers on the first r rows, b 1 , . . . , br the negative of the uppermost numbers on the first r columns, and r is the number of nodes on the 0-diagonal. For example, F0 Ž6, 4, 2, 1, 1. s Ž5, 2 N 4, 1., as shown by the left diagram of Fig. 3. We shall need a generalization of the Frobenius symbol due to Olsson w28x Žsee also w26x.. For k g N, we set Fk Ž l . s Ž a 1 , . . . , a k , a kq1 , . . . , a kqr N b 1 , . . . , br . ,

Ž 50 .

where the numbers a i , b j are now read on the diagram Yk Ž l.. Thus, F2 Ž6, 4, 2, 1, 1. s Ž7, 4, 1 N 2., as illustrated by the right diagram of Fig. 3. More formally, let l s l Ž l.. If l F k, then set l lq1 s ??? s l k s 0 and define Fk Ž l. s Ž a 1 , . . . , a k N., where a i s l i q k y i. If l ) k, then set m s Ž l kq 1 , . . . , l l . and define Fk Ž l. by Eq. Ž50., where a i s l i q k y i for i F k and Ž a kq 1 , . . . , a kqr N b 1 , . . . , br . s F0 Ž m .. DEFINITION 1. Let l be a partition and k g N. We put c k Ž l . s Ž a 1 q 1, . . . , a k q 1, a kq1 q 1, b 1 ,

a kq 2 q 1, b 2 , . . . , a kqr q 1, br . , where a i , b j are the components of the kth Frobenius symbol Fk Ž l..

FIG. 3.

F0 Ž6, 4, 2, 1, 1. s Ž5, 2 N 4, 1. and F2 Ž6, 4, 2, 1, 1. s Ž7, 4, 1 N 2..

Ž 51 .

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

117

Thus, proceeding with our example, we have c 0 Ž 6, 4, 2, 1, 1 . s Ž 6, 4, 3, 1 . ,

c 2 Ž 6, 4, 2, 1, 1 . s Ž 8, 5, 2, 2 . .

Finally, given a composition for c s Ž c1 , . . . , c r ., that is, a finite sequence of positive integers, we call splitting of c a pair Žg , g ., where g is a subsequence of c and g is its complement. Thus the splittings of Ž4, 2, 3. are

Ž Ž 4, 2, 3 . , Ž . . , Ž Ž 4, 2 . , Ž 3. . , Ž Ž 4, 3. , Ž 2. . , Ž Ž 2, 3 . , Ž 4 . . , Ž Ž 4. , Ž 2, 3 . . , Ž Ž 2 . , Ž 4, 3 . . , Ž Ž 3 . , Ž 4, 2. . , Ž Ž . , Ž 4, 2, 3 . . . We shall write Žg , g . * c to signify that Žg , g . is a splitting of c. We set k

Nk Ž g , g . s

Ý Ž k y i q 1.

Ž 52 .

is1 c ig g

and define the k-sign of a splitting by

e k Ž g , g . s Ž y1 .

Nk Žg , g .

.

Ž 53 .

For instance, the 2-signs of the previous splittings of Ž4, 2, 3. are, respectively, q, q, y, q, y, q, y, y. Note that e 0 Žg , g . s 1 for all splittings Žg , g . of c. We can now state the following. THEOREM 5.

Let k g N. The intertwining operator Hk is gi¨ en by Hk Ž sl . s

Ý

Žg , g .*c k Ž l .

e k Ž g , g . Pg m Pg .

Ž 54 .

For example, take l s Ž3, 1, 1., k s 0. Then, F0 Ž l. s Ž2 N 2., c 0 Ž l. s Ž3, 2., and H0 Ž sŽ3 , 1, 1. . s PŽ3 , 2. m 1 q PŽ3. m PŽ2. q PŽ2. m PŽ3. q 1 m PŽ3 , 2. . If now k s 2, then F2 Ž l. s Ž4, 1, 0 N 0., c 2 Ž l. s Ž5, 2, 1, 0. s Ž5, 2, 1., and H2 Ž sŽ3 , 1, 1. . s PŽ5 , 2, 1. m 1 q PŽ5 , 2. m PŽ1. y PŽ5 , 1. m PŽ2. y PŽ5. m PŽ2 , 1. q PŽ2, 1. m PŽ5. q PŽ2. m PŽ5 , 1. y PŽ1. m PŽ5 , 2. y 1 m PŽ5 , 2, 1. .

118

LECLERC AND LEIDWANGER

Finally, if k s 1, then F1Ž l. s Ž3, 0 N 1., c1Ž l. s Ž4, 1, 1. and H1 Ž sŽ3 , 1, 1. . s PŽ4 , 1, 1. m 1 q PŽ4 , 1. m PŽ1. q PŽ4 , 1. m PŽ1. q PŽ4. m PŽ1 , 1. y PŽ1, 1. m PŽ4. y PŽ1. m PŽ4 , 1. y PŽ1. m PŽ4 , 1. y 1 m PŽ4 , 1, 1. s 2 PŽ4, 1. m PŽ1. y 2 PŽ1. m PŽ4 , 1. . Proof. Define a linear map HkX from Sym to T m T by HkX Ž sl . s

Ý

Žg , g .*c k Ž l .

e k Ž g , g . Pg m Pg .

Ž 55 .

To prove that HkX s Hk , it is enough to prove that HkX Ž1. s Hk Ž1. s ¨ k and that HkX (p k Ž Fi` . s Fi` ( HkX Ž i g N. . Ž 56 . One has c k Ž0. s r k and e k Žg , g . s Žy1. < g < for all splittings Žg , g . of r k . Therefore, by Lemma 1 and Eq. Ž49., the first statement follows. To prove Ž56., we first note that in Eq. Ž41. l needs not be a partition, but may be an arbitrary sequence of positive integers. Thus one can compute Fi`Ž Pg m Pg . without reordering g or g , and it is not necessary to remove the summands Pg m Pg for which g or g contains two equal parts. This shows that if we denote by RŽ c k Ž l.. the right-hand side of Ž55., then for any composition c, Fi` R Ž c . s

Ý RŽ d . , d

where the sum is over all compositions d obtained from c by changing one part c j s i into d j s i q 1. Taking this into account and recalling the combinatorial description in Section 3 of p k Ž Fi` . sl , the proof of Ž56. reduces to checking that the Young diagram Yk Ž l. has one Žresp. two. addable nodes with label i or yi y 1 if and only if the composition c k Ž l. has one Žresp. two. partŽs. equal to i. This is a straightforward verification, and the theorem is proved. We end this section by noting that the limitation k G 0 in Theorem 5 is rather artificial. Indeed, the generalized Frobenius symbol Fk Ž l. is naturally defined for all k g Z. The reason why we imposed k G 0 is that we chose to realize LŽ L k . b` as LŽ L k .a` x for k ) 0 and LŽ2L 0 . b` as LŽ L 0 .a` x. More generally, one can define Hk for all k g Z as the unique b` -intertwiner from Sym s LŽ L k .a` x to T m2 such that Hk Ž 1 . s

½

¨k ¨ yk y1

Ž k G 0. , Ž k - 0. .

Ž 57 .

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SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

Then it follows from Ž35. that Hk Ž sl . s Hyky1 Ž slX . .

Ž 58 .

In other words, Theorem 5 is valid for k g Z provided that one defines c k Ž l . s cyky1 Ž lX .

Ž k - 0. .

Ž 59 .

Equivalently, c k Ž l. could also be defined for negative k - 0 in terms of the Frobenius symbol Fk Ž l.. 5. THE MAP h In this section we apply Theorems 4 and 5 to obtain a first set of relations between S-functions and P-functions. This contains as a particular case the formula proved by You w39x. To state it, we need the following doubling procedure on Young diagrams. Let l s Ž l1 , . . . , l r . be a bar partition. We denote by l the partition whose Frobenius symbol is F0 Ž l. s ŽŽ l1 y 1., . . . , Ž l r y 1. N l1 , . . . , l r .. THEOREM 6 w39x. defined by

Let h be the algebra homomorphism from Sym to T

h Ž p 2 kq1 . s 2 p 2 kq1 ,

h Ž p2 k . s 0

Ž k g N. .

Ž 60 .

For all bar partitions l there holds

h Ž sl . s 2 l Ž l. Pl2 .

Ž 61 .

From our point of view there is a more useful characterization of h , namely, this is the unique degree-preserving homomorphism of s-modules from Sym to T , where s acts on Sym by B2 kq1 f s Ž 2 k q 1 .

­f ­ p 2 kq1

,

By2 ky1 f s p 2 kq1 f ,

Kf s 2 f

Ž f g Sym.

Ž 62 .

and s acts on T by B2 kq1 g s k q

ž

1

/

­g

2 ­ p 2 kq1

,

By2 ky1 g s 2 p 2 kq1 g ,

Kg s 2 g

Ž g g T . . Ž 63 .

120

LECLERC AND LEIDWANGER

Note that the action on Sym is obtained by restricting to s the action of b` on any level 2 module LŽ L k .a` x. In contrast, the action on T is not the restriction of the level 1 action of b` on LŽ L 0 .. Its origin is revealed in the following. THEOREM 7. Let m : T m2 ª T denote the multiplication. Then m ( Hk s 0 for k / 0, y1 and m ( H0 s m ( Hy1 s h. Proof. By definition Hk is a b` -morphism from LŽ L k .a` x to LŽ L 0 .m2 and hence is a s-morphism. Now, by Eqs. Ž38., Ž63., for f, g g T s LŽ L 0 . we have

m Ž B2 kq1 Ž f m g . . s k q

ž ž

s kq

1 2 1 2

m

/ž /ž

­f ­ p 2 kq1 ­f

­ p 2 kq1

mgqfm

gqf

­g ­ p 2 kq1

­g ­ p 2 kq1

/

/

s B2 kq1 Ž fg . and

m Ž By2 ky1 Ž f m g . . s m Ž p 2 kq1 f m g q f m p 2 kq1 g . s 2 p 2 kq1 fg s By2 ky1 Ž fg . . Therefore m ( Hk is a s-morphism from Sym to T. Moreover, it follows from Theorem 5 that deg Ž m ( Hk Ž f . . s deg f q

k Ž k q 1. 2

Ž f g Sym. .

Ž 64 .

On the other hand the action Ž63. makes T an irreducible s-module. Therefore m ( Hk ŽSym. is equal to T or 0. If k / 0, y1 then, by Ž64., m ( Hk ŽSym. is strictly contained in T , and hence is 0. If k s 0, y1, let us consider the decomposition Sym s

[p l

2lT

Ž 65 .

of Sym into irreducible s-modules. Here l s Ž l1 , . . . , l r . runs over the set of all partitions and 2 l stands for Ž2 l1 , . . . , 2 l r .. Again for degree reasons m ( Hk Ž p 2 l T . s 0 for all nonempty partitions l, and clearly m ( Hk N T s h.

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

121

Using the description of H0 given in Theorem 5 we immediately obtain the following. THEOREM 8.

Let l be an arbitrary partition. Then

h Ž sl . s

Ý

Žg , g .*c 0 Ž l .

Pg Pg .

Ž 66 .

For example, if l s Ž2, 2. then c 0 Ž l. s Ž2, 1, 1. and Ž66. reads

h Ž sŽ2, 2. . s PŽ2 , 1, 1. PŽ0. q PŽ2 , 1. PŽ1. q PŽ2 , 1. PŽ1. q PŽ1 , 1. PŽ2. q PŽ2. PŽ1 , 1. q PŽ1. PŽ2 , 1. q PŽ1. PŽ2 , 1. q PŽ0. PŽ2 , 1, 1. s 4 PŽ2, 1. PŽ1. . If l s Ž2, 1, 1, 1. then c 0 Ž l. s Ž2, 3. and Ž66. reads

h Ž sŽ2, 1, 1, 1. . s PŽ2 , 3. PŽ0. q PŽ2. PŽ3. q PŽ3. PŽ2. q PŽ0. PŽ2 , 3. s y2 PŽ3, 2. q 2 PŽ3. PŽ2. . To recover Theorem 6, note that for a bar partition l s Ž l1 , . . . , l r ., c 0 Ž l . s Ž l1 , l1 , l 2 , l2 , . . . , l r , l r . .

Ž 67 .

Hence in this case the only nonzero summands in Ž66. are equal to Pl2 and there are 2 r such terms. 6. THE OPERATORS D k In w6x, Dynkin introduced the notion of subordinate g-modules over a semi-simple Lie algebra g. If LŽ L . and LŽ LX . are finite-dimensional irreducible g-modules with highest weights L and LX and highest weight vectors ¨L and ¨LX , respectively, then LŽ LX . is said to be subordinate to LŽ L . if there exists a UŽ gy. -homomorphism of LŽ L . onto LŽ LX . mapping ¨L onto ¨LX . Here, gy denotes the subalgebra of g generated by the lowering operators f i . It was proved by Dynkin that LŽ LX . is subordinate to LŽ L . if and only if LY s L y LX is a dominant integral weight. Indeed if this is the case then LŽ L . is isomorphic to the irreducible component of LŽ LX . m LŽ LY . with highest weight vector ¨LX m ¨LY , and one can construct the required UŽ gy. -homomorphism by ‘‘projecting’’ the tensor product on its first factor.

122

LECLERC AND LEIDWANGER

Adapting this idea to our situation and using the results of Section 4, one can obtain a family of UŽ b`y. -morphisms D k from Sym s LŽ L k .a` x to T s LŽ L 0 .. Let L low denote the sum of all weight spaces of LŽ L 0 . except the highest one spanned by ¨L 0 s 1, so that we have L Ž L 0 . s C ¨L 0 m L low ,

Ž 68 .

and by tensoring with LŽ L 0 ., LŽ L 0 .

m2

s L Ž L 0 . m C ¨L 0 [ L Ž L 0 . m L low .

Ž 69 .

Hence a tensor T g LŽ L 0 .m2 decomposes uniquely as T s G Ž T . m ¨L 0 q T X

Ž 70 .

where T X g LŽ L 0 . m L low . Clearly, G is a linear map from LŽ L 0 .m2 onto LŽ L 0 ., which satisfies for all i G 0 Fi`T s Fi` Ž G Ž T . . m ¨L 0 q T Y ,

Ž 71 .

with T Y g LŽ L 0 . m L low . Therefore G Ž Fi`T . s Fi` Ž G Ž T . .

Ž i g N.

Ž 72 .

and G is a UŽ b`y. -morphism. DEFINITION 2. Let k g Z and let Sym and T be identified with the b` -modules LŽ L k .a` x and LŽ L 0 ., respectively. We denote by D k the UŽ b`y. -morphism from Sym to T defined by D k s G( Hk .

Ž 73 .

Theorem 9 follows immediately from Eq. Ž54.: THEOREM 9. We ha¨ e D k Ž sl . s Pc k Ž l . .

Ž 74 .

For example, the values of D 0 Ž sl . and D 2 Ž sl . for all partitions l of 6 are given in Table I. The following property of D k is easily deduced from its definition. PROPOSITION 1. The map D k satisfies D k Ž fg . s fD k Ž g .

Ž f g T , g g Sym. .

In particular, D 0 is a projection, namely, D 0 ( D 0 s D 0 .

Ž 75 .

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

123

TABLE I The Operators D 0 and D 2 in Degree 6 sl

D 0 Ž sl .

D 2 Ž sl .

sŽ6. sŽ5, 1. sŽ4, 2. sŽ4, 1, 1. sŽ3, 3. sŽ3, 2, 1. sŽ3, 1, 1, 1. sŽ2, 2, 2. sŽ2, 2, 1, 1. sŽ2, 1, 1, 1, 1. sŽ1, 1, 1, 1, 1, 1.

PŽ6. PŽ5, 1. 0 PŽ4, 2. yPŽ3, 2, 1. PŽ3, 2, 1. 0 0 yPŽ3, 2, 1. yPŽ4, 2. yPŽ5, 1.

PŽ8, 1. PŽ7, 2. PŽ6, 3. PŽ6, 2, 1. PŽ5, 4. PŽ5, 3, 1. 0 PŽ4, 3, 2. 0 0 0

Proof. By linearity, it is enough to prove that D k Ž p 2 iq1 g . s p 2 iq1 D k Ž g .. But this is nothing else than D k Ž By2 iy1Ž g .. s By2 iy1Ž D k Ž g .., which follows from the fact that D k commutes with all lowering operators of b` . 7. PARTITIONS AND SYMMETRIC FUNCTIONS In this section, we collect a number of definitions and propositions relative to the combinatorics of partitions and the algebra of symmetric functions. Most of these are standard Žour main references being w9, 24, 29x. or elementary. This background will be useful in the following sections $ when we get to the description of the basic representation of sl2 and its connection with the operators D k . 7.1. Partitions and Sequences Let l be a partition and k be a fixed integer. We associate with l and k the infinite decreasing sequence

u k Ž l . s Ž l1 q k, l2 q k y 1, l3 q k y 2, . . . . .

Ž 76 .

This simple idea is quite classical, and variants of it appear in the literature under various names like b-numbers, partition sequences, or Maya diagrams Žsee, for example, w29x.. In our setting, it should be regarded as a combinatorial counterpart of the boson]fermion correspondence Žsee in Section 8 the proof of Proposition 4.. Note that a decreasing sequence of integers b s Ž bi . iG 1 is of the form u k Ž l. for some k if and only if bi s k y i q 1 for i large enough, and in this case l i s bi y k q i y 1.

124

LECLERC AND LEIDWANGER

Let u k Ž l. denote the decreasing sequence obtained from u k Ž l. by taking the complement in Z and changing all of the signs. For example, if l s Ž6, 4, 2, 1, 1. and k s 2, then

u 2 Ž l . s Ž 8, 5, 2, 0, y1, y3, y4, y5, . . . . and

u 2 Ž l . s Ž 2, y1, y3, y4, y6, y7, y9, y10, . . . . . An elementary but important property is that if lX denotes the conjugate of l, then

u k Ž l . s uyky1 Ž lX .

Ž 77 .

Žsee w24x I Ž1.7. and Ex. I.15, or w29x Ž2.2... Another worthy remark is that the composition c k Ž l. introduced in Section 4 is just a rearrangement of the nonnegative terms of the sequences u k Ž l. s Ž u ki . i G 1 and u k Ž l. s Ž u ki . iG 1. More precisely, if k G 0, then, in the notation of Definition 1, k q r is the number of positive terms of u k Ž l., r is the number of nonnegative terms of u k Ž l., and we have c k Ž l . s u k1 , . . . , u kk , u kkq 1 , u k1 , . . . , u kkq r , u kr .

ž

Ž 78 .

/

Similarly, if k - 0, then y1 1 yk qry1 c k Ž l . s u k1 , . . . , uyk , uyk , u kr , k k , uk , . . . , uk

ž

/

Ž 79 .

where r is the number of nonnegative terms of u k Ž l.. For example, c 2 Ž 6, 4, 2, 1, 1 . s Ž 8, 5, 2, 2 . s u 21 , u 22 , u 23 , u 21 .

ž

/

We now come to the notions of 2-quotient, 2-core, and 2-sign of a partition. To simplify notation, let us fix a partition l and write for short u 0 Ž l. s u . Let e s Ž e j . jG 1 Žresp. o s Ž oj . jG 1 . be the subsequence of u consisting of even Žresp. odd. terms. Then it is easily verified that e jr2 s t y j q 1 and Ž oj q 1.r2 s yt y j q 1 for j large enough and some t s t Ž l. g Z. Hence there exist partitions l0 and l1 defined by

uyt Ž l0 . s Ž Ž oj q 1 . r2 . jG1 ,

u t Ž l1 . s Ž e jr2 . jG1 .

Ž 80 .

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

125

The pair Ž l0 , l1 . is called the 2-quotient of l, and the partition

lŽ2. s

½

r2 t ry2 ty1

if t G 0 if t - 0

Ž 81 .

is the 2-core of l. Finally, the sequence

Ž e1 , e2 , . . . , e2 t , e2 tq1 , o1 , e2 tq2 , o 2 , . . . .

Ž 82 .

Ž o1 , o 2 , . . . , oy2 t , e1 , oy2 tq1 , e2 , oy2 tq2 , . . . .

Ž 83.

if t G 0 and

if t - 0 is a finite permutation w of u . The sign of w is called the 2-sign of l and is denoted by e 2 Ž l.. For example, let l s Ž8, 4, 4, 1.. Then

u s Ž 8, 3, 2, y2, y4, y5, . . . . ,

o s Ž 3, y5, y7, y9, . . . . ,

e s Ž 8, 2, y2, y4, y6, . . . . , and therefore

uyt Ž l0 . s Ž 2, y2, y3, y4, . . . . ,

u t Ž l1 . s Ž 4, 1, y1, y2, y3, . . . . ,

which shows that t s 1, lŽ2. s Ž2, 1., l0 s Ž3., and l1 s Ž3, 1.. Finally, Ž82. is equal in this case to Ž8, 2, y2, 3, y4, y5, . . . ., which is an even permutation of u , so that e 2 Ž l. s q1. The descriptions of lŽ2. , Ž l0 , l1 ., and e 2 Ž l. that we have just given are essentially those of James in terms of an abacus Žsee w9, p. 75x.. The original definitions of Robinson involve Young diagrams. Briefly, this goes as follows. The 2-core lŽ2. is the partition obtained by removing successively all possible dominoes from the Young diagram of l, starting from the rim. The 2-sign e 2 Ž l. is equal to q1 Žresp. y1. if the number of vertical dominoes that have been removed to reach the 2-core is even Žresp. odd.. Finally, the removed dominoes are divided into two groups D 0 and D 1 according to the content mod 2 of their lowest and rightmost node, and the partitions l0 and l1 are computed from D 0 and D 1 Žsee w3x for details.. This is illustrated in Fig. 4. It is known that given a staircase partition r s and an arbitrary pair of partitions Ž m 0 , m1 ., there exists a unique l such that lŽ2. s r s and Ž l0 , l1 . s Ž m0 , m1 .. Thus a partition is uniquely determined by its 2-core and 2-quotient and can be defined by specifying them. In this situation we shall use the notation l [ Ž lŽ2. ; l0 , l1 ..

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LECLERC AND LEIDWANGER

FIG. 4. The graphical calculation of Ž lŽ2. ; Ž l0 , l1 .. for l s Ž8, 4, 4, 1..

DEFINITION 3. Let l be a partition. We associate with it two partitions lU 0 and lU 1 by setting X

lU 0 [ Ž lŽ2. ; Ž l0 . , l1 . ,

X

lU 1 [ Ž lŽ2. ; l0 , Ž l1 . . .

Ž 84 .

For example, if l s Ž8, 4, 4, 1., then lU 0 s Ž8, 3, 1, 1, 1, 1, 1, 1. and lU 1 s Ž6, 4, 4, 3., as shown in Fig. 5. Now we have the crucial lemma. LEMMA 3. Let l be a partition with 2-core lŽ2. s r k and set i s k mod 2, j s k q 1 mod 2. Then c k Ž lU i . is a permutation wi of c k Ž l. and cyk y1Ž lU j . is a permutation wj of cyky1Ž l.. Moreo¨ er, sgn wi s e 2 Ž l . e 2 Ž lU i . Ž y1 .

< li <

Ž i s 0, 1 . .

Ž 85 .

Proof. Let us first assume that k s 0. We shall write n g u to signify that the integer n is a term of the decreasing sequence u . Then, 2 m q 1 g u 0 Ž l . m m g uy1 Ž l0 . m m g u 0 Ž Ž l0 .

X

.

by Ž 77 .

m 2 m q 1 g u 0 Ž lU 0 . . Similarly, 2 m q 1 g u 0 Ž l. m 2 m q 1 g u 0 Ž lU 0 .. On the other hand, 2 m g u 0 Ž l. m 2 m g u 0 Ž lU 0 . and 2 m g u 0 Ž l. m 2 m g u 0 Ž lU 0 .. Hence, in

FIG. 5. The partitions lU 0 and lU 1 for l s Ž8, 4, 4, 1..

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

127

view of Ž78., c 0 Ž lU 0 . is a permutation w 0 of c 0 Ž l.. To check that sgn w 0 s e 2 Ž l . e 2 Ž lU 0 . Ž y1 .

< l0 <

we use the graphical description in terms of dominoes. Indeed, when l1 s Ž0., the diagram of l decomposes uniquely into r domino hooks, where r is the number of nodes on the main diagonal of l. It is easily seen 0 that in this case sgn w 0 s Žy1. r s e 2 Ž l. e 2 Ž lU 0 .Žy1. < l < . Then one checks that the formula remains valid when dominoes of type 1 are added one by one. To prove the statement for cy1Ž lU 1 . and more generally for c k Ž lU i . and cyk y1Ž lU j . when k / 0, one uses the fact that u k Ž l. is obtained from u 0 Ž l. by merely adding k to all of the parts, and therefore the 2-quotient and 2-core of l can be computed in a similar way from u k Ž l., the only difference being a shift of the indices. We omit the details. To illustrate this, consider again l s Ž8, 4, 4, 1.. Then s s 2, lU 0 s Ž8, 3, 1, 1, 1, 1, 1, 1. and lU 1 s Ž6, 4, 4, 3.. On the other hand, c 2 Ž l. s Ž10, 5, 4, 1., c 2 Ž lU 0 . s Ž10, 4, 1, 5., and cy3 Ž l. s Ž6, 4, 3, 5, 2, 0., cy3 Ž lU 1 . s 0 Ž6, 5, 4, 3, 2, 0.. Finally, e 2 Ž l. s q1, e 2 Ž lU 0 . s y1, e 2 Ž lU 1 . s q1, Žy1. < l < 1 s q1, Žy1. < l < s y1, and both permutations w 0 and w 1 are even. 7.2. Symmetric Functions The combinatorial manipulations of the previous section are all connected with algebraic transformations in the ring of symmetric functions. Let t s Žt i . iG 1 be an infinite sequence of pairwise distinct integers such that t i s yi q 1 for i large enough. Then there is a finite permutation w such that w Žt . s u 0 Ž l. for some partition l. It is convenient to introduce the notation St s sgn wsl ,

Ž 86 .

and to extend this notation to sequences t such that t i s t j for some pair i, j, by putting St s 0 in this case. The classical formula for multiplying an S-function by a power sum Žsee, e.g., w24x I 3, Ex. 11. then reads pk St s

Ý Sj ,

Ž 87 .

j

where j runs over all of the sequences obtained from t by adding k to one of its parts. Clearly, only a finite number of these sequences have all of their parts pairwise distinct, and the above sum is finite. Thus, if t s Ž2, 0, y1, y3, y4, y5, . . . . and k s 3, the nonzero summands of Ž87.

128

LECLERC AND LEIDWANGER

correspond to the sequences

Ž 5, 0, y1, y3, y4, y5, . . . . ,

Ž 2, 3, y1, y3, y4, y5, . . . . ,

Ž 2, 0, y1, y3, y4, y2, . . . . , and going back to the standard notation, we get p 3 sŽ2 , 1, 1. s sŽ5 , 1, 1. y sŽ3 , 3, 1. q sŽ2 , 1, 1, 1, 1, 1. . The partitions m such that sm occurs in pk sl are those obtained from l by the addition of a ribbon Žor rim-hook. of length k. The sign of sm is then equal to Žy1. hy 1, where h is the height of the ribbon Žsee w24x I 3, Ex. 11.. Similarly, setting Dp k s k ­r­ pk , we have Dp k St s

Ý Sj ,

Ž 88 .

j

where j now runs over all of the sequences obtained from t by subtracting k to one of its parts. In general, given f s f Ž p1 , p 2 , . . . , pn , . . . . g Sym, we define the differential operator Df s f

ž

­ ­ p1

,2

­ ­ p2

,..., n

­ ­ pn

/

,... .

Ž 89 .

The operator Df is known to be the adjoint of the multiplication by f with respect to the scalar product ² ? , ? : for which the basis of S-functions is orthonormal. In other words, we have ² Df g , h: s ² g , fh:

Ž f , g , h g Sym. .

Ž 90 .

In particular, we have DsmŽ sl . s sl r m , the skew Schur function. The notions of 2-quotient, 2-core, and 2-sign have been used by Littlewood to evaluate the symmetric function c 2 Ž sl . w23x. Here, c 2 denotes the ring endomorphism of Sym which sends pi to p 2 i . Littlewood proved that

² sr c 2 Ž sl . , sm: s ² sl , e 2 Ž m . sm sm : s

½

0

1

0

if mŽ2. s r s , otherwise.

Ž 91 .

Equivalently, if one denotes by f 2 the adjoint of c 2 with respect to ² ? , ? :, one has

f 2 Ž sm r m Ž2. . s e 2 Ž m . sm 0 sm1 .

Ž 92 .

In a sense, Eq. Ž92. is the algebraic identity encoding the combinatorics of 2-cores and 2-quotients. However, some information is lost, since f 2 does

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

129

not distinguish between the two partitions m 0 and m1 of the 2-quotient. This suggests ‘‘polarizing’’ Ž92. by introducing a second set of variables. Let X 0 and X 1 be two countable sets of indeterminates and denote by SymŽ X 0 , X 1 . the C-algebra of functions symmetric in X 0 and X 1 separately. A linear basis of SymŽ X 0 , X 1 . is given, for example, by the products slŽ X 0 . smŽ X 1 .. The space SymŽ X 0 , X 1 . carries a scalar product defined by

² f Ž X 0 . g Ž X1 . , h Ž X 0 . l Ž X1 .: s ² f , h:² g , l :

Ž f , g , h, l g Sym. . Ž 93 .

DEFINITION 4. Let B s Cw q, qy1 x mC SymŽ X 0 , X 1 . and denote by F 2 the isomorphism of C-vector spaces given by F 2 : Sym Ž X . ª B sl Ž X . ª e 2 Ž l . qyt Ž l. m sl 0 Ž X 0 . sl1 Ž X 1 . ,

Ž 94 .

where t Ž l. is related to lŽ2. by Ž81.. Examples of values of F 2 Ž sl . are shown in Table II. It follows from the definition that F 2 is an isometric isomorphism if one defines the scalar product on B by

² q i m f Ž X 0 , X1 . , q j m g Ž X 0 , X1 .: s di j ² f Ž X 0 , X1 . , g Ž X 0 , X1 .: . Ž 95. Indeed, F 2 sends the orthonormal basis  sl4 onto an orthonormal basis of B.

TABLE II The Maps F 2 and n in Degree 6 sl

F 2 Ž sl .

n Ž sl .

sŽ6. sŽ5, 1. sŽ4, 2. sŽ4, 1, 1. sŽ3, 3. sŽ3, 2, 1. sŽ3, 1, 1, 1. sŽ2, 2, 2. sŽ2, 2, 1, 1. sŽ2, 1, 1, 1, 1. sŽ1, 1, 1, 1, 1, 1.

1 m sŽ3.Ž X 1 . y1 m sŽ3.Ž X 0 . 1 m sŽ1.Ž X 0 . sŽ2.Ž X1 . y1 m sŽ2, 1.Ž X 1 . y1 m sŽ2.Ž X 0 . sŽ1.Ž X1 . qy2 m 1 1 m sŽ2, 1.Ž X 0 . 1 m sŽ1.Ž X 0 . sŽ1, 1.Ž X 1 . y1 m sŽ1, 1.Ž X 0 . sŽ1.Ž X 1 . 1 m sŽ1, 1, 1.Ž X 1 . y1 m sŽ1, 1, 1.Ž X 0 .

sŽ6. ysŽ1, 1, 1, 1, 1, 1. ysŽ4, 2. sŽ4, 1, 1. sŽ2, 2, 1, 1. sŽ3, 2, 1. ysŽ3, 1, 1, 1. ysŽ2, 2, 2. sŽ3, 3. sŽ2, 1, 1, 1, 1. ysŽ5, 1.

130

LECLERC AND LEIDWANGER

To state the main properties of F 2 , we need some notation. For f g Sym and i s 0, 1, we introduce the operators Df Ž X i . and fˆŽ X i . Žmultiplication by f Ž X i .., which act on B in an obvious way: Df Ž X i . Ž q j m g Ž X 0 , X 1 . . s q j m Df Ž X i . g Ž X 0 , X 1 . ,

Ž 96 .

fˆŽ X i . Ž q j m g Ž X 0 , X 1 . . s q j m f Ž X i . g Ž X 0 , X 1 . .

Ž 97 .

For f g Sym, let f Ž X 0 q X 1 . denote the image of f by the ring homomorphism from Sym to SymŽ X 0 , X 1 . sending pk on pk Ž X 0 q X 1 . [ pk Ž X 0 . q pk Ž X 1 .. It is known Žsee, e.g., w24x I 5, Ex. 25. that f Ž X 0 q X1 . s

Ý sl Ž X 0 . Ž Ds f . Ž X1 . , l

l

Ž 98 .

the sum being over all partitions l. The subring SymŽ X 0 q X 1 . of SymŽ X 0 , X 1 . consisting of the functions of the form f Ž X 0 q X 1 . is in fact the ring of symmetric functions in the whole set of variables X 0 j X 1. Similarly, we set Dp k Ž X 0qX 1 . [ Dp k Ž X 0 . q Dp k Ž X 1 . . Now since f 2 is a ring homomorphism such that

f 2 Ž p 2 k . s 2 pk s pk Ž X q X . ,

Ž 99 .

we have

f2 Ž c2 Ž f . g . s f Ž X q X . f2 Ž g .

Ž f , g g Sym. .

Ž 100.

The analogue of this result for F 2 is contained in the next proposition. PROPOSITION 2.

For f, g g Sym and k g N, we ha¨ e

F 2 Ž Dp 2 k g . s Dp k Ž X 0qX 1 .F 2 Ž g . ,

$

F 2 Ž p 2 k g . s pk Ž X 0 q X 1 . F 2 Ž g . .

Ž 101. This implies that F 2 Ž c 2 Ž f . g . s fˆŽ X 0 q X 1 . F 2 Ž g . ,

Ž 102.

F 2 Ž sr k c 2 Ž f . . s qyt Ž r k . m f Ž X 0 q X 1 . .

Ž 103.

and in particular,

Proof. By linearity, it is enough to prove Eq. Ž101. for g s sl. Using Eqs. Ž87., Ž88., we obtain the combinatorial description of p 2 k sl and Dp 2 k sl in terms of the sequence u 0 Ž l.. Then Ž101. follows by comparison with the definitions Ž80., Ž81. of the 2-quotient and 2-core of l. Note that the signs

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

131

agree because the addition of a 2 k-ribbon R of height h to l corresponds to the addition of a k-ribbon RX , either to l0 or to l1 , of height hX s h y ¨ , where ¨ denotes the number of vertical dominoes in the unique domino tiling of R. Finally, we introduce the fundamental involution of Sym. Let f ŽyX . denote the image of f Ž X . under the ring automorphism sending pk Ž X . onto pk ŽyX . [ ypk Ž X .. Then it is well known Žw24x I 3, Ex. 23. that sl Ž yX . s Ž y1 .

< l<

slX Ž X . .

Ž 104.

One can combine the previous transformations and define, for instance, f ŽyX 0 q X 1 . as the image of f under the morphism defined by pk ŽyX 0 q X 1 . [ ypk Ž X 0 . q pk Ž X 1 .. Then f Ž yX 0 q X 1 . s

< < Ý Ž y1. l sl Ž X 0 . Ž Ds f . Ž X1 . . X

l

l

Ž 105.

This notation is convenient and consistent in the sense that identities such as f Ž X 1 . s f Ž yX 0 q Ž X 0 q X 1 . . s

< < Ý Ž y1. l sl Ž X 0 . Ž Ds f . Ž X 0 q X1 . . X

l

l

Ž 106. are satisfied. In particular, < < Ý Ž y1. l sl Ž X . Ž Ds f . Ž X . s f Ž X y X . s f Ž 0. , X

l

l

Ž 107.

the constant term of f Ži.e., the degree 0 term.. PROPOSITION 3. We ha¨ e F 2 Ž T . s C q, qy1 m Sym Ž yX 0 q X 1 . , F2

ž[ kG0

sr k c 2 Ž Sym. s C q, qy1 m Sym Ž X 0 q X 1 . .

/

Ž 108. Ž 109.

Proof. Equation Ž109. is an obvious consequence of Proposition 2. To prove Ž108., we shall use the fact that F 2 is isometric. Let I denote the ideal of Sym generated by even power sums p 2 k Ž k ) 0.. Then, since pm forms an orthogonal linear basis of Sym, we have T s I H , and therefore F 2 Ž T . s F 2 Ž I . H . By Proposition 2, F 2 Ž I . is spanned by symmetric

132

LECLERC AND LEIDWANGER

functions of the type q j m f Ž X 0 q X 1 . g Ž X 0 , X 1 . with f Ž0. s 0. Now

² q i m h Ž yX0 q X1 . , q j m f Ž X 0 q X1 . g Ž X 0 , X1 .: s d i j ² h Ž yX 0 q X 1 . , f Ž X 0 q X 1 . g Ž X 0 , X 1 .: s d i j ² h Ž X 0 q X 1 . , f Ž yX 0 q X 1 . g Ž yX 0 , X 1 .: s d i j ² h Ž X . , f Ž 0 . g Ž yX , X .: s 0. Here we have used the following properties of the scalar product:

² f Ž X 0 , X1 . , g Ž X 0 , X1 .: s ² f Ž yX 0 , X1 . , g Ž yX 0 , X1 .: , ² f Ž X 0 q X1 . , g 0 Ž X 0 . g 1Ž X1 .: s ² f Ž X . , g 0 Ž X . g 1Ž X .: . The first one comes from the fact that f Ž X . ¬ f ŽyX . is isometric, whereas the second one results from Ž98. Žsee w24x I 5, Ex. 25.. Hence we have obtained that E [ C q, qy1 m Sym Ž yX 0 q X 1 . ; F 2 Ž I .

H

.

Finally, since the products plŽyX 0 q X 1 . pmŽ X 0 q X 1 . form a linear basis of SymŽ X 0 , X 1 ., we have E [ F 2 Ž I . s B, which proves the reverse inclusion. $

8. THE BASIC REPRESENTATION OF sl2 $

The Lie algebra sl2 s AŽ1. is the simplest infinite-dimensional 1 Kac]Moody algebra. Its basic representation can be obtained in several ways w22, 7, 11x. It turns out that the principal realization of Lepowsky and Wilson and the homogeneous realization of Frenkel and Kac can both be constructed in Sym. The intertwiner between these two realizations is a distinguished involution n of Sym that will play a major role in our second description of the operators D k in Section 9. $ Following Date, Jimbo, Kashiwara, and Miwa, we shall first describe sl2 as a subalgebra of a` . Let sl2 be the subalgebra of a` , consisting of the matrices A s Ž a i j . such that a iq2 , jq2 s a i j ,

a2 iq1, 2 jq1 q a2 iq2, 2 jq2 s 0

Ž i , j g Z. . Ž 110.

The central extension sl2 [ Cc with Lie bracket defined by Eqs. Ž14.,$Ž15. $X $ is denoted by sl2 Žand called the derived algebra of sl2 .. The algebra sl2 is

133

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

$X

obtained by adjoining to sl2 the degree operator Dsy

i

Ý igZ

2

Eii .

Ž 111.

The Chevalley generators e0 , e1 , f 0 , f 1 , h 0 , h1 are ei s

Ý

Ek , kq1 ,

Ž 112.

Ekq1, k ,

Ž 113.

k'i mod 2

fi s

Ý k'i mod 2

h i s w ei , fi x s

Ž Ek , k y Ekq1, kq1 . q d i , 0 c.

Ý

Ž 114.

k'i mod 2

$X

Equivalently, sl2 can be constructed as the central extension ŽCw t, ty1 x mC sl 2 . [ Cc of the loop algebra of sl 2 , the bracket being given by t k m x, t l m y s t kql m Ž xy y yx . q k d k , yl tr Ž xy . c

Ž x, y g sl 2 , k, l g Z.

Ž 115.

Žsee w15x.. The isomorphism i between the two realizations is

i Ž t j m e. s

Ý

Ek , 2 jqkq1 ,

Ž 116.

Ekq 1, 2 jqk ,

Ž 117.

k odd

iŽt j m f . s

Ý k odd

i Ž t j m h. s

Ý Ž Ek , 2 jqk y Ekq1, 2 jqkq1 . ,

Ž 118.

k odd

where es

ž

0 0

1 , 0

/

fs

ž

0 1

0 , 0

/

0 . y1

Ž 119.

h1 s i Ž 1 m h .

Ž 120.

hs

ž

1 0

/

In particular, one has e1 s i Ž 1 m e . ,

f1 s i Ž 1 m f . ,

and e0 s i Ž t m f . ,

f 0 s i Ž ty1 m e . ,

h 0 s i Ž c y 1 m h . . Ž 121 .

134

LECLERC AND LEIDWANGER

In this setting, the degree operator acts by D, i Ž t k m x . s k i Ž t k m x .

Ž x g sl 2 , k g Z. ;

Ž 122.

hence its name. $ Let L 0 and L 1 denote the fundamental weights of sl2 . The fundamental representations LŽ L 0 . and LŽ L 1 . can be obtained by restriction from the corresponding representations of a` . They are transformed into each other $X by the automorphism of sl2 exchanging e0 with e1 and f 0 with f 1 , so we shall only describe LŽ L 0 .. First we note that because of Ž110., the principal Heisenberg subalgebra $ of sl2 coincides with that of b` , namely, $

s s s a l sl2 s

ž[ kgZ

C B2 kq1 [ Cc.

/

Ž 123.

One then checks that the even Heisenberg algebra generated by B2 k $X Ž k g ZU . commutes with sl2 :

w B2 k , e i x s w B2 k , f i x s 0

Ž i s 0, 1, k g ZU . .

Ž 124.

2lT

Ž 125.

This implies that the decomposition Sym s

[p l

already encountered in Ž65. can be interpreted as the decomposition into $X irreducible sl2-modules of the restricted module LŽ L 0 .a` x.$In particular, T is identified with the basic representation LŽ L 0 . of sl2 , and in this construction the principal subalgebra s acts by B2 kq1 f s Ž 2 k q 1 .

­f ­ p 2 kq1

,

By2 ky1 f s p 2 kq1 f ,

cf s f

Ž f g T , k g N . . Ž 126. Thus we have obtained the Lepowsky]Wilson principal realization of LŽ L 0 . w22x. $ Let us denote by r P the above representation of sl2 in Sym , LŽ L 0 .a` x. By combining Theorem 1 and Eqs. Ž112. ] Ž114., one easily obtains the combinatorial description of r P , namely,

r P Ž e i . sl s

Ý sm , m

r P Ž f i . sl s

Ý sn ,

Ž 127.

n

where m Žresp. n . runs through the partitions obtained from l by removing Žresp. adding. a node with content d ' i mod 2.

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

135

We observe that P-functions are not, in general, weight vectors of this representation. For example, PŽ3. s sŽ3. q sŽ2, 1. q sŽ1, 1, 1. decomposes into $

the sum of sŽ3. q sŽ1, 1, 1. and sŽ2, 1. , which are sl2-weight vectors of weight L 0 y 2 a 0 y a 1 and L 0 y a 0 y 2 a 1 , respectively, where a 0 and a 1 are the simple roots. $ There is yet another Heisenberg subalgebra of sl2 called the homogeneous Heisenberg subalgebra. This is the subalgebra h generated by the elements hŽ k . s i Ž t k m h .

Ž k g ZU . ,

Ž 128.

which satisfy by Ž115. the commutation relations

w hŽ k . , hŽ l . x s d k , yl 2 kc.

Ž 129. $

By means of the map F 2 introduced in the previous section, the action of sl2 $ on Sym can be carried to B. For x g sl2 , put r XH Ž x . s F 2 ( r P Ž x .( Fy1 2 . PROPOSITION 4.

For k g N, we ha¨ e

r XH Ž hŽ k . . s Dp k Ž X 0yX 1 . ,

$

r XH Ž hŽyk . . s pk Ž X 0 y X 1 . .

Ž 130.

Proof. The easiest way to prove this is to use the infinite-wedge representation of Kac]Peterson Žsee w15x, 14.9.. This is an alternative realization of LŽ L 0 .a` in a space F with basis vectors u I s u i1 n u i 2 n ??? n u i k n ??? , where I s Ž i k . k G 1 runs through all decreasing sequences of integers such that i k s yk q 1 for k sufficiently large. Here, u i denotes the canonical basis of C Z on which the matrix units operate by Ei j u k s d jk u i . The Lie algebra A` of Z = Z-matrices with a finite number of nonzero entries acts on F by derivation, that is, Ei j Ž u i1 n u i 2 n ??? . s Ž Ei j u i1 . n u i 2 n ??? q u i1 n Ž Ei j u i 2 . n ??? q ??? , and this extends uniquely to a projective representation of a` , and hence to a linear representation of a` . The unique isomorphism between the ‘‘bosonic’’ realization Sym and the ‘‘fermionic’’ realization F is the so-called boson]fermion correspondence s : Sym ª F. It sends the S-function sl onto the infinite wedge

s Ž sl . s uu 0 Ž l. .

136

LECLERC AND LEIDWANGER

Now, by Ž118., hŽ j. acts on a wedge u I by hŽ j. u I s

Ý u J y Ý uK ,

where J Žresp. K . runs over all sequences obtained from I by replacing an odd Žresp. even. term i s by i s q 2 j. The desired result then follows from Eqs. Ž80., Ž87., Ž88., Ž94.. Recall that, by Proposition 3, F 2 Ž T . s C q, qy1 m Sym Ž yX 0 q X 1 . ,

Ž 131.

and note that Cw q, qy1 x can be identified with the group algebra Cw Q x of the root lattice Q s Z a of sl 2 . Furthermore, Proposition 4 shows that the homogeneous subalgebra h acts on F 2 Ž T . according to the canonical commutation relations representation associated with the generators pj Ž X 0 y X 1 . of the polynomial ring SymŽyX 0 q X 1 .. Hence the restriction of r XH to F 2 Ž T . is a version of the Frenkel]Kac realization of the basis $ sl2-module w7x. Now it is easy to carry back the homogeneous picture from B to Sym. Define a linear involution à on B by

à Ž qyi m f Ž X 0 , X 1 . . s

½

qyi m f Ž yX 0 , X 1 . yi

q

m f Ž X 0 , yX1 .

Ž i G 0. Ž i - 0. .

Ž 132.

Then

à ( F 2 Ž T . s C q, qy1 m Sym Ž X 0 q X 1 . ,

Ž 133.

and h acts on qy1 Cw qy1 x m SymŽ X 0 q X 1 . Žresp. Cw q x m SymŽ X 0 q X 1 .. via the canonical commutation relations representation associated this time with the variables pk Ž X 0 q X 1 . Žresp. ypk Ž X 0 q X 1 ... This makes it natural to introduce DEFINITION 5. Let n be the involution of Sym given by n s Fy1 2 ( Ã ( F 2 . In other words, n is the linear map:

¡e Ž l. e Ž l n Ž s . s~ ¢e Ž l. e Ž l

U0

2

2

l

U1

2

2

< < . Ž y1. l slU 0 0

. Ž y1.

< l1 <

slU 1

Ž t Ž l. G 0. Ž t Ž l. - 0. ,

Ž 134.

where the notations lU 0 , lU 1, and t Ž l. have been explained in Section 7, Eqs. Ž81., Ž84..

137

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

For example, the values of F 2 Ž sl . and n Ž sl . in degree 6 are displayed in Table II. Finally, put $

rH Ž x . s n ( rP Ž x . (n

Ž x g sl2 . .

Ž 135.

We deduce Proposition 5 immediately from the previous discussion and from Proposition 2 and Proposition 3. PROPOSITION 5. The in¨ olution n satisfies

nŽT. s

[s kG0

rkc 2

Ž Sym. .

Ž 136.

Moreo¨ er, h acts on sr k c 2 ŽSym. ¨ ia

r H Ž hŽ j. . s Ž y1 .

ky 1

Dp 2 j ,

r H Ž hŽyj. . s Ž y1 .

ky1

ˆp 2 j

Ž j ) 0. . Ž 137.

Proposition 5 shows that r H provides another version of the Frenkel]Kac homogeneous realization of LŽ L 0 ., whose carrier space is this time the subspace n Ž T . of Sym instead of the subspace Cw q, qy1 x m SymŽ X 0 q X 1 . of B. Thus the involution n of Sym may be viewed as the intertwining operator interchanging the principal and homogeneous pictures, both realized in Sym. 9. ANOTHER APPROACH TO THE OPERATORS D k We shall now make use of the maps n and F 2 introduced in Section 7 and Section 8 to obtain different descriptions of D k . Since by Eqs. Ž58., Ž73., D k Ž sl . s Dyky1Ž slX ., it is enough to consider the case k G 0. Our first result is THEOREM 10.

Let f g Sym and k g N. Then D k Ž c 2 Ž f . . s n Ž sr k c 2 Ž f . . .

Ž 138.

This will be a consequence of PROPOSITION 6.

Let l be a partition with 2-core lŽ2. s r k . Then D k Ž sl . s D k Ž n Ž sl . . .

Ž 139.

138

LECLERC AND LEIDWANGER

Proof. By Theorem 9, D k Ž sl . s Pc k Ž l. . Now using Lemma 3, one finds that Pc k Ž l. s e 2 Ž l . e 2 Ž lU i . Ž y1 .

< li <

Pc k Ž lU i . ,

Ž 140.

where i s k mod 2. Hence, by Definition 5, Pc k Ž l . s D k Ž n Ž sl ... Proof of Theorem 10. By linearity we may assume that f s sl. It follows from Ž87. and the definition of the 2-core of a partition that sr k c 2 Ž sl . is a linear combination of sm with mŽ2. s r k . On the other hand, we deduce easily from Ž88. that Dp 2 j sr k s 0 for any j, which means that sr k g T. Therefore by Proposition 1 and Proposition 6, D k Ž c 2 Ž sl . . s

1 sr k

D k Ž sr k c 2 Ž sl . . s

1 sr k

D k n Ž sr k c 2 Ž sl . . .

ž

/

Ž 141.

Now n Ž sr k c 2 Ž sl .. g T by Ž136.; hence using Proposition 1 again, we have D k n Ž sr k c 2 Ž sl . . s n Ž sr k c 2 Ž sl . . D k Ž 1 . s n Ž sr k c 2 Ž sl . . sr k , Ž 142 .

ž

/

and the theorem is proved. Observe that Proposition 1 and Theorem 10 provide a characterization $ of the operators D k that does not involve the algebra b` , but rather sl2 . Namely, D k is the unique linear map from Sym to Sym such that, for f g T and g g Sym, D k Ž fg . s fD k Ž g . ,

Ž 143.

D k Ž c 2 Ž g . . s n Ž sr k c 2 Ž g . . .

Ž 144.

This property will now be used to derive an expression of D k as a differential operator of infinite order. First we need some notation. Given a partition l and an integer k g N, we define a new partition Jk Ž l. by specifying its 2-core and 2-quotient: Jk Ž l . s THEOREM 11.

½

Ž r k ; l , Ž 0. . , Ž r k ; Ž 0. , l. ,

if k is even if k is odd.

Ž 145.

For k g N, we ha¨ e Dk s

< <

Ý Ž y1. l e 2 Ž Jk Ž lX . . s J Ž l . Dc Ž s . , l

k

X

2

l

Ž 146.

where the sum is o¨ er all partitions l, and for f g Sym, the notation Df is defined by Eq. Ž89..

139

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

Proof. Let us define an endomorphism D k of the vector space Sym by Dk s

< <

Ý Ž y1. l e 2 Ž Jk Ž lX . . s J Ž l . Dc Ž s . . X

k

l

2

l

We shall prove that D k satisfies Eq. Ž143., Ž144.. First we note that since c 2 Ž sl . is a polynomial in the even power sums p 2 j , the differential operator Dc 2 Ž sl . involves only derivations with respect to the even power sums, and one has Dc 2 Ž sl .Ž fg . s fDc 2 Ž sl . g for f g T. Hence Ž143. is obviously satisfied. On the other hand, using Eq. Ž103. and the relation n s Fy1 2 (à ( F 2 , condition Ž144. can be translated into yt Ž r k . F 2 ( D k ( Fy1 m g Ž X 0 q X 1 . . . Ž 147 . 2 Ž1 m g Ž X 0 q X1 . . s à Ž q

At this stage it is convenient to introduce the ring automorphism f ¬ f Ž Xr2. of Sym defined by pk Ž Xr2. [ pk Ž X .r2. This transformation may be combined with the formal sum and difference introduced in Section 7. Thus, the following identity holds: f

ž

yX q Y 2

s f yX q

/ ž

XqY

ž

2

//

s

Ý sl Ž yX . Ž Ds f . l

l

ž

XqY 2

/

.

Ž 148. LEMMA 4.

For f, g g Sym, we ha¨ e Dc 2 Ž f . Ž c 2 Ž g Ž Xr2 . . . s c 2 Ž Ž Df g . Ž Xr2 . . .

Ž 149.

Proof of Lemma 4. By linearity we may assume that f s pl and g s pm . Clearly, we have c 2 Ž pl . s p 2 l , where for l s Ž l1 , . . . , l r . we write 2 l s Ž2 l1 , . . . , 2 l r .. Similarly,

c 2 Ž pm Ž Xr2 . . s

1 2

l Ž m.

p2 m Ž X . .

Then Dc 2 Ž pl . Ž c 2 Ž pm Ž Xr2 . . . s s

1 2

l Ž m.

Dp 2 l p 2 m s

l1 ??? l r 2

l Ž m .y l Ž l .

s c2

ž

c2

ž

1 2

l Ž m .y l Ž l .

l1 ??? l r 2

­ ­ pl

l Ž m .y l Ž l .

pm

­ ­ p2 l

p2 m

/

Dpl pm s c 2 Ž Ž Dpl pm . Ž Xr2 . . .

/

Now taking f s sl in Lemma 4 and using Ž103., we obtain the following.

140

LECLERC AND LEIDWANGER

LEMMA 5.

For g g Sym,

Dc 2 Ž sl . ( Fy1 1mg 2

ž ž

X 0 q X1 2

s Fy1 1 m Ž Dsl g . 2

//

ž

ž

X 0 q X1 2

//

.

It follows from Proposition 2 and the definition of Jk Ž l. that we also have LEMMA 6.

Let i s k mod 2. Then

e 2 Ž Jk Ž l . . s J k Ž l.Fy1 1mg 2

ž ž

X 0 q X1 2

s Fy1 qytŽ r k . m sl Ž X i . g 2

ž

ž

//

X 0 q X1 2

//

.

We can now prove that D k satisfies Ž147.. Indeed, using again the notation i s k mod 2 and j s k q 1 mod 2, we have X 0 q X1

F 2 ( D k ( Fy1 1mg 2 s

ž ž ½ ½

Ž a.

//

< < Ý F 2 Ž y1. l e 2 Ž Jk Ž lX . . s J Ž l . Dc Ž s . ( Fy1 2

l

s

2

k

2

l

< < Ý F 2 Ž y1. l e 2 Ž Jk Ž lX . . s J Ž l .Fy1 2

l

Ž b.

s qytŽ r k . m

žÝ žÝ l

Žc.

s qytŽ r k . m

l

Žd.

s qytŽ r k . m g

ž

< <

sl Ž yX i . Ž Dsl g

yX i q X j 2

ž

2

X 0 q X1 2

//5

//5

2

X 0 q X1 2

/

X 0 q X1 2

X 0 q X1

X 0 q X1

Ž y1. l slX Ž X i . Dsl g

s à qytŽ r k . m g

ž

1 m Dsl g

X

k

1mg

ž ž ž Ž .ž Ž .ž // .ž // X

//

.

Here, equality Ža. follows from Lemma 5, equality Žb. from Lemma 6, equality Žc. from Ž104., and equality Žd. from Ž148.. Finally, since f ¬ f Ž Xr2. is an automorphism of Sym, this identity is equivalent to Ž147., and the theorem is proved.

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

141

To illustrate Theorem 11, let us compute D 0 sŽ4, 1, 1. . We have D 0 s 1 q sŽ1 , 1. Dc 2 Ž s Ž1. . q sŽ1 , 1, 1 , 1. Dc 2 Ž s Ž2. . y sŽ3 , 1. Dc 2 Ž s Ž1 , 1. . q sŽ1, 1, 1, 1, 1, 1. Dc 2 Ž s Ž3. . y sŽ3 , 1, 1, 1. Dc 2 Ž s Ž2 , 1. . q sŽ5 , 1. Dc 2 Ž s Ž1 , 1 , 1. . q ??? The terms we have not written annihilate symmetric functions of degree less than 8, so this truncated expansion is enough for our purpose. Applying this operator to sŽ4, 1, 1. , we obtain D 0 sŽ4 , 1, 1. s sŽ4 , 1, 1. q sŽ1 , 1. Ž ysŽ4. q sŽ2 , 1, 1. . q sŽ1 , 1, 1, 1. Ž ysŽ2. . y sŽ3, 1. Ž ysŽ2. . q sŽ3 , 1, 1, 1. s sŽ4 , 2. q sŽ4 , 1, 1. q sŽ3 , 3. q 2 sŽ3 , 2, 1. q sŽ3 , 1, 1, 1. q sŽ2 , 2, 2. q sŽ2 , 2, 1, 1. s PŽ4, 2. , as predicted by Theorem 9. In general, comparison of Theorem 9 and Theorem 11 gives rise to various expansions of P-functions as quadratic polynomials in S-functions. Thus, D 1 s sŽ1. q sŽ1 , 1, 1. Dc 2 Ž s Ž1. . q sŽ1 , 1, 1, 1, 1. Dc 2 Ž s Ž2. . q sŽ2 , 2, 1. Dc 2 Ž s Ž1 , 1. . q sŽ1, 1, 1, 1, 1, 1, 1. Dc 2 Ž s Ž3. . q sŽ2 , 2, 1, 1, 1. Dc 2 Ž s Ž2 , 1. . y sŽ4 , 2, 1. Dc 2 Ž s Ž1 , 1 , 1. . q ??? and PŽ4, 2. s D 1 sŽ3 , 2. s sŽ1. sŽ3 , 2. q sŽ1 , 1, 1. sŽ3. q sŽ2 , 2, 1. sŽ1. . At this point it is worth recalling that the S-function expansion of Dc 2 Ž sl . sm m is explicitly known. Indeed, define coefficients dln by writing Dc 2 Ž sl . sm s

Ý e 2 Ž mrn . dlnm sn .

Ž 150.

n

It has been shown in w3x that the generalized Littlewood]Richardson m coefficient dln is equal to the number of Yamanouchi domino tableaux of shape mrn and weight l. Therefore the formula for D k sm obtained by application of Theorem 11 has an explicit combinatorial description. This leads to interesting combinatorial identities relating numbers of generalized Young tableaux of various kinds, which are discussed in w21x.

10. CONCLUDING REMARKS As mentioned in the Introduction, the set of formulas discovered in w16x constitutes the starting point of this work. Let us recall this result.

142

LECLERC AND LEIDWANGER

THEOREM 12 w16x. Then

Let l be a bar partition of length k and set l s r k q m. Pl s

Ý sŽ r q2 n . Dc Ž s . sm . X

k

n

Ž 151.

n

2

In other words, defining a differential operator Dk by Dk s

Ý sŽ r q2 n . Dc Ž s . , n

k

X

2

Ž 152.

n

one has Dk sm s Pr kq m for all partitions m of length F k. It is easy to see that the condition on m ensures that c k Ž m . s l and that no partition n of length ) k will give a nonzero contribution to Eq. Ž151.. On the other hand, when l Ž n . F k, one has < <

Ž y1. n e 2 Ž Jk Ž n X . . s J k Ž n X . s sŽ r kq2 n . X ,

Ž 153.

and therefore Dk sm s D k sm . However, this is no longer true in general when l Ž m . ) k, and one can check, for example, that Dk ŽSym. is not contained in T. We wish to mention that the second part of this work generalizes nicely to n ) 2. Indeed, let r denote an n-core partition and let i be a fixed integer between 0 and n y 1. Then, given a partition l, one can define a partition Jr Ž l, i . by specifying its n-core and n-quotient: Jr Ž l , i . [ Ž r ; Ž 0 . , . . . , l , . . . , Ž 0 . . .

Ž 154.

Here l appears as the ith coordinate of the n-quotient of Jr Ž l, i .. Now define Dr Ž i . [

< <

Ý Ž y1. l e n Ž Jr Ž lX , i . . s J Ž l , i. Dc Ž s . . l

r

X

n

l

Ž 155.

Then, using the methods of Section 9, it can be shown that D r Ž i . maps Sym to the subalgebra T Ž n. s Cw pi ; i f 0 mod n x. The images of S-functions under D r Ž i . do not seem to belong to known families of symmetric functions, and, for instance, their relation to the basis of Hall]Littlewood functions at an nth root of 1 is not as simple as in the case n s 2. We think that it might be an interesting problem to study these functions in more detail.

SCHUR FUNCTIONS AND AFFINE LIE ALGEBRAS

143

ACKNOWLEDGMENTS We thank A. Lascoux and J.-Y. Thibon for numerous and fruitful discussions on the matters of this paper. We are also grateful to P. Pragacz for his interest in this work and for sending us his preprint w32x. The computer algebra systems SCHUR w38x, developed by B. Wybourne, and ACE w36, 37x, developed by S. Veigneau, have been very helpful at various stages of this work. We would like to thank S. Veigneau for his help in using ACE.

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