A Determinantal Formula for Supersymmetric Schur Polynomials

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A Determinantal Formula for Supersymmetric Schur Polynomials

E.M. MOENS ElsM.Moens@UGent.be

J. VAN DER JEUGT Joris.VanderJeugt@UGent.be

Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium

Received January 8, 2002; Revised September 25, 2002

Abstract. We derive a new formula for the supersymmetric Schur polynomials_λ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebragl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression fors_λ(x/y).

This new expression gives rise to a determinantal formula fors_λ(x/y). In particular, the denominator identity forgl(m/n) corresponds to a determinantal identity combining Cauchy’s double alternant with Vandermonde’s determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.

Keywords: supersymmetric Schur polynomials, Lie superalgebragl(m/n), characters, covariant tensor repre- sentations, determinantal identities

1. Introduction

This paper deals with a new formula for supersymmetric Schur polynomials s_λ(x/y), parametrized by a partitionλ, and symmetric in two sets of variablesx = (x1, . . . ,xm) andy=(y1, . . . ,yn). Supersymmetric Schur polynomials, or S-functions, appeared for the first time in the work of Berele and Regev [3], who also showed that these polynomials are closely related to characters of certain representations of the Lie superalgebragl(m/n) orsl(m/n). More precisely, the characters of irreducible covariant tensor representations of gl(m/n) are supersymmetric S-functions (just as the characters of irreducible covariant tensor representations of gl(m) are ordinary Schur functions). It is in the context of gl(m/n) representations that our new formula for supersymmetric S-functions was discov- ered, as a consequence of a certain character formula of Kac and Wakimoto [6]. In their general study of characters for classical Lie superalgebras and affine superalgebras, Kac and Wakimoto [6] developed the idea of so-called tame representations. These tame representations allow for the construction of an explicit character formula. Unfortunately, it is not easy to give a simple characterization of which representations are tame. In this paper, we show that the covariant tensor representations of gl(m/n) are indeed tame. As a consequence we can apply the character formula of Kac and Wakimoto, and obtain a new expression for supersymmetric S-functions. This new expression is not particularly elegant, but after certain manipulations we show that it is equivalent to a nice determinantal formula. Once

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this determinantal formula fors_λ(x/y) was found, we realized that its validity could also be proved independently. Indeed, in [10, Section I.3, Exercise 23] or [13] it is shown that the supersymmetric S-functions are characterized by four properties. Using our determinantal expression, it is possible to show that these four properties are indeed satisfied. Finally, one can also prove the determinantal formula using a double Laplace expansion and a special case of the so-called Sergeev-Pragacz formula [22].

The structure of this paper is as follows. In the first section, we fix the notation and recall some known formulas for ordinary and supersymmetric Schur functions. The simplest form of our new determinantal formula fors_λ(x/y) is already given in (1.17). Furthermore, we point out that the case λ = 0 gives rise to an interesting determinantal identity, which can be called “the denominator identity forgl(m/n)”. Section 2 is devoted to showing that covariant tensor representations of the Lie superalgebragl(m/n) are tame, and to applying the Kac-Wakimoto character formula. This section uses a lot of representation theory, and closes with a formula for the character, i.e. a formula fors_λ(x/y). This formula is not in an optimal form, and in Section 3 we use a number of intricate but elementary manipulations to derive from this form our main result, Theorem 3.4, giving the determinantal expression fors_λ(x/y). In Section 4, we show that the determinantal expression satisfies the four characteristic properties of supersymmetric S-functions, thus yielding an independent proof of the determinantal formula. Finally, in Section 5, we provide a straightforward proof, without using representation theory, based on the formula of Sergeev-Pragacz and Laplace’s theorem. So in fact we provide three proofs: one in the context of representation theory ofgl(m/n), one by means of the characteristic properties, and the last one using the formula of Sergeev-Pragacz. A reader who is not so familiar with representation theory can easily skip Section 2 (apart from Definition 2.2). In fact, we could have pre- sented our main result without any reference to representation theory. However, we did not want to hide the natural background of this new formula, and therefore we have chosen to present also its representation theoretic origin. This is also clear from the determinantal formula itself: in this expression, the so-called (m,n)-indexk of λ is crucial; this definition ofk has a natural interpretation in representation theory ofgl(m/n) (in terms of atypicality).

Letλ = (λ1, . . . , λp,0,0, . . .) be a partition of the nonnegative integerN, withλ1 ≥

· · · ≥λp>0 and

iλi = |λ| = N. The numberp =(λ) is the length ofλ. The Young diagramF^λof shapeλis the set of left-adjusted rows of squares withλisquares (or boxes) in theith row (reading from top to bottom). For example, the Young diagram of (5,2,1,1) is given by:

F^λ =

(1.1) As usual,λ denotes the conjugate toλ; e.g. ifλ = (5,2,1,1) thenλ = (4,2,1,1,1).

Denote byS(x) the ring of symmetric functions inmindependent variablesx1, . . . ,xm[10].

The Schur functions or S-functionss_λform aZ-basis ofS(x). There are various ways to define the S-functions [10]. For a partition λwith(λ) ≤ m, there is the determinantal

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formula (as a quotient of two alternants):

s_λ(x)= det

x_i^λ^j⁺^m⁻^j

1≤i,j≤m

det x^m_i ⁻^j

1≤i,j≤m

. (1.2)

The numerator can be rewritten as det

x_i^λ^j⁺^m⁻^j

1≤i,j≤m=det(x^λ+δ^m)=

w∈Sm

ε(w)w(x^λ+δ^m), (1.3)

whereSmis the symmetric group acting onx=(x1, . . . ,xm),ε(w) the signature ofw, and

δm=(m−1,m−2, . . . ,1,0). (1.4)

Thus (1.2) is essentially equivalent to Weyl’s character formula for the Lie algebragl(m), where an irreducible representation ofgl(m) is characterized by a partitionλand its character is given by s_λ(x). The denominator on the right hand side of (1.2) is the Vandermonde determinant, equal to the product

1≤i<j≤m(xi−xj); this is Weyl’s denominator formula forgl(m).

Whenλ=(r),s_λ(x) is the complete symmetric functionhr(x), and whenλ=(1^r),s_λ(x) is the elementary symmetric functioner(x). The Jacobi-Trudi formula and the N¨agelsbach- Kostka formula gives_λin terms of these [10]:

s_λ(x)=det

h_λ_i₋i+j(x)

1≤i,j≤(λ)=det

e_λ_i₋i+j(x)

1≤i,j≤(λ). (1.5) Other determinantal formulas fors_λare Giambelli’s formula [10] and the ribbon formula [9].

Finally, recall there is a combinatorial formula fors_λ(x) as a sum of monomials summed over all column-strict Young tableaux of shapeλ[10].

All these formulas, except (1.2), have their extensions to skew Schur functionss_λ/µ(x), whereλandµare two partitions withλi ≥µifor alli.

Let us now recall some notions ofsupersymmetric S-functions[3, 7, 17]. The ring of doubly symmetric polynomials inx = (x1, . . . ,xm) andy = (y1, . . . ,yn) isS(x,y) = S(x)⊗ZS(y). An element p ∈ S(x,y) has thecancellation property if it satisfies the following: when the substitutionx1 =t,y1 = −t is made in p, the resulting polynomial is independent oft. We denoteS(x/y) the subring ofS(x,y) consisting of the elements satisfying the cancellation property. The elements ofS(x/y) are the supersymmetric polynomials [17].

The complete supersymmetric functionshr(x/y) belong toS(x/y), and are defined by hr(x/y)=

r k=0

hr−k(x)ek(y). (1.6)

The following gives a first formula for the supersymmetric S-functions:

s_λ(x/y)=det

h_λ_i₋i+j(x/y)

1≤i,j≤(λ). (1.7)

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The polynomialss_λ(x/y) are identically zero whenλm+1 >n. Denote byHm,nthe set of partitions withλm+1 ≤n, i.e. the partitions (with their Young diagram) inside the (m,n)- hook. Stembridge [17] showed that the set ofs_λ(x/y) withλ ∈ Hm,n form aZ-basis of S(x/y).

There exist a number of other formulas for the supersymmetric S-functions. One is a combinatorial formula in terms of supertableaux of shapeλ, see [3]. From the combinatorial formula, one finds expansions ofs_λ(x/y) in terms of ordinary S-functions:

s_λ(x/y)=

µ

s_µ(x)s(λ/µ)(y)=

µ,ν

c^λ_µ,νs_µ(x)s_ν(y), (1.8)

wherec^λ_µ,νare the Littlewood-Richardson coefficients [10].

Just as the functions s_λ(x) are characters of simple modules of the Lie algebragl(m), the supersymmetric S-functions are characters of (a class of) simple modules of the Lie superalgebra gl(m/n) [3]. In this context, a different formula for s_λ(x/y) was found by Sergeev (see [12]) and in [19]; the first proof of this formula was given by Pragacz [12]. To describe the so-called Sergeev-Pragacz formula, letλbe a partition withλm+1≤n. Consider the Young diagram F^λ, letF^κbe the part ofF^λthat falls within them×nrectangle, and let F^τ, resp. F^η, be the remaining part to the right, resp. underneath this rectangle; i.e.

λ=(κ+τ)∪η. This is illustrated, form =5,n =8 andλ=(11,9,4,3,2,2,2,1), as follows:

F^λ= hence

κ = (8,8,4,3,2)

τ = (3,1)

η = (2,2,1) (1.9)

Then, the Sergeev-Pragacz formula fors_λ(x/y) is given by

s_λ(x/y)=D₀⁻¹

w∈S_m×S_n

ε(w)w

x^τ+δ^my^η^+δⁿ

(i,j)∈F^κ

(xi+yj) , (1.10) where (i,j) ∈ F^κ if and only if the box with row-indexi (read from left to right) and column-index j (read from top to bottom) belongs toF^κ, and

D0=

1≤i<j≤m

(xi−xj)

1≤i<j≤n

(yi−yj). (1.11)

This formula is useful for the computation ofs_λ(x/y), and even for the computation of Littlewood-Richardson coefficients [2, 18]. Note that for the special case that λm ≥ n, (1.10) becomes:

s_λ(x/y)=s_τ(x)s_η(y) m i=1

n j=1

(xi+yj). (1.12)

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This is the case offactorization. This formula was first derived by Berele and Regev [3, Theorem 6.20] and will referred to as the Berele-Regev formula. Furthermore, from (1.10) one easily deduces duality:

s_λ(x/y)=s_λ(y/x). (1.13)

Observe thatD0in (1.10) is just Weyl’s denominator forgl(m)⊕gl(n). So whenλ=0, (1.10) does not yield a new denominator identity related to gl(m/n); it only gives the denominator identity forgl(m) andgl(n).

In this paper, we shall give a new formula fors_λ(x/y). In its simplest form, this yields a new determinantal formula for supersymmetric S-functions. Furthermore, this formula yields a genuine denominator identity related togl(m/n). Let us briefly describe one of the forms of the new formula. First, we introduce some new notations. Let

D(x)=

1≤i<j≤m

(xi−xj) and E(x,y)= m i=1

n j=1

(xi−yj) (1.14)

and defineDby

D=

1≤i<j≤m(xi−xj)

1≤i<j≤n(yi−yj) m

i=1n

j=1(xi+yj) = D(x)D(y)

E(x,−y). (1.15)

Letλbe a partition withλm+1≤n, i.e.λ∈Hm,n, and put

k=min{j |λj+m+1−j ≤n}; (1.16)

sinceλm+1≤n, we have that 1≤k≤m+1. Then the new formula reads

s_λ(x/y)=(−1)^mn⁻^m⁺^k⁻¹D⁻¹det

R X_λ Y_λ 0

, (1.17)

where the (rectangular) blocks of the determinant are given by

R= 1

xi+yj

1≤i≤m,1≤j≤n

, X_λ=

x_i^λ^j⁺^m⁻ⁿ⁻^j

1≤i≤m,1≤j≤k−1, Y_λ=

y^λ_jⁱ⁺ⁿ⁻^m⁻ⁱ

1≤i≤n−m+k−1,1≤j≤n.

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For example, letm =3, n = 5 andλ = (7,2,2,2,1). Thenκ = (5,2,2),τ = (2), η=(2,1) andk=2. Thus, according to formula (1.17),

s(7,2,2,2,1)= −Ddet







1 x1+y1

1 x1+y2

1 x1+y3

1 x1+y4

1 x1+y5 x₁⁴

1 x2+y1

1 x2+y2

1 x2+y3

1 x2+y4

1 x2+y5 x₂⁴

1 x₃+y₁

1 x₃+y₂

1 x₃+y₃

1 x₃+y₄

1 x₃+y₅ x₃⁴ y₁⁶ y₂⁶ y₃⁶ y⁶₄ y₅⁶ 0 y₁⁴ y₂⁴ y₃⁴ y⁴₄ y₅⁴ 0 y₁⁰ y₂⁰ y₃⁰ y⁰₄ y₅⁰ 0







. (1.18)

Whenλ =0 it follows from (1.7) or (1.10) thats_λ(x/y)=1. The new formula (1.17) gives rise to a denominator identity forgl(m/n). Supposem≤n(m≥nis similar); when λ=0, it follows from (1.16) thatk=1. So theX_λ-block and 0-block disappear in (1.17).

Changing the order of theR-block andY_λ-block, implies

det







y₁ⁿ⁻^m⁻¹ · · · y_nⁿ⁻^m⁻¹

... ...

y₁⁰ · · · y_n⁰

1

x1+y1 · · · _x₁₊¹_y_n

... ...

1

x_m+y₁ · · · _x_m¹₊_y_n







=

1≤i<j≤m(xi−xj)

1≤i<j≤n(yi−yj) m

i=1

n

j=1(xi+yj) . (1.19)

Clearly, when m = n, this is simply Cauchy’s double alternant; whenm = 0, it is just Vandermonde’s determinant. When 0<m<n, it is a combination of the two. These type of determinants have already been encountered in a different context [1] (we found this reference in [8]); here they are for the first time related to a denominator identity.

2. Covariant modules for the Lie superalgebragl(m/n)

For general theory on classical Lie superalgebras and their representations, we refer to [4, 5, 14]; for representations of the Lie superalgebragl(m/n), see [19–21].

Letg=gl(m/n),h⊂gits Cartan subalgebra, andg=g₋₁⊕g₀⊕g₊₁the consistent Z-grading. Note thatg₀ = gl(m)⊕gl(n), and putg⁺ = g₀⊕g₊₁ andg⁻ = g₀⊕g₋₁. The dual spaceh^∗ has a natural basis{1, . . . , m, δ1, . . . , δn}, and the roots ofgcan be expressed in terms of this basis. Letbe the set of all roots,0the set of even roots, and 1the set of odd roots. One can choose a set of simple roots (or, equivalently, a triangular decomposition), but note that contrary to the case of simple Lie algebras not all such choices are equivalent. The so-calleddistinguished choice[4] for a triangular decomposition g=n⁻⊕h⊕n⁺is such thatg₊₁⊂n⁺andg₋₁⊂n⁻. Thenh⊕n⁺is the corresponding distinguished Borel subalgebra, and₊ the set of positive roots. For this choice we have

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explicitly:

0,+= {i−j|1≤i < j ≤m} ∪ {δi−δj |1≤i < j ≤n},

(2.1) 1,+= {βi j =i−δj |1≤i ≤m, 1≤ j ≤n},

and the corresponding set of simple roots (the distinguished set) is given by

= {1−2, . . . , m−1−m, m−δ1, δ1−δ2, . . . , δn−1−δn}. (2.2) Thus in the distinguished basis there is only one simple root which is odd. As usual, we put

ρ0= 1 2

α∈0,+

α , ρ1= 1 2

α∈1,+

α , ρ=ρ0−ρ1. (2.3)

There is a symmetric form (,) onh^∗induced by the invariant symmetric form ong, and in the natural basis it takes the form (i, j)=δi j, (i, δj)=0 and (δi, δj)= −δi j. The odd roots are isotropic: (α, α)=0 ifα∈1.

The Weyl group ofgis the Weyl groupWofg₀, hence it is the direct product of symmetric groupsSm×Sn. Forw∈W, we denote byε(w) its signature.

Let∈h^∗; theatypicalityof, denoted by atyp(), is the maximal number of linearly independent rootsβisuch that (βi, βj)=0 and (, βi)=0 for alliand j [6]. Such a set {βi}is called a-maximal isotropic subset of.

Given a set of positive roots+of, and a simple odd rootα, one may construct a new set of positive roots [6, 11] by

₊=(₊∪ {−α})\ {α}. (2.4)

The set₊is called a simple reflection of+. Since we use only simple reflections with respect to simple odd roots,0,+remains invariant, but1,+will change and the newρis given by:

ρ=ρ+α. (2.5)

LetV be a finite-dimensional irreducibleg-module. Such modules areh-diagonalizable with weight decomposition V = ⊕µV(µ), and the character is defined to be chV =

µdimV(µ)e^µ, wheree^µ(µ∈h^∗) is the formal exponential. Consider such a moduleV. If we fix a set of positive roots+, we may talk about the highest weightofV and about the correspondingρ. If₊is obtained from+by a simpleα-reflection, whereαis odd, anddenotes the highest weight ofV with respect to₊, then [6]

=−αif (, α)=0; =if (, α)=0. (2.6)

Ifαis asimpleodd root from+then (ρ, α)= ¹₂(α, α)= 0 [6, p. 421], and therefore, following (2.5) and (2.6):

+ρ=+ρ if (+ρ, α)=0,

(2.7) +ρ=+ρ+α if (+ρ, α)=0.

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From this, one deduces that for theg-moduleV, atyp(+ρ) is independent of the choice of+; then atyp(+ρ) is referred to as the atypicality ofV (if atyp(+ρ)=0,V is typical, otherwise it is atypical). If one can choose a (+ρ)-maximal isotropic subsetS in₊such thatS ⊂⊂₊(is the set of simple roots with respect to₊), then the g-moduleVis calledtame, and a character formula is known due to Kac and Wakimoto [6].

It reads:

chV = j⁻¹e^−ρR⁻¹

w∈W

ε(w)w

e^+ρ

β∈S

(1+e^−β)⁻¹ , (2.8)

where

R=

α∈0,+

(1−e^−α)

α∈1,+

(1+e^−α) (2.9)

and j is a normalization coefficient to make sure that the coefficient ofe on the right hand side of (2.8) is 1.

The rest of this section is now devoted to a particular class of finite-dimensional irreducible g-modules, thecovariantmodules, and to showing that these modules are tame. Berele and Regev [3], and Sergeev [15], showed that the tensor product of N copies of the natural (m+n)-dimensional representation ofg=gl(m/n) is completely reducible, and that the irreducible componentsV_λcan be labeled by a partitionλof N such thatλis inside the (m,n)-hook, i.e. such thatλm+1 ≤ n. These representations V_λ are known as covariant modules. Let us first considerV_λin the distinguished basis fixed by (2.2). Then, the highest weightλofV_λin the standard-δ-basis is given by [19]

_λ=λ11+ · · · +λmm+ν₁δ1+ · · · +ν_nδn, (2.10) whereν_j =max{0, λ_j−m}for 1≤ j ≤ n. Let us consider the atypicality ofV_λ, in the distinguished basis. For this purpose, it is sufficient to compute the numbers (λ+ρ, βi j), withβi j = i −δj, for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and count the number of zeros.

It is convenient to put the numbers (λ +ρ, βi j) in a (m ×n)-matrix (the atypicality matrix [19, 20]), and give the matrix entries in the (m,n)-rectangle together with the Young frame ofλ. This is illustrated here for example (1.9):

7 9

–5 –4 –3 –2 –1 0 3 5

–3 –2 –1 0 1 2 5 7

–1 0 1 2 3 4 7 9

5 6 8 10

9 10 11 12 13 16 18

13 15

8

(2.11)

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So for this example, atyp(λ+ρ)=3, since the atypicality matrix contains three zeros. In the following, we shall sometimes refer to (i,j) as the position ofβi j; the position is also identified by a box in the (m,n)-rectangle. The row-indexkof the zero with smallest row index will play an important role (see Definition 2.2).

The maximal isotropic subsetS_λconsist now of those odd rootsβi jwith (_λ+ρ, βi j)= 0, i.e. it corresponds to the zeros in the atypicality matrix. From the combinatorics of atypicality matrices [20], it follows that

S_λ = {βi,λi+m+1−i |1≤i ≤m, 1≤λi+m+1−i ≤n}.

That is to say, one finds the zeros in the atypicality matrix as follows: on rowmin column λm+1; on rowm−1 in columnλm−1+2; in general one has, see also (2.11):

Proposition 2.1 The atypicality matrix has its zeros on row i and in columnλi+m−i+1 (i =m,m−1, . . .)as long as these column indices are not exceeding n.

Clearly, S_λ is in general not a subset of the set of simple roots(sincecontains only one odd root,m−δ1). So formula (2.8) cannot be applied. The purpose is to show that there exists a sequence of simple oddα-reflections such that for the new₊, where the moduleV_λhas highest weight, there exists a (+ρ)-maximal isotropic subsetS withS ⊂⊂₊.

Definition 2.2 Forλ∈Hm,n,the (m,n)-index ofλis the number

k=min{i |λi+m+1−i ≤n}, (1≤k≤m+1). (2.12) In what follows,kwill always denote this number; it will be a crucial entity for our devel- opments. By Proposition 2.1,m−k+1 is the atypicality ofV_λ. Ifk=m+1, thenS_λ = ∅ andV_λis typical and trivially tame. Thus in the following, we shall assume thatk≤m. To begin with,+corresponds to the distinguished choice, andis the distinguished set of simple roots (2.2). The highest weight ofV_λis given byλ. Denote⁽¹⁾=λ,ρ⁽¹⁾ =ρ and⁽¹⁾ =. Now we perform a sequence of simple oddα⁽ⁱ⁾-reflections; each of these reflections preserve 0,+ but may change ⁽ⁱ⁾+ρ⁽ⁱ⁾ and⁽ⁱ⁾. Denote the sequence of reflections by:

⁽¹⁾+ρ⁽¹⁾, ⁽¹⁾−→α⁽¹⁾⁽²⁾+ρ⁽²⁾, ⁽²⁾−→ · · ·α⁽²⁾ −→α⁽^f⁾+ρ, (2.13) where, at each stage, α⁽ⁱ⁾ is an odd root from⁽ⁱ⁾. For givenλ, consider the following sequence of odd roots (with positions on rowm, rowm−1,. . ., rowk):

rowm: βm,1, βm,2, . . . , βm,λk−k+m

rowm−1 : βm−1,1, βm−1,2, . . . , βm−1,λk−k+m−1

... ...

rowk: βk,1, βk,2, . . . , βk,λk

(2.14)

in this particular order (i.e. starting withβm,1and ending withβk,λk). Then we have:

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Lemma 2.3 The sequence(2.14)is a proper sequence of simple odd reflections forλ, i.e.α⁽ⁱ⁾is a simple odd root from⁽ⁱ⁾. At the end of the sequence,one finds:

=

1−2, 2−3, . . . , k−2−k−1, k−1−δ1, δ1−δ2, δ2−δ3, . . . , δλk−1−δλk, δ_λk −k, k−δ_λk+1, δ_λk+1−k+1, k+1−δ_λk+2, . . . , δ_λk+m−k−m,

m−δλk+m+1−k, δλk+m+1−k−δλk+m+2−k, . . . , δn−1−δn

. (2.15)

Furthermore,

+ρ=_λ+ρ+ m i=k+1

λk−k+i j=λi+1

βi,j. (2.16)

Proof: Let us consider, in the first stage, the reflections with respect to the roots in rowm.

Clearly,α⁽¹⁾ =βm,1 =m−δ1is an odd root from⁽¹⁾ =. Performing the reflection with respect toβm,1implies that⁽²⁾containsm−1−δ1, δ1−m, m−δ2as simple odd roots. Thus⁽²⁾containsα⁽²⁾ =βm,2. A reflection with respect toβm−2implies that⁽³⁾ containsm−1−δ1, δ2−m, m−δ3 as simple odd roots. So this process continues, and afterλk−k+msuch reflections (i.e. at the end of rowm), we have

⁽^λ^k⁻^k⁺^m⁺¹⁾=

1−2, . . . , m−2−m−1, m−1−δ1,

δ1−δ2, . . . , δλk−k+m−1−δλk−k+m, δλk−k+m−m, m−δ_λk−k+m+1, δ_λk+m+1−k−δ_λk+m+2−k, . . . , δn−1−δn

. (2.17)

Observe that this process can continue sinceλk−k+m<nby definition of the (m,n)-index kofλ. So after the first stage (i.e. after the reflections with respect to odd roots of rowm) there are three odd roots in⁽^λ^k⁻^k⁺^m⁺¹⁾, and the set is ready to continue the reflections with respect to the elements of rowm−1, sinceβm−1,1 belongs to⁽^λ^k⁻^k⁺^m⁺¹⁾. Since at each stageα⁽ⁱ⁾is a simple odd root, (2.7) implies

⁽ⁱ⁺¹⁾+ρ⁽ⁱ⁺¹⁾=⁽ⁱ⁾+ρ⁽ⁱ⁾ if

⁽ⁱ⁾+ρ⁽ⁱ⁾, α⁽ⁱ⁾

=0, ⁽ⁱ⁺¹⁾+ρ⁽ⁱ⁺¹⁾=⁽ⁱ⁾+ρ⁽ⁱ⁾+α⁽ⁱ⁾ if

⁽ⁱ⁾+ρ⁽ⁱ⁾, α⁽ⁱ⁾

=0. Examining this explicitly for the elements of rowmyields

⁽^λ^k⁻^k⁺^m⁺¹⁾+ρ⁽^λ^k⁻^k⁺^m⁺¹⁾=_λ+ρ+^λ^k⁻

k+m

j=λm+1

βm,j. (2.18)

Ifk=mthe lemma follows. Ifk<mthe process continues; suppose this is the case. But now we are in a situation where the elements of rowm−1 play completely the same role as those of rowmin the first stage. This means that at the end of the second stage, the new

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set of simple roots is given by ⁽ⁱ⁾ =

1−2, . . . , m−3−m−2, m−2−δ1, δ1−δ2, . . . , δ_λk−k+m−2−δ_λk−k+m−1, δλk−k+m−1−m−1, m−1−δλk−k+m, δλk−k+m−m, m−δλk−k+m+1, δλk+m+1−k−δλk+m+2−k, . . . , δn−1−δn

, (2.19)

and the new⁽ⁱ⁾+ρ⁽ⁱ⁾by

⁽ⁱ⁾+ρ⁽ⁱ⁾ =λ+ρ+^λ^k⁻

k+m

j=λm+1

βm,j+^λ^k⁻

k+m−1

j=λm−1+1

βm−1,j (2.20)

(the last addition follows by inspecting the atypicality matrix). Continuing with the remaining stages (i.e. rows in (2.14)) leads to (2.15) and (2.16).

Corollary 2.4 The covariant module V_λis tame.

Proof: Having performed the simple odd reflections (2.14), one can compute the atypicality matrix for+ρusing (2.16). This gives:

(+ρ, βi j)=0 for all (i,j) withk≤i≤m, λk+1≤ j ≤λk+m+1−k.

(2.21) Therefore the set

S =

k−δλk+1, k+1−δλk+2, . . . , m−δλk+m+1−k

(2.22)

is a (+ρ)-maximal isotropic subset. Furthermore,S ⊂, see (2.15).

Let us illustrate some of these notions for Example (1.9).

(2.23)

In (2.23)(a) the positions marked with “i” refer to the (+ρ)-maximal isotropic set (2.22).

The first element, with row indexk, is simply the position of the box just to the right of the Young diagramF^λin rowk. For the remaining positions one continues in the direction of the diagonal until one reaches rowm. For convenience, let us refer to these positions as

“the isotropic diagonal.” The positions of the odd roots that have been used for the sequence

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of reflections to go from_λandtoandare marked by “x” in (2.23)(a). So, they are simply all positions to the left of the isotropic diagonal. Finally, (2.23)(b) shows the positions of thoseβi jthat appear on the right hand side of (2.16); they are marked by “*”.

These are all positions to the left of the isotropic diagonal that are not insideF^λ. One can see from these examples that the (m,n)-indexkdetermines all other necessary ingredients.

We are now in a position to evaluate the character formula (2.8), chV_λ= j⁻¹e^−ρR⁻¹

w∈W

ε(w)w

e^+ρ

β∈S

(1+e^−β)⁻¹ , (2.24)

where R is given by (2.9) with1,+ replaced by1,+ (0,+ remains unchanged). The mn elements of1,+ are±βi j, where one must take the minus-sign ifβi j appears in the list (2.14) (i.e. if its position is marked by “x” in (2.23)(a)) and the plus-sign otherwise.

However, all this information is not necessary here, since by definition ofρandR e^−ρR⁻¹=e^−ρR⁻¹.

Putting, as usual in this context,

xi=eⁱ, yj=e^δ^j (1≤i ≤m,1≤ j ≤n), (2.25) one has

e^−ρR⁻¹=D⁻¹ m i=1

x_i^(m⁻ⁿ⁻¹⁾^/² n

j=1

y⁽ⁿ_j⁻^m⁻¹⁾^/²,

withDgiven in (1.15). Using (2.16) and (2.22), one finds e^+ρ

β∈S

(1+e^−β)⁻¹ = m i=1

x_i⁽¹⁻^m⁺ⁿ⁾^/² n j=1

y⁽¹_j⁺^m⁻ⁿ⁾^/²

k−1

i=1

x_i^λⁱ⁺^m⁻ⁱ⁻ⁿ

× m i=k

x_i^λ^k⁺^m⁺¹⁻^k⁻ⁿ

λk

j=1

y^λ

j+n−j−m j

λk+m+1−k j=λk+1

yⁿ_j^−λ^k⁻^m⁻¹⁺^k

× n j=λk+m+2−k

yⁿ_j⁻^j _m

i=k

(xi+y_λ_k₊i+1−k). (2.26) In order to rewrite this in a more appropriate form, let us introduce some further notation related to the partitionλ∈Hm,n. Clearly, the (m,n)-indexkdefined in (2.12) plays again an essential role. Related to this, let us also put

l=λk+1, r =n−m+k−l. (2.27)

Now we have

chV_λ= j⁻¹D⁻¹

w∈W

ε(w)w(t_λ),