ORTHOGONAL DIVIDED DIFFERENCES AND SCHUBERT

POLYNOMIALS, P̃-FUNCTIONS, AND VERTEX OPERATORS

Alain Lascoux and Piotr Pragacz

Dedicated to Bill Fulton on his 60th birthday

Abstract

Using divided differences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided differences on some distinguished symmetric polynomials (P̃- polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials, generalizing P̃-polynomials, and well-suited to intersection theory computations, are also studied.

Contents: Introduction. 1.Divided differences. 2.Bases of polynomial rings. 3.Vertex operators. 4.Applications to P̃-polynomials and orthogonal Schubert polynomials. Appendix: results in type B. References.

Introduction

Divided differences were introduced by Newton in his famous interpolation formula (cf. [N, pp.481–483], and [L] for some historical comments).

Their importance in geometry was shown by Bernstein-Gel’fand-Gel’fand [BGG] and Demazure [D1,D2] in the context of Schubert calculus for generalized flag varieties associated with semisimple algebraic groups in the early 1970’s. More recently, simple divided differences, interpreted as correspondences in flag bundles, were extensively used in the sequence of papers [F1,F2,F3] by Fulton in the context of degeneracy loci associated with classical groups. Still another interpretation of divided differences, as Gysin morphisms in the cohomology of flag bundles associated with semisimple algebraic groups, was discussed in [P2, Sect.4] and [PR, Sect.5]. We refer to the lecture notes [FP] for an introduction.

The case of SL(n) has been developed by the first author and Schützenberger (see e.g. [LS1,LS2,LS3] and [M2]).

For other classical groups, parallel studies were done by Billey-Haiman [BH], Fomin-Kirillov [FK], Ratajski and the second author [PR], and by the authors [LP1].

Typeset by AMS-TEX

This research was supported by grant No. 5031 of French-Polish cooperation C.N.R.S.–P.A.N.

and KBN grant No. 2P03A 05112.

The present paper is a continuation of [LP1]. Here we study divided differences associated with the orthogonal groups SO(2n) and SO(2n + 1) (i.e. for types D and B). The results for type B are an immediate adaptation of the results for type C given in [LP1]. We summarize them in the appendix. We add, however, a certain new result (Theorem 9) for type B, whose C-analogue was not needed in our former paper [LP1].

Our results for type D require some new computations with vertex operators, that are furnished in Section 3 of the present paper and summarized in our main Theorem 11. In order to simplify the computations with divided differences, we display them as planar arrays, which allows us to perform some kind of “ jeu de taquin”. This offers a certain technical novelty w.r.t. our former paper [LP1]. In type C_{n} (or B_{n}), the key rôle was/is played by the divided differences of the form

(∂_{0}∂_{1} ⋯ ∂_{n−1}) ⋯ (∂_{0}∂_{1} ⋯ ∂_{n−k}) ,  (∗)

where k ≤ n. It appears that in type D_{n} a similar rôle is played by the divided differences of the form

(∂_{♡}∂_{2} ⋯ ∂_{n−1}∂_{1}∂_{2} ⋯ ∂_{n−2}) ⋯ (∂_{♡}∂_{2} ⋯ ∂_{n−2k+1}∂_{1}∂_{2} ⋯ ∂_{n−2k}) ,  (∗∗)

where k ≤ n/2. Here ∂_{i}, for i > 0, are Newton’s (simple) divided differences:

f ∂_{i} := \frac{f − f(…, x_{i+1}, x_{i}, …)}{x_{i} − x_{i+1}} ,

and moreover we set

f ∂_{0} := \frac{f − f(−x_{1}, x_{2}, …)}{−x_{1}} ,

f ∂_{♡} := \frac{f − f(−x_{2}, −x_{1}, x_{3}, …)}{−x_{1} − x_{2}} .

We compose the simple orthogonal divided differences in (*) and (**) from left to right. As the the Weyl group of type D is naturally embedded in the Weyl group of type B, the divided difference (**) can be expressed in terms of (*). Such basic relations are given in Proposition 6 and Corollary 8.

The symmetric functions which are most adapted to orthogonal divided differences are P̃-polynomials [PR], which are a variant of Schur P -polynomials.

Our paper is of an algebro-combinatorial nature but its motivation comes from geometry. The algebro-combinatorial properties studied here should be useful in Schubert calculus associated with orthogonal groups and the related degeneracy loci.

The computations of this paper are closely related to the ones in [LLT1]; we plan to develop this link in some future publication.

The algebro-combinatorial techniques, used in the present paper, are chosen to be as elementary as possible. This should help the reader, with more geometric and less algebro-combinatorial background, to read the paper. We mention however, that several results, used in the proof of Theorem 9, in the appendix, are particular instances of more general properties of Hall–Littlewood polynomials (see [LLT2] and [LP2]).

Let us remark that there is also an interesting algebra and combinatorics of “isobaric divided differences”, with associated Grothendieck polynomials (cf. [FL]).

This work has extensively used ACE ([V]) for explicit computations.

It is our pleasure and honor to dedicate the present article to the mathematician whose recent work has illuminated important connections between geometry and combinatorics.

NOTATION AND CONVENTIONS :

A vector (of length m) is a sequence [v_{1}, …, v_{m}] ∈ ℤ^{m}. We will compare vectors of the same lengths, writing

[v_{1}, …, v_{m}] ⊆ [u_{1}, …, u_{m}]

if v_{i} ≤ u_{i} for all i = 1, …, m.

Given a vector α = [α_{1}, …, α_{m}], we will write |α| for the sum of its components. A partition is an equivalence class of sequences [i_{1} ≥ ⋯ ≥ i_{m}] ∈ ℕ^{m}, where we

identify the sequences [i_{1}, …, i_{m}] with [i_{1}, …, i_{m}, 0]. We denote the corresponding partition by I = (i_{1}, …, i_{m}), by taking any representative sequence.

A part of a partition I, is a nonzero component of any sequence that represents I.

The length of a partition I is the number of its nonzero parts, denoted ℓ(I). We call a partition strict if all its parts are different. We write I ⊆ J for two partitions I and J (of possibly different lengths) if the

same relation holds for any pair of the same length representing them. All operators act, in this paper, on their left. Polynomials are usually treated as operators acting by multiplication.

1. Divided differences

Let n be a fixed (throughout the paper) positive integer.

The symmetric group (i.e. the Weyl group of type A) 𝔖_{n} is the group with generators s_{1}, …, s_{n−1} subject to the relations

s_{i}^{2} = 1,  s_{i−1} s_{i} s_{i−1} = s_{i}s_{i−1} s_{i} , s_{i}s_{j} = s_{j}s_{i} ∀i, j : |i − j| > 1 . (1.1)

We shall call s_{1}, …, s_{n−1} simple transpositions of 𝔖_{n}.

The hyperoctahedral group (i.e. the Weyl group of type B) 𝔅_{n} is an extension of 𝔖_{n} by an element s_{0} such that

s_{0}^{2} = 1, s_{0} s_{1} s_{0} s_{1} = s_{1} s_{0} s_{1} s_{0}, s_{0} s_{i} = s_{i} s_{0} for i ≥ 2.  (1.2)

The Weyl group 𝔇_{n} of type D is the extension of 𝔖_{n} by an element s_{♡} such that

s_{♡}^{2} = 1, s_{1}s_{♡} = s_{♡} s_{1}, s_{♡} s_{2} s_{♡} = s_{2}s_{♡} s_{2}, s_{♡} s_{i} = s_{i} s_{♡} for i > 2. (1.3)

The group 𝔇_{n} can be thought as a subgroup of 𝔅_{n} by sending s_{♡} to s_{0} s_{1} s_{0}.

The above three groups act on vectors of length n by

[v_{1}, …, v_{n}] s_{i} := [v_{1}, …, v_{i−1}, v_{i+1}, v_{i}, v_{i+2}, …, v_{n}]
[v_{1}, …, v_{n}] s_{0} := [−v_{1}, v_{2}, …, v_{n}]
[v_{1}, …, v_{n}] s_{♡} := [−v_{2}, −v_{1}, v_{3}, …, v_{n}]

The orbit of the vector v = [1, …, n] under W = 𝔖_{n}, 𝔅_{n} or 𝔇_{n}, is in bijection with the elements of W, and we shall code each w ∈ W by the vector [1, …, n] w, writing \overline{ı} instead of −i.

The three groups W = 𝔖_{n}, 𝔅_{n}, 𝔇_{n} also act on the ring of polynomials in n indeterminates X = {x_{1}, …, x_{n}}: the simple transposition s_{i}, i ≥ 1, exchanges x_{i} and x_{i+1}, s_{0} sends x_{1} to −x_{1}, s_{♡} sends x_{1} to −x_{2} and x_{2} to −x_{1}, the action being trivial in the non-listed cases. We shall denote by f^{w} the image of a polynomial f ∈ ℤ[X] under w ∈ W, and write x^{α}, with α = [α_{1}, …, α_{n}] ∈ ℕ^{n}, for the monomial x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}.

Let Pol be the ring of polynomials in X with coefficients in ℤ[\frac{1}{2}] (we need only division by 2). For any m ≤ n, let Sym(m|n − m) denote the subring of Pol consisting of polynomials invariant under all s_{i}, 1 ≤ i ≤ n − 1, i ≠ m, and let Sym(n) = Sym(n|0) = Sym(0|n) be the ring of symmetric polynomials. It contains as subrings Sym^{B}(n), the ring of polynomials invariant under 𝔅_{n}, and Sym^{D}(n), the invariants of 𝔇_{n}. It is easy to see that Pol is a free module over these different rings, generated by x^{α}, α ⊆ [n − 1, …, 0] (or α ⊆ [0, …, n − 1]) over Sym(n), by x^{α}, α ⊆ [2n − 1, 2n − 3, …, 1] (or α ⊆ [1, …, 2n − 3, 2n − 1]) over Sym^{B}(n), and by x^{α}, α ⊆ [2n − 2, 2n − 4, …, 2, 0] (or α ⊆ [0, 2, …, 2n − 2]) over Sym^{D}(n).

The respective elements of maximal length in each of the groups are

ω := [n, …, 1] for 𝔖_{n} ,
w_{0}^{B} := [\overline{1}, …, \overline{n}] for 𝔅_{n} ,

w_{0}^{D} := \{ \begin{matrix}[\overline{1}, & …, & \overline{n}] & n & even \\ [1, & \overline{2}, & …, & \overline{n}] & n & odd \end{matrix}

for 𝔇_{n}. We shall also need the following element of 𝔇_{n}:

υ := ωw_{0}^{D} = \{ \begin{matrix}[\overline{n}, & …, & \overline{1}] & n & even \\ [n, & \overline{n − 1}, & …, & \overline{1}] & n & odd . \end{matrix}

Relations between reduced decompositions in W can be represented planarly. By definition, a planar display will be identified with its reading from left to right and top to bottom (row–reading). We shall also use column–reading, that is, reading successive columns downwards, from left to right.

For example, we will write

2 1 2 ≡ 1 2 1

for the following equality for simple transpositions:

s_{2} s_{1} s_{1} = s_{1} s_{2} s_{1} .

Suppose that a rectangle is filled row-wise from left to right, and column-wise from bottom to top with consecutive numbers from {1, …, n − 1}.

Then one easily checks that its row–reading and column–reading produce two words which, interpreted as words in the s_{i}, are congruent modulo the Coxeter relations.

Here is an example of such a congruence :

3456 2345 1234 ≡ 3 2 1 · 4 3 2 · 5 4 3 · 6 5 4

the congruence class being conveniently denoted by the rectangle 3 4 5 6 2 3 4 5 1 2 3 4 .

More generally, the planar arrays that we shall write, will have the property that their row–reading and column–reading are congruent modulo Coxeter relations (cf. [LS2,LS3] or [EG] for a “jeu de taquin” on reduced decompositions). In this notation, one has, for any integers a, b, c, d, k : 1 ≤ a < b, c < d ≤ n, a + d = b + c, k < d − b, the congruence

\begin{matrix}b + 1 & ⋯ & b + k \\ b & ⋯ & ⋯ & ⋯ & d \\ ⋮ & & & & ⋮ \\ a & ⋯ & ⋯ & ⋯ & c \end{matrix} ≡ \begin{aligned}b & ⋯ & ⋯ & ⋯ & d \\ ⋮ & & & & ⋮ \\ a & ⋯ & ⋯ & ⋯ & c \\ & & c − k & ⋯ & c − 1 \end{aligned}  (1.4)

It is convenient to work in the group algebra of W = 𝔖_{n}, 𝔅_{n} or 𝔇_{n}. The works of Young and Weyl have stressed the rôle of the alternating sum of elements of these groups. Let, for W = 𝔖_{n}, 𝔅_{n} or 𝔇_{n},

Ω^{W} := ∑_{w∈W} (−1)^{ℓ(w)}w .  (1.5)

Using that 𝔅_{n} (resp. 𝔇_{n}) is isomorphic to the semi-direct product 𝔖_{n} ⋉ ℤ_{2}^{n} (resp. 𝔖_{n} ⋉ ℤ_{2}^{n−1}), one obtains the following factorizations in the group algebra:

Ω^{𝔅_{n}} = Ω^{𝔖_{n}} ∏_{1≤i≤n}(1 − τ_{i}) = ∏_{1≤i≤n}(1 − τ_{i}) Ω^{𝔖_{n}} ,  (1.6)

Ω^{𝔇_{n}} = \frac{1}{2} Ω^{𝔖_{n}} ( ∏_{1≤i≤n}(1 + τ_{i}) + ∏_{1≤i≤n}(1 − τ_{i}))

= \frac{1}{2}( ∏_{1≤i≤n}(1 + τ_{i}) + ∏_{1≤i≤n}(1 − τ_{i})) Ω^{𝔖_{n}} ,  (1.7)

where τ_{1} := s_{0}, and τ_{i} = s_{i−1} τ_{i−1} s_{i−1} for i > 1. The elements Ω^{W}, as operators on the ring of polynomials Pol, can be obtained from the cases of 𝔖_{2}, 𝔅_{1}, 𝔇_{2}. To see this, we first need to define simple divided differences :

Pol ∋ f ↦ f ∂_{i} := (f − f^{s_{i}})/(x_{i} − x_{i+1}), i ≥ 1 ,  (1.8)
Pol ∋ f ↦ f ∂_{0} := (f − f^{s_{0}})/(−x_{1}) ,  (1.9)
Pol ∋ f ↦ f ∂_{♡} := (f − f^{s_{♡}})/(−x_{1} − x_{2}) .  (1.10)

The ∂_{i}, ∂_{0}, ∂_{♡} satisfy the Coxeter relations (1.1)–(1.3), together with the relations

∂_{♡}^{2} = 0 = ∂_{i}^{2} for 0 ≤ i < n .  (1.11)

Therefore, to any element w of the group W, there corresponds a divided difference ∂_{w}. Any reduced decomposition s_{i_{1}} s_{i_{2}} ⋯ s_{i_{ℓ}} = w of w gives rise to a factorization ∂_{i_{1}} ∂_{i_{2}} ⋯ ∂_{i_{ℓ}} of ∂_{w} (cf. [BGG] and [D1,D2]).

We shall display divided differences planarly according to the same conventions as for products of s_{i}’s.

For example, the divided difference

∂_{0}∂_{1}∂_{2}∂_{3}∂_{0}∂_{1}∂_{2}∂_{0}∂_{1}

will be displayed as

∂_{0} ∂_{1} ∂_{2} ∂_{3}
∂_{0} ∂_{1} ∂_{2}
∂_{0} ∂_{1}

As said before, the displays that we write have the property that their row–reading is congruent to their column–reading, and thus the preceding one encodes the equality

∂_{0}∂_{1}∂_{0}∂_{2}∂_{1}∂_{0}∂_{3}∂_{2}∂_{1} = ∂_{0}∂_{1}∂_{2}∂_{3}∂_{0}∂_{1}∂_{2}∂_{0}∂_{1} .

We shall especially need the maximal divided differences ∂_{ω}, ∂_{w_{0}^{B}}, and ∂_{w_{0}^{D}}. To describe them using alternating sums of group elements, we define

Δ := ∏_{1≤i<j≤n}(x_{i} − x_{j}) = x_{1}^{n−1} ⋯ x_{n}^{0} Ω^{𝔖_{n}} ,  (1.12)

Δ^{B} := ∏_{i=1}^{n} x_{i} ∏_{n≥i>j≥1}(x_{i}^{2} − x_{j}^{2}) = \frac{1}{2^{n−1}} x^{[1,3,... ,2n−1]} Ω^{𝔅_{n}} ,  (1.13)

and Δ^{D} := ∏_{n≥i>j≥1}(x_{i}^{2} − x_{j}^{2}) = \frac{1}{2^{n−1}} x^{[2,4,... ,2n−2]} Ω^{𝔇_{n}} .  (1.14)

The Weyl character formula for types A, B, and D can be written as Lemma 1. For each of the groups W = 𝔖_{n}, 𝔅_{n} or 𝔇_{n}, the alternating sum Ω^{W}, as an operator on the ring of polynomials Pol, is related to the maximal divided difference by

Ω^{𝔖_{n}} \frac{1}{Δ} = ∂_{ω} , Ω^{𝔅_{n}} \frac{1}{Δ^{B}} = (−1)^{\binom{n}{2}} ∂_{w_{0}^{B}} , and Ω^{𝔇_{n}} \frac{1}{Δ^{D}} = (−1)^{\binom{n}{2}} ∂_{w_{0}^{D}}.

Indeed, all the operators in Lemma 1 commute with multiplication by polynomials which are invariant under W. Moreover, they decrease degree by the length of the maximal element of the group. Since Pol is a module over Sym^{W} (n) with a basis of monomials of degree strictly less than this length, except for a single monomial, it remains only to check that the actions of Ω’s and ∂’s agree on this monomial, which offers no difficulty.

2. Bases of polynomial rings

The monomials mentioned in the previous section are not an appropriate basis, when interpreted in terms of cohomology classes for the flag variety. Define, for the rest of this paper, the vector

ρ := [n − 1, …, 1, 0] .  (2.1)

Motivated by geometry, one defines recursively Schubert polynomials Y_{α}, for any sequence α ∈ ℕ^{n}, with α ⊆ ρ, by

Y_{α} ∂_{i} = Y_{β} , if α_{i} > α_{i+1} ,  (2.2)

where

β = [α_{1}, …, α_{i−1}, α_{i+1}, α_{i} − 1, α_{i+2}, …, α_{n}] ,

starting from Y_{ρ} = x^{ρ} (cf. [LS1], [M2]).

In particular, if α ∈ ℕ^{n} is weakly decreasing, then Y_{α} is equal to the monomial x^{α}.

If, on the contrary, α_{1} ≤ ⋯ ≤ α_{k} and α_{k+1} = ⋯ = α_{n} = 0, for some k ≤ n, then Y_{α} coincides with the Schur polynomial s_{λ}(x_{1}, …, x_{k}), where λ = (α_{k}, …, α_{1}).

CONVENTION: Let α ∈ ℕ^{k}. Then we shall write Y_{α} for Y_{α,0,... ,0}.

We also record, for later use, the following equality: for α = [α_{1}, …, α_{k}] ∈ ℕ^{k},

Y_{α} x_{1} ⋯ x_{k} = Y_{[α_{1}+1,... ,α_{k}+1]}.  (2.3)

On Pol there is a scalar product:

( , ) : Pol × Pol → Sym(n),

defined for f, g ∈ Pol by

(f, g) := fg ∂_{ω}.  (2.4)

There exists an involution^{1} α ↦ α′ such that

(Y_{α}^{ω}, Y_{β′}) = (−1)^{|α|}δ_{αβ} .  (2.5)

Moreover, when α, β ⊆ ρ are such that |α| + |β| = |ρ|, then one has

(Y_{α}, Y_{[n−1−β_{1},n−2−β_{2},... ,0−β_{n}]}) = δ_{αβ}.  (2.6)

We also will need Q̃-polynomials of [PR]. We set Q̃_{i} := e_{i} = e_{i}(X), the i-th elementary symmetric polynomial in X . Given two nonnegative integers i ≥ j, we adapt Schur’s definition of his Q-functions by putting

Q̃_{i,j} := Q̃_{i}Q̃_{j} + 2 ∑_{p=1}^{j}(−1)^{p}Q̃_{i+p}Q̃_{j−p} .  (2.7)

Given any partition I = (i_{1}, …, i_{k}), where we can assume k to be even, we set

Q̃_{I} := Pfaffian(M) ,  (2.8)

where M = (m_{p,q}) is the k × k skew-symmetric matrix with m_{p,q} = Q̃_{i_{p},i_{q}} for 1 ≤ p < q ≤ k.

Equivalently, for any partition I = (i_{1} ≥ i_{2} ≥ … ≥ i_{ℓ} > 0) , the polynomial Q̃_{I} = Q̃_{I}(X) is defined recurrently on ℓ by putting for odd ℓ,

Q̃_{I} := ∑_{j=1}^{ℓ}(−1)^{j−1}Q̃_{i_{j}} Q̃_{(i_{1},... ,i_{j−1},i_{j+1},... ,i_{ℓ})}  (2.9)

and for even ℓ,

Q̃_{I} := ∑_{j=2}^{ℓ}(−1)^{j}Q̃_{i_{1},i_{j}} Q̃_{(i_{2},... ,i_{j−1},i_{j+1},... ,i_{ℓ})} .  (2.10)

For any positive integer k, let ρ(k) denote the partition

ρ(k) := (k, k − 1, …, 1) .  (2.11)

The ring Sym(n) is a free module over the ring of polynomials symmetric in x_{1}^{2}, …, x_{n}^{2}, with a basis provided by the Q̃_{I}(X), where I ⊆ ρ(n) ranges over strict partitions.

As functions of x_{1}, …, x_{m}, the Q̃-polynomials can also be defined recursively by induction on m, involving now all partitions without restriction, as for Hall-Littlewood polynomials: for any strict partition I, one has

Q̃_{I}(x_{1}, …, x_{m}) = ∑_{j=0}^{ℓ(I)} x_{m}^{j}( ∑_{|I|−|J|=j} Q̃_{J}(x_{1}, …, x_{m−1})),  (2.12)

1This involution is: code(w) = α ↦ α′ = code(wω) (cf. [M2]).

where the sum is over all (i.e. not necessarily strict) partitions J ⊆ I such that I/J has at most one box in every row (cf. [PR, Prop. 4.1]). Moreover, given a partition I′ = (…, i, j, j, k, …) and denoting I = (…, i, k, …), one has the factorization property

Q̃_{I′} = Q̃_{j,j}Q̃_{I}.  (2.13)

We define, for a strict partition I,

P̃_{I} := 2^{−ℓ(I)} Q̃_{I} .  (2.14)

The ring Sym(n) is a free module over Sym^{D}(n) with a basis provided by the P̃_{I}, where I ranges over strict partitions contained in ρ(n − 1).

Now we will need the following divided difference:

∂_{v} = (∂_{♡}∂_{2} ⋯ ∂_{n−1} ∂_{1} ⋯ ∂_{n−2}) ⋯ (∂_{♡}∂_{2}∂_{3} ∂_{1}∂_{2}) ∂_{♡} n even  (2.15)

and

∂_{v} = (∂_{♡}∂_{2} ⋯ ∂_{n−1} ∂_{1} ⋯ ∂_{n−2}) ⋯ (∂_{♡}∂_{2}∂_{3}∂_{4} ∂_{1}∂_{2}∂_{3}) (∂_{♡}∂_{2}∂_{1}) n odd (2.16)

Denote by

⟨ , ⟩ : Sym(n) × Sym(n) → Sym^{D}(n)

the scalar product defined for f, g ∈ Sym(n) by

⟨f, g⟩ := fg ∂_{v}.  (2.17)

For strict partitions I, J ⊆ ρ(n − 1), one has

⟨P̃_{I}, P̃_{ρ(n−1)∖J}⟩ = (−1)^{\binom{n}{2}} δ_{IJ},  (2.18)

where ρ(n − 1) ∖ I is the strict partition whose parts complement the parts of I in {n − 1, n − 2, …, 1} (cf. [PR]).

Consequently, the polynomial ring Pol = ℤ[\frac{1}{2}][x_{1}, …, x_{n}] is a free Sym^{D}(n)- module with a basis Y_{α} P̃_{I}, where α ranges over subsequences contained in ρ and I runs over all strict partitions contained in ρ(n − 1). Note that the element of maximal degree of this basis is x^{ρ}P̃_{ρ(n−1)}. Let

[ , ] : Pol × Pol → Sym^{D}(n)

be a scalar product, defined for f, g ∈ Pol by

[f, g] := fg ∂_{w_{0}^{D}}.  (2.19)

One has, for α, β ⊆ ρ and strict partitions I, J ⊂ ρ(n − 1),

[Y_{α}^{ω} P̃_{I}, Y_{β′} P̃_{ρ(n−1)∖J}] = (−1)^{|α|+\binom{n}{2}} δ_{αβ} δ_{IJ} .  (2.20)

(See (2.5).)

Let Y = {y_{1}, …, y_{n}} be a second set of indeterminates of cardinality n. The symbol ≡ will mean: “congruent modulo the ideal generated by the relations f(x_{1}^{2}, …, x_{n}^{2}) = f(y_{1}^{2}, …, y_{n}^{2}), f ∈ Sym(n), together with x_{1} ⋯ x_{n} = y_{1} ⋯ y_{n}”.

Following Fulton [F2,F3], define

\begin{aligned}F (X , Y) & := |P̃_{n+j−2i}(X) + P̃_{n+j−2i}(Y)|_{1≤i,j≤n−1} \\ & = | \begin{matrix}P̃_{n−1}(X) + P̃_{n−1}(Y) & 0 & … \\ P̃_{n−3}(X) + P̃_{n−3}(Y) & P̃_{n−2}(X) + P̃_{n−2}(Y) & … \\ ⋮ & ⋮ & ⋮ \\ & & 1 & P̃_{1}(X) + P̃_{1}(Y) \end{matrix} | . \end{aligned}

(2.21)

Following [PR], define

P̃(X , Y) := ∑ P̃_{I}(X) P̃_{ρ(n−1)∖I}(Y) ,  (2.22)

where the summation is over all strict partitions I ⊆ ρ(n − 1). The reasoning in [LP1, Sect.2] made for case C_{n} adapts to case D_{n} and furnishes:

Proposition 2. We have (i)

F (X , Y) ≡ P̃(X , Y) .  (2.23)

(ii) For every w ∈ 𝔇_{n} ∖ 𝔖_{n},

P̃(X^{w}, X) = 0 ,  (2.24)

and for every w ∈ 𝔖_{n},

P̃(X^{w}, X) = P̃(X , X) = s_{ρ(n−1)}(X) .  (2.25)

(iii) For every f ∈ Sym(n),

⟨f(X) , F (X , Y)⟩ ≡ (−1)^{\binom{n}{2}} f(Y) .  (2.26)

(iv) For every f ∈ Pol,

[f(X) , ∏_{n≥i>j≥1}(x_{i} − y_{j}) F (X , Y) ] ≡ f(Y) .  (2.27)

In other words, F (X , Y) is a reproducing kernel for the scalar product ⟨ , ⟩, and ∏_{i>j}(x_{i} − y_{j}) F (X , Y) is a reproducing kernel for [ , ]. One can show that the “vanishing property” (ii) characterizes P̃(X , Y) up to ≡. The congruence (i) can be also derived from geometry by comparing the classes of diagonals in flag bundles associated with SO(2n) given in [F2,F3] and [PR] (see also [G]).

LASCOUX AND PRAGACZ 11

3. Vertex Operators

In this section we shall mainly make computations using the following two divided differences: Definition 3. For k ≤ n, we set

∇_{k}^{B}(n) := (∂_{0}∂_{1} ⋯ ∂_{n−1}) ⋯ (∂_{0}∂_{1} ⋯ ∂_{n−k}) .  (3.1)

For k ≤ n/2, we put

∇_{k}^{D}(n) := (∂_{♡}∂_{2} ⋯ ∂_{n−1}∂_{1}∂_{2} ⋯ ∂_{n−2}) ⋯ (∂_{♡}∂_{2} ⋯ ∂_{n−2k+1}∂_{1}∂_{2} ⋯ ∂_{n−2k}). (3.2)

We shall need the following fact from [LP1], quoted in the appendix: Fact 4. Let k ≤ n and let α = [α_{1} ≤ ⋯ ≤ α_{k}] ∈ ℕ^{k} with α_{k} ≤ n − k. Suppose that I ⊆ ρ(n) is a strict partition. Then the image of Q̃_{I} Y_{α} under ∇_{k}^{B}(n) is 0 unless n − 0 − α_{1}, …, n − (k − 1) − α_{k} are parts of I. In this case, the image is (−1)^{k(n−1)+s}2^{k}Q̃_{J}, where J is the strict partition with parts

{i_{1}, …, i_{ℓ(I)}} ∖ {n − 0 − α_{1}, …, n − (k − 1) − α_{k}},

and s is the sum of positions of the parts erased in I.

Example 5. For n = 7 and k = 2 , we have

\begin{matrix}Q̃_{(5,4,3,2,1)} Y_{[2,5]} ∇_{2}^{B}(7) \\ = Q̃_{(5,4,3,2,1)} Y_{[2,5]} (∂_{0}∂_{1}∂_{2}∂_{3}∂_{4}∂_{5}∂_{6})(∂_{0}∂_{1}∂_{2}∂_{3}∂_{4}∂_{5}) = 4Q̃_{(4,3,2)} , \end{matrix}

and for k = 3 , we have

Q̃_{(7,5,4,3,1)} Y_{[2,3,4]}∇_{3}^{B}(7) = −8Q̃_{(7,4)} .

The following result establishes a basic relation between the ∇^{D}’s and ∇^{B}’s: Proposition 6. Let k be a positive integer. As operators on Sym(2k),

∇_{k}^{D}(2k) = x_{1} ⋯ x_{2k} ∇_{2k}^{B}(2k) + x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(2k) .  (3.3)

Before proving (3.3), we illustrate it by the following examples:

Example 7. As operators on Sym(2),

∇_{1}^{D}(2) = ∂_{♡} = x_{1}x_{2}∂_{0}∂_{1}∂_{0} + x_{1}∂_{0}∂_{1}.

As operators on Sym(4),

\begin{matrix}∇_{2}^{D}(4) & = (∂_{♡}∂_{2}∂_{3}∂_{1}∂_{2})∂_{♡} = \\ x_{1}x_{2}x_{3}x_{4} (∂_{0}∂_{1}∂_{2}∂_{3})(∂_{0}∂_{1}∂_{2})(∂_{0}∂_{1})∂_{0} + x_{1}x_{2}x_{3} (∂_{0}∂_{1}∂_{2}∂_{3})(∂_{0}∂_{1}∂_{2})(∂_{0}∂_{1}). \end{matrix}

The RHS of the last equation is depicted planarly as

x_{1}x_{2}x_{3}x_{4} \begin{aligned}∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} \\ & ∂_{0} & ∂_{1} & ∂_{2} \\ & & ∂_{0} & ∂_{1} \\ & & & ∂_{0} \end{aligned} + x_{1}x_{2}x_{3} \begin{aligned}∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} \\ & ∂_{0} & ∂_{1} & ∂_{2} \\ & & ∂_{0} & ∂_{1} \end{aligned}

Proof of the proposition. In this proof, let X := {x_{1}, …, x_{2k}}. Both sides of (3.3) are Sym^{B}(2k)-linear. The operator ∇_{k}^{D}(2k) sends all Q̃_{I}(X), I ⊆ ρ(2k − 1) to 0, except for Q̃_{ρ(2k−1)} which is sent to (−1)^{k}2^{2k−1} (cf.(2.18)). Thus ∇_{k}^{D}(2k) annihilates all Q̃_{I}(X), I ⊆ ρ(2k), except for I = ρ(2k − 1) which is sent to (−1)^{k}2^{2k−1} and I = ρ(2k) which is sent to (−1)^{k}2^{2k−1}x_{1} ⋯ x_{2k}.

The action of

x_{1} ⋯ x_{2k}∇_{2k}^{B}(2k)

is given by Fact 4. Only Q̃_{ρ(2k)}(X) survives and is sent to (−1)^{k}2^{2k}x_{1} ⋯ x_{2k}. We will now calculate the action of

x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(2k)

on the Q̃_{I}(X), where I ⊆ ρ(2k) is a strict partition. We set, temporarily in this proof,

∇ := ∇_{2k−1}^{B}(2k) and ∇′ := ∇_{2k−1}^{B}(2k − 1) ,

so that

∇ = ∇′∂_{2k−1} ⋯ ∂_{1}.

Let X′ := {x_{1}, …, x_{2k−1}}. We decompose Q̃_{I}(X) as a sum of products of powers of x_{2k} times some Q̃_{J}(X′), according to the formula (2.12):

Q̃_{I}(X) = ∑ Q̃_{J}(X′) x_{2k}^{m_{J}} .

Let \overline{J} be the strict partition obtained from a partition J by subtracting all the pairs of equal parts. We have three cases to examine:

1. Let i_{1} ≤ 2k − 2. Then for each J, |\overline{J}| + m_{J} + 2k − 1 < deg∇, and hence

x_{1} ⋯ x_{2k−1}Q̃_{I}(X)∇ = 0.

2. Let i_{1} = 2k − 1. For degree reasons, Q̃_{I}(X)x_{1} ⋯ x_{2k−1}∇ ≠ 0 is possible only if I = ρ(2k − 1) (I being a strict partition).

Claim. We have

x_{1} ⋯ x_{2k−1}Q̃_{J}(X′)∇′ ≠ 0  (3.4)

only if J = \overline{J} = ρ(2k − 2).

Indeed, suppose first that j_{1} = 2k − 1. Then

x_{1} ⋯ x_{2k−1}Q̃_{J}(X′) = P · Q̃_{H}(X′),

where P is a polynomial symmetric in x_{1}^{2}, …, x_{2k−1}^{2}, and a strict partition H has no part equal to 2k − 1. Since this expression is annihilated by ∇′, we cannot have (3.4). So, for degree reasons, (3.4) holds only if j_{1} = 2k − 2. Suppose now that j_{2} = 2k − 2. We get

x_{1} ⋯ x_{2k−1}Q̃_{J}(X′) = P · Q̃_{H}(X′),

where P is a polynomial symmetric in x_{1}^{2}, …, x_{2k−1}^{2}, and a strict partition H has no part equal to 2k − 2. Since this expression is annihilated by ∇′, we cannot have (3.4). So, for degree reason, (3.4) holds only if j_{2} = 2k − 3. Continuing this way, we get the claim.

For J = ρ(2k − 2), we compute

x_{1} ⋯ x_{2k−1}Q̃_{J}(X′)x_{2k}^{2k−1}∇ = Q̃_{ρ(2k−1)}(X′)∇′x_{2k}^{2k−1} ∂_{2k−1} ⋯ ∂_{1} = (−1)^{k−1}2^{2k−1}.

3. Let i_{1} = 2k. Then (i_{2}, …) ⊆ ρ(2k − 1). We have

\begin{matrix}x_{1} ⋯ x_{2k−1}Q̃_{I}(X)∇ = x_{1}^{2} ⋯ x_{2k−1}^{2}x_{2k}Q̃_{(i_{2},...)}(X)∇ \\ = & Q̃_{(i_{2},...)}(X)∇′x_{1}^{2} ⋯ x_{2k−1}^{2}x_{2k} ∂_{2k−1} ⋯ ∂_{1} . \end{matrix}

Now for H ⊆ (i_{2}, …) ⊆ ρ(2k − 1), Q̃_{H}(X′)∇′ ≠ 0 iff H = ρ(2k − 1), so iff (i_{2}, …) = ρ(2k − 1). We have

Q̃_{ρ(2k−1)}(X)∇′ = Q̃_{ρ(2k−1)}(X′)∇′ = (−1)^{k}2^{2k−1}

and

x_{1}^{2} ⋯ x_{2k−1}^{2}x_{2k} ∂_{2k−1} ⋯ ∂_{1} = −x_{1} ⋯ x_{2k}.

Summarizing,

Q̃_{I}(X)x_{1} ⋯ x_{2k−1}∇ ≠ 0

only if I = ρ(2k − 1), when we get (−1)^{k−1}2^{2k−1}; or I = ρ(2k), when we get (−1)^{k−1}2^{2k−1}x_{1} ⋯ x_{2k}.

Finally, comparing the computed values of the Q̃_{I}(X) under the operators:

∇_{k}^{D}(2k) , x_{1} ⋯ x_{2k} ∇_{2k}^{B}(2k), and x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(2k),

that are possibly nonzero only for I = ρ(2k) and ρ(2k − 1), we get the desired formula (3.3). (Note that we have also used the equality 2^{p−1} = 2^{p} − 2^{p−1}.) □

Corollary 8. Let k be a positive integer such that k ≤ n/2. As operators on the ring Sym(2k | n − 2k),

∇_{k}^{D}(n) = x_{1} ⋯ x_{2k} ∇_{2k}^{B}(n) + x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(n) ∂_{1} ⋯ ∂_{n−2k} .  (3.5)

This property is obtained from Proposition 6 by composing the expression for the operator ∇_{k}^{D}(2k) with the divided difference

(∂_{2k} ⋯ ∂_{n−1}) ⋯ (∂_{2} ⋯ ∂_{n−2k+1})(∂_{1} ⋯ ∂_{n−2k}) = \begin{matrix}∂_{2k} & ⋯ & ∂_{n−1} \\ ⋮ & & ⋮ \\ ∂_{2} & ⋯ & ∂_{n−2k+1} \\ ∂_{1} & ⋯ & ∂_{n−2k} \end{matrix}

In the proof of Theorem 11, we will need the following supplement to Fact 4:

Theorem 9. Let k ≤ n and let α = [α_{1} ≤ ⋯ ≤ α_{k}] ∈ ℕ^{k} with α_{k} = n − k + 1. Suppose that I ⊆ ρ(n) is a strict partition. Then the image of Q̃_{I} Y_{α} under ∇_{k}^{B}(n) is 0 unless ℓ(I) ≢ n (mod 2) and n − 0 − α_{1}, …, n − (k − 2) − α_{k−1} are parts of I. In this case, the image is (−1)^{(k−1)(n−1)+1+s}2^{k}Q̃_{J}, where J is the strict partition with parts

{i_{1}, …, i_{ℓ(I)}} ∖ {n − 0 − α_{1}, …, n − (k − 2) − α_{k−1}} ,

and s is the sum of positions of the parts erased in I.

The proof of this theorem will be given in the appendix.

Example 10.

(i) For n = 5 and k = 1, we have

x_{1}^{5}Q̃_{(5,3,2,1)}∂_{0}∂_{1}∂_{2}∂_{3}∂_{4} = −2Q̃_{(5,3,2,1)} and x_{1}^{5}Q̃_{(5,2,1)}∂_{0}∂_{1}∂_{2}∂_{3}∂_{4} = 0.

(ii) For n = 7 and k = 2, we have

Q̃_{(7,6,4,1)} Y_{[1,6]} ∇_{2}^{B}(7) = −4Q̃_{(7,4,1)}

and

Q̃_{(7,6,4,3,1)} Y_{[1,6]} ∇_{2}^{B}(7) = 0.

(iii) For n = 7 and k = 4, we have

Q̃_{(7,6,4,3,2,1)} Y_{[1,2,2,4]} ∇_{4}^{B}(7) = 16Q̃_{(7,2,1)}

and

Q̃_{(7,6,4,3,2,1)} Y_{[1,3,4,4]} ∇_{4}^{B}(7) = −16Q̃_{(7,4,2)}.

The following theorem is the main result of the present paper:

Theorem 11. Let k be a positive integer such that k ≤ n/2. Suppose that I ⊆ ρ(n − 1) is a strict partition. Let α = [α_{1} ≤ α_{2} ≤ ⋯ ≤ α_{2k}] ∈ ℕ^{2k} with α_{2k} ≤ n − 2k. Then the image of P̃_{I} Y_{α} under ∇_{k}^{D}(n) is 0 unless all the integers n − 1 − α_{1}, …, n − 2k − α_{2k} belong to {i_{1}, …, i_{ℓ(I)}, 0}. In this case, the image is (−1)^{s}P̃_{J}, where J is the strict partition with parts

{i_{1}, …, i_{ℓ(I)}} ∖ {n − 1 − α_{1}, …, n − 2k − α_{2k}} .

Moreover, let s′ be the sum of positions of the parts erased in I, and s′′ := ℓ(I) + 1. Then s = s′ if α_{2k} < n − 2k, and s = s′ + s′′ if α_{2k} = n − 2k.^{2}

Example 12. (i) For n = 7 and k = 1, we have

P̃_{(5,4,3,2,1,0)} Y_{[1,3]}∇_{1}^{D}(7) = P̃_{(5,4,3,2,1,0)} Y_{[1,3]} ∂_{♡}∂_{2}∂_{3}∂_{4}∂_{5}∂_{6}∂_{1}∂_{2}∂_{3}∂_{4}∂_{5} = −P̃_{(4,3,1)} and

2It is convenient to treat here 0 = n − 2k − α_{2k} as an “extra part” of I, and take s to be the sum of positions of all the parts erased in I, including the extra part.

P̃_{(6,4,3,2,1,0)} Y_{[2,5]} ∇_{1}^{D}(7) = P̃_{(6,3,2,1)}.

(ii) For n = 7 and k = 2, we have

P̃_{(6,5,4,3,2,1,0)} Y_{[1,1,1,2]} ∇_{2}^{D}(7) = −P̃_{(6,2)}

and

P̃_{(6,5,4,3,2,1,0)} Y_{[1,1,1,3]} ∇_{2}^{D}(7) = P̃_{(6,2,1)} .

Proof of the theorem. To compute the action of ∇_{k}^{D}(n), one uses its decomposition into a sum of two operators, given in Eq.(3.5).

The image of Q̃_{I}(X) Y_{α} under the first operator

Ω_{1} := x_{1} ⋯ x_{2k} ∇_{2k}^{B}(n)

is given by Fact 4 combined with Eq.(2.3) if α_{2k} < n − 2k, and by Theorem 9 combined with (2.3) in the case α_{2k} = n − 2k.

Since x_{2k} appears in Y_{α}, the same results, however, do not directly furnish the value of Q̃_{I}(X_{n}) Y_{α} under the second operator

Ω_{2} := x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(n) ∂_{1} ⋯ ∂_{n−2k} .

To end this computation, we proceed as follows. For simplicity of indices, let us take temporarily n = 7 and k = 2. Suppose that α = [α_{1} ≤ α_{2} ≤ α_{3} ≤ α_{4}] ∈ ℕ^{4} is such that α_{4} ≤ 3. We want to compute

Q̃_{I} Y_{α} x_{1}x_{2}x_{3} \begin{aligned}∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} & ∂_{6} \\ & ∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} \\ & & ∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} \\ & & & & ∂_{1} & ∂_{2} & ∂_{3} \end{aligned}

Now, thanks to the relations (1.4), one has

∂_{0} ∂_{1} ∂_{2} ∂_{3} ∂_{4} ∂_{5} ∂_{6}
∂_{0} ∂_{1} ∂_{2} ∂_{3} ∂_{4} ∂_{5}
∂_{0} ∂_{1} ∂_{2} ∂_{3} ∂_{4}
∂_{1} ∂_{2} ∂_{3}

= ( \begin{matrix}∂_{0} & ∂_{1} & ∂_{2} \\ & ∂_{0} & ∂_{1} \\ & & ∂_{0} \end{matrix} ) ( \begin{matrix}∂_{3} & ∂_{4} & ∂_{5} & ∂_{6} \\ ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} \\ ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} \\ & ∂_{1} & ∂_{2} & ∂_{3} \end{matrix} ) = ( \begin{matrix}∂_{0} & ∂_{1} & ∂_{2} \\ & ∂_{0} & ∂_{1} \\ & & ∂_{0} \end{matrix} ) ( \begin{matrix}∂_{4} & ∂_{5} & ∂_{6} \\ ∂_{3} & ∂_{4} & ∂_{5} & ∂_{6} \\ ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} \\ ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} \end{matrix} )

= ( \begin{matrix}∂_{0} & ∂_{1} & ∂_{2} \\ & ∂_{0} & ∂_{1} \\ & & ∂_{0} \end{matrix} ) (∂_{4}∂_{5}∂_{6}) ( \begin{matrix}∂_{3} & ∂_{4} & ∂_{5} & ∂_{6} \\ ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} \\ ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} \end{matrix} )

Since ∂_{4}∂_{5}∂_{6} commutes with the divided differences on its left, the last expression is rewritten as

∂_{4}∂_{5}∂_{6} \begin{aligned}∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} & ∂_{6} \\ & ∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} & ∂_{5} \\ & & ∂_{0} & ∂_{1} & ∂_{2} & ∂_{3} & ∂_{4} \end{aligned} = ∂_{4}∂_{5}∂_{6} ∇_{3}^{B}(7)

Since ∂_{4}∂_{5}∂_{6} commutes with x_{1}x_{2}x_{3} and Q̃_{I} = Q̃_{I}(x_{1}, …, x_{7}), the polynomial to be computed is equal to

Y_{α} ∂_{4}∂_{5}∂_{6} Q̃_{I} x_{1}x_{2}x_{3} ∇_{3}^{B}(7) .

However, using (2.2), the image of Y_{[α_{1},α_{2},α_{3},α_{4}]} under ∂_{4}∂_{5}∂_{6} is

Y_{[α_{1},α_{2},α_{3},0,0,0,α_{4}−3]} = Y_{[α_{1},α_{2},α_{3}]}

if α_{4} = 3, and 0 otherwise. Hence, by (2.3), the polynomial to be computed is equal to

Y_{[α_{1}+1,α_{2}+1,α_{3}+1]}Q̃_{I}∇_{3}^{B}(7) .

In general, arguing along these lines, we evaluate

Q̃_{I} Y_{α} x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(n) ∂_{1}∂_{2} ⋯ ∂_{n−2k} ,

where α = [α_{1} ≤ α_{2} ≤ ⋯ ≤ α_{2k}] ∈ ℕ^{2k} is such that α_{2k} ≤ n − 2k. By the relations (1.4), this amounts to evaluating

Q̃_{I} Y_{α} x_{1} ⋯ x_{2k−1} ∂_{2k}∂_{2k+1} ⋯ ∂_{n−1} ∇_{2k−1}^{B}(n) .

Since ∂_{2k}∂_{2k+1} ⋯ ∂_{n−1} commutes with x_{1} ⋯ x_{2k−1} and Q̃_{I} = Q̃_{I}(x_{1}, …, x_{n}), the polynomial to be computed is equal to

Y_{α} ∂_{2k}∂_{2k+1} ⋯ ∂_{n−1}Q̃_{I}x_{1} ⋯ x_{2k−1} ∇_{2k−1}^{B}(n) .

However, using (2.2), the image of Y_{α} under ∂_{2k}∂_{2k+1} ⋯ ∂_{n−1} is

Y_{[α_{1},... ,α_{2k−1},0^{n−2k},α_{2k}−(n−2k)]} .  (3.6)

The expression (3.6) equals 0 unless α_{2k} = n−2k, when it is equal to Y_{[α_{1},... ,α_{2k−1}]} . Hence, by (2.3), the polynomial to be computed is equal to

Q̃_{I}Y_{[α_{1}+1,... ,α_{2k−1}+1]} ∇_{2k−1}^{B}(n) .

Since α_{2k−1} + 1 ≤ n − (2k − 1), Fact 4 provides the end of the computation with the operator Ω_{2} .

Note that in the case when we have a contribution from both operators Ω_{1} and Ω_{2}, we also use the equality 2^{p−1} = 2^{p} − 2^{p−1} . □

Example 13. (i) For n = 5, k = 1, we have

P̃_{(3,2)} Y_{[1,3]} ∇_{1}^{D}(5) = P̃_{2}

and this comes from the contribution of both operators Ω_{1}, Ω_{2}:

Q̃_{(3,2)} Y_{[1,3]} x_{1}x_{2}∇_{2}^{B}(5) = Q̃_{(3,2)} Y_{[2,4]} ∇_{2}^{B}(5) = 4Q̃_{2}

by Theorem 9, and

Q̃_{(3,2)} Y_{[1,3]} x_{1}∇_{1}^{B}(5) ∂_{1}∂_{2}∂_{3} = Q̃_{(3,2)} Y_{[2]} ∇_{1}^{B}(5) = −2Q̃_{2} .

by Fact 4. (ii) For n = 7, k = 2, we have

P̃_{(6,5,4,3,2,1,0)} Y_{[0,1,2,2]} ∇_{2}^{D}(7) = −P̃_{(5,3)}

and only the operator Ω_{1} gives the contribution:

Q̃_{(6,5,4,3,2,1)} Y_{[0,1,2,2]} x_{1}x_{2}x_{3}x_{4}∇_{4}^{B}(7) = Q̃_{(6,5,4,3,2,1)} Y_{[1,2,3,3]} ∇_{4}^{B}(7) = −16Q̃_{(5,3)}

by Fact 4. (iii) For n = 7, k = 2, we have

P̃_{(6,5,4,3,2,1,0)} Y_{[1,1,2,3]} ∇_{2}^{D}(7) = −P̃_{(6,3,1)}

and the contribution comes from both operators Ω_{1}, Ω_{2}:

Q̃_{(6,5,4,3,2,1)} Y_{[1,1,2,3]} x_{1}x_{2}x_{3}x_{4} ∇_{4}^{B}(7) = Q̃_{(6,5,4,3,2,1)} Y_{[2,2,3,4]} ∇_{4}^{B}(7) = −16Q̃_{(6,3,1)}

by Theorem 9, and

\begin{aligned}Q̃_{(6,5,4,3,2,1)} Y_{[1,1,2,3]} & x_{1}x_{2}x_{3} ∇_{3}^{B}(7) ∂_{1}∂_{2}∂_{3} \\ & = Q̃_{(6,5,4,3,2,1)} Y_{[2,2,3]} ∇_{3}^{B}(7) = 8Q̃_{(6,3,1)} . \end{aligned}

by Fact 4.

4. Applications to P̃-polynomials and orthogonal Schubert polynomials

The following presentation of a P̃-polynomial in the form

P̃_{I} = x^{α(I)} P̃_{ρ(n−1)} Ω(I) ,

where α(I) ⊆ ρ and Ω(I) is a divided difference operator, appears to be quite useful:

Lemma 14. Let I = (i_{1}, …, i_{ℓ} > 0) ⊆ ρ(n − 1) be a strict partition. If n and ℓ are of the same parity, we set h := n − ℓ, and {j_{1} < ⋯ < j_{h}} := {1, …, n} ∖ {i_{1} + 1, …, i_{ℓ} + 1}. If n and ℓ are of different parity, we set h := n − ℓ − 1, and {j_{1} < ⋯ < j_{h}} := {1, …, n} ∖ {i_{1} + 1, …, i_{ℓ} + 1, 1}.

Then for α(I) := [n − j_{1}, …, n − j_{h}, 0, …, 0] and k := h/2,

x^{α(I)} P̃_{ρ(n−1)} ∂_{[2k,... ,1]}∇_{k}^{D}(n) = (−1)^{s} P̃_{I} ,  (4.1)

where s is the number of positions of the parts erased in ρ(n − 1) in order to get the partition I.

The assertion of the lemma is a direct consequence of Theorem 11 and the definition of a Schur S-polynomial via the Jacobi symmetrizer.

Now, with every strict partition I = (i_{1}, …, i_{ℓ} > 0) , we associate the following element v(I) ∈ 𝔇_{n}. If n − ℓ is even, we set

v(I) := [i_{1} + 1, i_{2} + 1, …, i_{ℓ} + 1, \overline{j_{1}}, …, \overline{j_{h}}] ,  (4.2)

and if n − ℓ is odd,

v(I) := [i_{1} + 1, i_{2} + 1, …, i_{ℓ} + 1, 1, \overline{j_{1}}, …, \overline{j_{h}}] .  (4.3)

(The notation is the same as in Lemma 14.)

Theorem 15. For a strict partition I ⊆ ρ(n − 1),

x^{ρ}P̃_{ρ(n−1)} ∂_{v(I)} = (−1)^{|I|+\binom{n}{2}}P̃_{I}.  (4.4)

The proof of this result is analogous to the proof of [LP1, Thm. A.6]. Using the notation of Lemma 14, we have

∂_{v(I)} = ∂_{σ} ∂_{[k,k−1,... ,1]} ∇_{k}^{D}(n) ,  (4.5)

where

σ = \{ \begin{matrix}[j_{1}, …, j_{h}, i_{1} + 1, …, i_{ℓ} + 1], & n & and & ℓ & of the same parity \\ [j_{1}, …, j_{h}, i_{1} + 1, …, i_{ℓ} + 1, 1], & n & and & ℓ & of different parity . \end{matrix}

Note that x^{ρ} ∂_{σ} = x^{α(I)}, and hence the assertion follows by Lemma 14.

This result leads to the following characterization of P̃-polynomials via orthogonal divided differences:

Corollary 16. For a strict partition I ⊆ ρ(n − 1), we set w(I) := v(I)^{−1}w_{0}^{D}. More explicitly, for even ℓ, w(I) = [\overline{i_{1} + 1}, …, \overline{i_{ℓ} + 1}, j_{1}, …, j_{h}]^{−1}, and for odd ℓ, w(I) = [\overline{i_{1} + 1}, …, \overline{i_{ℓ} + 1}, \overline{1}, j_{1}, …, j_{h}]^{−1}. Then w = w(I) is the unique element of 𝔇_{n} such that ℓ(w) = |I| and P̃_{I} ∂_{w} ≠ 0. In fact, P̃_{I} ∂_{w(I)} = (−1)^{|I|}.

This can be also seen by geometric considerations (see [P1] and [LP1]), with the help of the characteristic map ([B], [D1,D2]).

More generally, consider, for any w ∈ 𝔇_{n}, the orthogonal Schubert polynomial

X_{w}^{D} = X_{w}^{D}(n) = x^{ρ}P̃_{ρ(n−1)}∂_{w_{0}^{D}w}  (4.6)

of degree ℓ(w). Arguing in the same way as in [LP1, pp.33–36], one shows that these Schubert polynomials have the stability property in the sense that for w ∈ 𝔇_{n} ⊂ 𝔇_{n+1},

X_{w}^{D}(n + 1)|_{x_{n+1}=0} = X_{w}^{D}(n).  (4.7)

Together with the “maximal Grassmannian property” from Theorem 15, asserting that, for even ℓ,

X_{[\overline{i_{1}+1},... ,\overline{i_{ℓ}+1},j_{1},... ,j_{h}]}^{D} = (−1)^{|I|+\binom{n}{2}}P̃_{I} ,  (4.8)

and for odd ℓ,

X_{[\overline{i_{1}+1},... ,\overline{i_{ℓ}+1},\overline{1},j_{1},... ,j_{h}]}^{D} = (−1)^{|I|+\binom{n}{2}}P̃_{I} ,  (4.9)

this shows that they provide a natural tool for the cohomological study of Schubert varieties for the orthogonal group SO(2n) and the related degeneracy loci.

We also record

Proposition 17. For a strict partition I = (i_{1}, i_{2}, i_{3}, i_{4}, …) ⊆ ρ(n − 1),

P̃_{I} ∂_{♡}∂_{2} ⋯ ∂_{i_{1}}∂_{1}∂_{2} ⋯ ∂_{i_{2}} = (−1)^{i_{1}+i_{2}} P̃_{(i_{3},i_{4},...)}.  (4.10)

To see this, we argue in a manner similar to the proof of [LP1, Prop. 5.12]. For J = (i_{3}, i_{4}, …), we choose the presentation from Lemma 14:

±P̃_{I} = x^{α(I)} P̃_{ρ(n−1)}∂_{u} , ±P̃_{J} = x^{α(J)} P̃_{ρ(n−1)}∂_{v}

for appropriate u, v ∈ 𝔇_{n}. Let σ ∈ 𝔖_{n} be the permutation such that

v = σ u s_{♡}s_{2} ⋯ s_{i_{1}}s_{1}s_{2} ⋯ s_{i_{2}} .

The assertion now follows from x^{α(I)}∂_{σ} = x^{α(J)}.

Appendix: results in type B

In this appendix we give a summary of the results for type B_{n}. They are obtained directly from the results for type C_{n} in [LP1], by changing ∂_{0} in loc.cit. to −2∂_{0}, and read as follows: writing ∇ := ∇_{n}^{B}(n), we have

Theorem 18. (i) For α ∈ ℕ^{n}, α ⊆ ρ,

Y_{α} P̃_{ρ(n)} ∇ = (−1)^{|α|+\binom{n+1}{2}}Y_{α}^{ω}.  (5.1)

(ii) For strict I ⊊ ρ(n), and α ⊆ ρ,

Y_{α} P̃_{I} ∇ = 0 .  (5.2)

Denote by

⟨ , ⟩ : Sym(n) × Sym(n) → Sym^{D}(n)

the scalar product defined for f, g ∈ Sym(n) by

⟨f, g⟩ := fg ∇ .  (5.3)

For strict partitions I, J ⊆ ρ(n), one has

⟨P̃_{I}, P̃_{ρ(n)∖J}⟩ = (−1)^{\binom{n+1}{2}} δ_{IJ},  (5.4)

where ρ(n) ∖ J is the strict partition whose parts complement the parts of J in {n, n − 1, …, 1} (cf. [PR]).

Consequently, the polynomial ring Pol = ℤ[\frac{1}{2}][x_{1}, …, x_{n}] is a free Sym^{B}(n)- module with basis Y_{α} P̃_{I}, where α ranges over subsequences contained in ρ and I runs over all strict partitions contained in ρ(n). Note that the element of the maximal degree of this basis is x^{ρ}P̃_{ρ(n)}. Let

[ , ] : Pol × Pol → Sym^{B}(n)

be a scalar product, defined for f, g ∈ Pol by

[f, g] := fg ∂_{w_{0}^{B}}.  (5.5)

One has, for α, β ⊆ ρ and strict partitions I, J ⊂ ρ(n),

[Y_{α}^{ω} P̃_{I}, Y_{β′} P̃_{ρ(n)∖J}] = (−1)^{|α|+\binom{n+1}{2}} δ_{αβ} δ_{IJ} .  (5.6)

(See (2.5).)

Let Y = {y_{1}, …, y_{n}} be a second set of indeterminates of cardinality n. The symbol ≡ will mean: “congruent modulo the ideal generated by the relations f(x_{1}^{2}, …, x_{n}^{2}) = f(y_{1}^{2}, …, y_{n}^{2}), where f ∈ Sym(n).”

Following Fulton [F2,F3], define

\begin{aligned}F (X , Y) & := |P̃_{n+1+j−2i}(X) + P̃_{n+1+j−2i}(Y)|_{1≤i,j≤n} \\ & = | \begin{matrix}P̃_{n}(X) + P̃_{n}(Y) & 0 & … \\ P̃_{n−2}(X) + P̃_{n−2}(Y) & P̃_{n−1}(X) + P̃_{n−1}(Y) & … \\ ⋮ & ⋮ & ⋮ \\ & & 1 & P̃_{1}(X) + P̃_{1}(Y) \end{matrix} | . \end{aligned}

(5.7)

Following [PR], define

P̃(X , Y) := ∑ P̃_{I}(X) P̃_{ρ(n)∖I}(Y) ,  (5.8)

where the summation is over all strict partitions I ⊆ ρ(n). The reasoning in [LP1, Sect.2] made for case C_{n} adapts to case B_{n} and furnishes:

Proposition 19. We have (i)

F (X , Y) ≡ P̃(X , Y).  (5.9)

(ii) For every w ∈ 𝔅_{n} ∖ 𝔖_{n},

P̃(X^{w}, X) = 0 ,  (5.10)

and for every w ∈ 𝔖_{n},

P̃(X^{w}, X) = P̃(X , X) = s_{ρ(n)}(X) .  (5.11)

(iii) For every f ∈ Sym(n),

⟨f(X) , F (X , Y)⟩ ≡ (−1)^{\binom{n+1}{2}} f(Y) .  (5.12)

(iv) For every f ∈ Pol,

[f(X) , ∏_{n≥i>j≥1}(x_{i} − y_{j}) F (X , Y) ] ≡ f(Y) .  (5.13)

In other words, F (X , Y) is a reproducing kernel for the scalar product ⟨ , ⟩, and ∏_{i>j}(x_{i} − y_{j}) F (X , Y) is a reproducing kernel for [ , ]. One can show that the “vanishing property” (ii) characterizes P̃(X , Y) up to ≡. The congruence (i) can be also derived from geometry by comparing the classes of diagonals in flag bundles associated with SO(2n + 1) given in [F2,F3] and [PR] (see also [G]).

Proposition 20. Suppose n ≥ p > 0. Let IpJ ⊆ ρ(n) be a strict partition and H ⊆ ρ(n) a strict partition not containing p. Then

x_{1}^{n−p}P̃_{IpJ} ∂_{0}∂_{1} ⋯ ∂_{n−1} = (−1)^{ℓ(I)+n}P̃_{IJ}  (5.14)

and

x_{1}^{n−p}P̃_{H} ∂_{0}∂_{1} ⋯ ∂_{n−1} = 0 .  (5.15)

More generally,

Theorem 21. Let 0 < k ≤ n and let α = [α_{1} ≤ ⋯ ≤ α_{k}] ∈ ℕ^{k} be such that α_{k} ≤ n − k. Suppose that I ⊆ ρ(n) is a strict partition. Then the image of P̃_{I} Y_{α} under ∇_{k}^{B}(n) is 0 unless n − α_{1} − 0, …, n − α_{k} − (k − 1) are parts of I. In this case, the image is (−1)^{k(n−1)+s}P̃_{J}, where J is the strict partition with parts

{i_{1}, …, i_{ℓ(I)}} ∖ {n − α_{1} − 0, …, n − α_{k} − (k − 1)},

and s is the sum of positions of the parts erased in I.

(This is a restatement of [LP1, Prop. 5.9].)

Proposition 22. For a strict partition I = (i_{1}, i_{2}, …),

P̃_{I} ∂_{0}∂_{1} ⋯ ∂_{i_{1}−1} = (−1)^{i_{1}} P̃_{(i_{2},...)} .  (5.16)

Now, let us associate with every strict partition I = (i_{1}, …, i_{ℓ} > 0) the following element of 𝔅_{n}:

v(I) := [i_{1}, …, i_{ℓ}, \overline{j_{1}}, …, \overline{j_{h}}] ,  (5.17)

where j_{1} < ⋯ < j_{h}.

Theorem 23. For every strict partition I ⊆ ρ(n),

x^{ρ}P̃_{ρ(n)} ∂_{v(I)} = (−1)^{|I|+\binom{n+1}{2}}P̃_{I}.  (5.18)

This leads to the following characterization of P̃-polynomials via divided differences:

Corollary 24. For any strict partition I, let w(I) := v(I)^{−1}w_{0}^{B}, that is w(I) = [\overline{i_{1}}, …, \overline{i_{ℓ}}, j_{1}, …, j_{h}]^{−1} . Then w = w(I) is the unique element of 𝔅_{n} such that ℓ(w) = |I| and P̃_{I} ∂_{w} ≠ 0. In fact, P̃_{I} ∂_{w(I)} = (−1)^{|I|}.

This can be also seen by geometric considerations (see [P1] and [LP1]), with the help of the characteristic map ([B], [D1,D2]).

More generally, consider, for any w ∈ 𝔅_{n}, the orthogonal Schubert polynomial

X_{w}^{B} = X_{w}^{B}(n) = x^{ρ}P̃_{ρ(n)}∂_{w_{0}^{B}w}  (5.19)

of degree ℓ(w). Arguing in the same way as in [LP1, pp.33–36], one shows that these Schubert polynomials have the stability property in the sense that for w ∈ 𝔅_{n} ⊂ 𝔅_{n+1},

X_{w}^{B}(n + 1)|_{x_{n+1}=0} = X_{w}^{B}(n).  (5.20)

Together with the “maximal Grassmannian property” from Theorem 23, asserting that

X_{[\overline{i_{1}},... ,\overline{i_{ℓ}},j_{1},... ,j_{h}]}^{B} = (−1)^{|I|+\binom{n+1}{2}}P̃_{I} ,  (5.21)

this shows that they provide a natural tool for the cohomological study of Schubert varieties for the orthogonal group SO(2n + 1) and the related degeneracy loci.

We now give Proof of Theorem 9

Given a symmetric function f, let D_{f} be the Foulkes derivative i.e. the adjoint operator to the multiplication by f w.r.t. the standard scalar product on the ring Sym of symmetric functions in a countable number of variables (cf. [M1]). We use the following vertex operators on Sym:

U^{s} := 1 − D_{P_{1}}s_{1} + D_{P_{2}}s_{2} − ⋯,  (5.22)
U^{e} := 1 − D_{P_{1}}e_{1} + D_{P_{2}}e_{2} − ⋯,  (5.23)

and

V^{e} := 1 − D_{e_{1}}P_{1} + D_{e_{2}}P_{2} − · · · .  (5.24)

We refer to [LP1, p.24] for the definitions of Schur P -functions P_{I} [S]. In loc. cit. the reader can also find a definition of Q′-functions Q′_{I} [LLT2], used in the following proposition:

Proposition 25. Let I be a strict partition. We have the following identities of symmetric functions in Sym:

Q̃_{I} U^{s} = \{ \begin{matrix}Q̃_{I}, & ℓ(I) & even \\ 0, & ℓ(I) & odd ; \end{matrix}  (5.25)

Q′_{I} U^{e} = \{ \begin{matrix}Q′_{I}, & ℓ(I) & even \\ 0, & ℓ(I) & odd ; \end{matrix}  (5.26)

and

P_{I} V^{e} = \{ \begin{matrix}P_{I}, & ℓ(I) & even \\ 0, & ℓ(I) & odd . \end{matrix}  (5.27)

Proof. First of all, arguing as in [LP1, pp.24–27], with the help of the operators V^{e}, U^{s}, and U^{e} instead of V_{k}^{e}, U_{k}^{s}, and U_{k}^{e} in loc. cit., we note that the equalities (5.25), (5.26), and (5.27) are equivalent.

We show (5.27). It suffices to prove the statement when the set of indeterminates {x_{1}, …, x_{n}} is of finite cardinal n > |I|.

Besides the well-known equality: for k > 0,

x_{1}^{k} ∏_{2≤i≤n}(x_{1} + x_{i}) ∂_{1}∂_{2} ⋯ ∂_{n−1} = P_{k}(x_{1}, …, x_{n}) ,  (5.28)

we also need the following formula from [P2]:

Fact 26. For a strict partition I,

\begin{aligned}P_{I}(x_{2}, …, x_{n}) & ∏_{2≤i≤n}(x_{1} + x_{i}) ∂_{1}∂_{2} ⋯ ∂_{n−1} \\ & = \{ \begin{matrix}(−1)^{n−1}P_{I}(x_{1}, …, x_{n}), & n − ℓ(I) & odd \\ 0, & n − ℓ(I) & even. \end{matrix} \end{aligned}  (5.29)

(More precisely, Eq.(5.29) is a special case of the following formula given in [P2, Prop.1.3(ii)]: Let q, r, k, and h be integers such that 0 < q < n, n = q + r, 0 ≤ k ≤ q, and 0 ≤ h ≤ r. Suppose I = (i_{1}, …, i_{k}) ∈ ℕ^{∗k} and J = (j_{1}, …, j_{h}) ∈ ℕ^{∗h}. Then

P_{I}(x_{1}, …, x_{q}) P_{J}(x_{q+1}, …, x_{n}) ∏_{1≤i≤q<j≤n}(x_{i} + x_{j}) \begin{matrix}∂_{q} & ⋯ & ∂_{n−1} \\ ⋮ & & ⋮ \\ ∂_{2} & ⋯ & ∂_{r+1} \\ ∂_{1} & ⋯ & ∂_{r} \end{matrix}

= d · P_{(i_{1},... ,i_{k},j_{1},... ,j_{h})}(x_{1}, …, x_{n}) ,  (5.30)

where d is zero if (q − k)(r − h) is odd and

d = (−1)^{(q−k)r}\binom{⌊(n − k − h)/2⌋}{⌊(q − k)/2⌋}  (5.31)

otherwise.

We get Eq.(5.29) as Eq.(5.30) specialized to q = 1 and k = 0.)

To end the proof of (5.27), we first write

P_{I}(x_{2}, …, x_{n}) = P_{I} − P_{I}D_{e_{1}} · x_{1} + P_{I}D_{e_{2}} · x_{1}^{2} − P_{I}D_{e_{3}} · x_{1}^{3} + ⋯,  (5.32)

where the RHS is evaluated in the first n variables. Then we multiply both sides of (5.32) by

(x_{1} + x_{2})(x_{1} + x_{3}) ⋯ (x_{1} + x_{n})

and apply the operator ∂_{1}∂_{2} ⋯ ∂_{n−1}. We get the following equalities of symmetric polynomials in the first n variables. If n is odd, the RHS of the so-obtained equality becomes

P_{I} − P_{I}D_{e_{1}} · P_{1} + P_{I}D_{e_{2}} · P_{2} − P_{I}D_{e_{3}} · P_{3} + ⋯

by (5.28) and (5.29), and its LHS is equal to

\{ \begin{matrix}P_{I}, & ℓ(I) & even \\ 0, & ℓ(I) & odd \end{matrix}

by (5.29). This shows (5.27) for odd n. If n is even, the RHS of the obtained equality becomes

0 − P_{I}D_{e_{1}} · P_{1} + P_{I}D_{e_{2}} · P_{2} − P_{I}D_{e_{3}} · P_{3} + ⋯

by (5.28) and (5.29), and its LHS is equal to

\{ \begin{matrix}0, & ℓ(I) & even \\ −P_{I}, & ℓ(I) & odd \end{matrix}

by (5.29). This shows (5.27) for even n.

Thus the proposition has been proved. □

Let now ∂_{0}^{C} be the divided difference defined by

Pol ∋ f ↦ f ∂_{0}^{C} := (f − f^{s_{0}})/(2x_{1}) .  (5.33)

Arguing similarly as in [LP1, pp.27–28], using

∂_{0}^{C} = D_{P_{1}} − D_{P_{2}} x_{1} + D_{P_{3}} x_{1}^{2} − ⋯,  (5.34)

and the formula

x_{1}^{p} ∂_{1} ⋯ ∂_{n−1} = s_{p−n+1}(x_{1}, …, x_{n}) ,  (5.35)

one shows Lemma 27. As operators on Sym, evaluated in symmetric polynomials in the first n variables,

1 − U^{s} = ∂_{0}^{C} x_{1}^{n} ∂_{1} ⋯ ∂_{n−1} .  (5.36)

Equations (5.25) and (5.36), together with the equalities

x_{1}^{2m} ∂_{0}^{C} = ∂_{0}^{C} x_{1}^{2m} and x_{1}^{2m+1} ∂_{0}^{C} = −∂_{0}^{C} x_{1}^{2m+1} + x_{1}^{2m} ,  (5.37)

applied for even n = 2m or odd n = 2m + 1 accordingly, imply

Proposition 28. Let I ⊆ ρ(n) be a strict partition. We have

Q̃_{I} x_{1}^{n} ∂_{0}∂_{1} ⋯ ∂_{n−1} = \{ \begin{matrix}−2Q̃_{I}, & n + ℓ(I) & odd \\ 0, & n + ℓ(I) & even . \end{matrix}  (5.38)

Eq.(5.38) is the content of Theorem 9 for k = 1. For higher k, one gets the desired assertion by [LP1, Thm 5.1], Proposition 28, and [LP1, Lemma 5.10]. (Note that this last fact holds true for any nonnegative integer α_{1}, in the notation of loc. cit., as is clear from its proof.)

This ends the proof of Theorem 9.

Finally, we take this opportunity to correct some misprints in [LP1]: – should read: p.11_{11} “... a partition ...”

p.13_{2} “ ⟨ , ⟩ : SP(X) × SP(X) → SP(X^{2}) ”

p.36_{2} “... ∂′_{u}C_{w} = C_{v} ...”

p.37^{4} “... is ∇_{k} ∘ ∂′_{ω^{(k)}} ...”

p.37_{10} “... ∂′_{w_{I}} (Q̃_{I}(X)) = 1 ... ”

Moreover, in Example 5.11, the sequence of successive signs is: +, +, –, – .

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A.L. : C.N.R.S., Institut Gaspard Monge, Université de Marne-la-Vallée, 5 Bd Descartes, Champs sur Marne, 77454 Marne La Vallée Cedex 2, France e-mail: Alain.Lascoux@univ-mlv.fr

P.P. : Mathematical Institute of Polish Academy of Sciences, Chopina 12, 87-100 Toruń, Poland e-mail: pragacz@mat.uni.torun.pl