Polynomial identities related to Special Schubert varieties

Let $\mathcal S$ be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincar\'{e} polynomials of the intersection cohomology of $\mathcal S$ by means of the Poincar\'{e} polynomials of its strata, obtaining interesting polynomial identities relating Poincar\'{e} polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.


Introduction
In the paper [10], it was shown how one can obtain suitable polynomial identities from the study of the intersection cohomology of Schubert varieties with two strata (compare with [10, p. 115]). The aim of our work is to extend the same approach to Special Schubert varieties with an arbitrary number of strata, by showing that the Poincaré polynomials of their intersection cohomology naturally leads to a class of tricky polynomial identities. In the final Appendix, we provide some of the numerical tests for the polynomial identities that we obtained in the meantime, and a symbolic study of a particular case.
The starting point of our analysis is the main result of the paper [13], which we now summarize. Let S be a single condition Schubert variety or special Schubert variety of dimension n (see [5, p. 328] and [18,Example 8.4.9]). As it is well known, S admits two standard resolutions: a small resolution ξ : D → S [18,Definition 8.4.6] and a (usually) non-small one π : S → S [20, §3.4 and Exercise 3.4.10]. We will describe both resolutions π and ξ in §2. 4. By [15, §6.2] and [18,Theorem 8.4.7], we have where IC • S denotes the intersection cohomology complex of S [12, p. 156], and D b c (S) is the constructible derived category of sheaves of Q-vector spaces on S.
By the celebrated Decomposition theorem [2,3,4,22], the intersection cohomology complex of S is also a direct summand of Rπ * Q S [n] in D b c (S). Specifically, the Decomposition theorem says that there is a decomposition in D b c (S) [4, Theorem 1.6.1] where p H i (Rπ * Q S [n]) denote the perverse cohomology sheaves [4, §1.5]. Furthermore, the perverse cohomology sheaves p H i (Rπ * Q S [n]) are semisimple, i.e. direct sum of intersection cohomology complexes of semisimple local systems, supported in the smooth strata of S. In the paper [13], the summands involved in (2) are explicitly described. It turns out that the semisimple local systems involved in the decomposition are constant sheaves supported in the smooth strata of π. In other words, the decomposition (2) takes the following form for suitable multiplicities m pq ∈ N 0 (that are computed in [13,Theorem 3.5]) and where the strata ∆ p are special Schubert varieties, as well.
Following the same lines as in [10, section 4], our main aim is to deduce some classes of polynomial identities from the isomorphism (3). Specifically, we are going to prove a class of local identities as well as a class of global identities.
Our first task is accomplished in Theorem 3.1. In a nutshell, the argument behind our local polynomial identity rests on the remark that each summand of (3) is a direct sum of shifted trivial local systems in D b c (∆ 0 p ), when restricted to the smooth part ∆ 0 p of each stratum ∆ p . This fact follows by applying the Leray-Hirsch theorem (see [23,Theorem 7.33], [8,Lemma 2.5]) to the summands, that are described on ∆ 0 p by means of suitable Grassmann fibrations. This implies that we are allowed to associate a Poincaré polynomial to each summand of (3), thus providing our local identity in the stratum ∆ 0 p (for more details compare with §3). As for the global polynomial identities, the idea is very similar to that of [10, section 4] (compare with Theorem 4.2). From (3) we deduce an isomorphism among the i-th hypercohomology spaces that leads to an equality of the corresponding Poincaré polynomials Again, all summands of (5) are determined by means of Leray-Hirsch theorem as Poincaré polynomials of suitable Grassmann fibrations. We also observe that an explicit inductive algorithm for the computation of the Poincaré polynomials of the intersection cohomology of Special Schubert varieties straightforwardly follows from our results (see Corollary 4.3 and Remark 4.4). Although these Poincaré polynomials are already known, the availability of an algorithm for their computation could be the starting point for obtaining an analogous algorithm for all Schubert varieties in a future paper, being an explicit formula in this general case not known yet.
In the Appendix, we give an example of proof of the global polynomial identity of Theorem 4.2 in a particular case, by algebraic manipulation only, with a divide and conquer strategy.

Basic facts and notations
2.1. Preliminaries. Throughout the paper, we shall work with Q-coefficients cohomology groups; that is, for any complex variety V and any integer k, Decomposition theorem. The Decomposition theorem, which was proved by A. Beilinson, J. Bernstein and P. Deligne in [2], is a tool of paramount importance: most of our results descend from it directly.
Furthermore, the perverse sheaves p H i (Rf * IC X ) are semisimple; i.e., there is a decomposition into finitely many disjoint locally closed and nonsingular subvarieties Y = S β and a canonical decomposition into a direct sum of intersection complexes of semisimple local systems In the literature one can find different approaches to the Decomposition Theorem [2], [3], [4], [22], [24], which is a very general result but also rather implicit. On the other hand, there are many special cases for which the Decomposition Theorem admits a simplified and explicit approach. One of these is the case of varieties with isolated singularities [21,9,11]. For instance, in the work [9], a simplified approach to the Decomposition Theorem for varieties with isolated singularities is developed, in connection with the existence of a natural Gysin morphism, as defined in [7,Definition 2.3] (see also [6] for other applications of the Decomposition Theorem to the Noether-Lefschetz Theory).

2.3.
Grassmannians and Poincaré polynomials. We shall denote by G k (C n ) the Grassmannian of k-vector subspaces of C n ; that is, the set of all k-dimensional subspaces of C n . More in general, we can extend this definition by replacing C n with any complex vector space V (see [16, §6]  Let X be a topological space. The Poincaré polynomial H X of its cohomology and the Poincaré polynomial IH X of its intersection cohomology (later on, they will be simply called Poincaré polynomials) are given by respectively. When X = G k (C l ), we have the following explicit formula of the Poincaré polynomial (see [5, p. 328 where we assume P 0 = 1 and take

Special Schubert Varieties.
In this subsection we collect some facts concerning special Schubert varieties and their resolutions. For more details and explanations we refer the reader to [13, §2.2 -2.6]. Let i, j, k, l be integers such that and fix a j-dimensional subspace F ⊆ C l . We are working with single condition (or special) Schubert varieties and we are considering the Whitney stratification where, for any p, is a special Schubert variety, as well, and ∆ p = Sing ∆ p+1 . For any 0 < q < p ≤ r + 1 there is a commutative diagram is a resolution of singularities, and the function The resolutions π p are small when k ≤ c (see [

Local polynomial identities
Before we give the proof of the first theorem, we shall fix some notations in order to make it more readable. For any pair of integers (p, q) with 0 < q < p, we set Theorem 3.1. For any pair of integers (p, q) with 0 < q < p there is a local polynomial identity Proof. By the Decomposition theorem [4, Theorem 1.6.1], we know that In [13,Remark 3.1] it is shown how the Leray-Hirsch theorem implies that (for a generalization of the Leray-Hirsch theorem in a categorical framework we refer to [23,Theorem 7.33] and [8, Lemma 2.5]). In addition, in [13,Theorem 3.5] it is proved that where i pr : ∆ r ֒→ ∆ p is an inclusion. By [13, Remark 3.3], we also have Combining these results, we obtain where r ∈ {q, . . . , p} because ∆ r \∆ 0 q = ∅ whenever r < q. Since the γ-th cohomology group of a topological space is trivial when γ < 0, we obtain The right-hand complex can be rewritten as follows and, for any γ, its γ-th term is The isomorphism (6) implies that the γ-th terms of those complexes are isomorphic for any We shall observe one last thing before we compute Poincaré polynomials. For any n ∈ N, G 0 (C n ) = {0}, as it is the space of 0-dimensional subspaces through the origin. As a consequence, Therefore, when r = p, Similarly, when r = q, we have In conclusion, for any γ ≥ −m p , we have Let s = γ + m p and recall that m p − m r − δ pr = 2d pr (see [13, §2.6]).
for any s ≥ 0. At long last, if we denote by we obtain identities If we formally multiply both sides by t s , and if we take the sum over s ∈ Z s a s pq · t s = p−1 We are done, because, by definition of Poincaré polynomials, the above equality becomes that is,

Global polynomial identities
We shall begin by introducing some further notations: Recall that we defined a small resolution ξ p : D p → ∆ p as provides an isomorphism between G and G k−ip (C l−j ). Therefore, we recognize D p as the Grassmannian of subspaces of dimension k of the restriction of the tautological bundle S over G k+j−ip (C l ) to the subspace G: By the Leray-Hirsch theorem [23,Theorem 7.33], we have (compare also with [8, §2] for a discussion of the Leray-Hirsch theorem in a context which is closely related with that considered here).
The following formula, which represents the main result if this paper, provides a strong generalization of [10, §2, Remark 4.2].
Theorem 4.2. For any q < p, we have: Proof. The first part of the proof is similar to that of Theorem 3.1. We combine the Decomposition theorem [4, Theorem 1.6.1] and [13, Theorem 3.5] so as to obtain that is, We have already met the term D δpp+α pp and the second summand of the right-hand side is the zero complex since ∆ 0 = ∅. Hence, we have where we have also taken account of m p − m r − δ pr = 2d pr (see [13, §2.6]).
If we apply hypercohomology, we obtain (for any β ∈ Z) where IC • ∆q → I • is an injective resolution of IC • ∆q and Γ is the global section functor. Similarly, where Q∆ p → I • is an injective resolution and the last equality is [17, Theorem 7.12, p. 242]. Substituting in the above isomorphism, we obtain As we did in the proof of Theorem 3.1, we conclude which can be compactly rewritten as Again, Leray-Hirsch theorem implies that∆ p has the same Poincaré polynomial as G ip (F ) × G k−ip (C l−ip ). Thus, the left-hand side is By virtue of [13,Formula (19)] and Remark 4.1, we have in other words, Adopting the same notations as §2.3, we have .

Formula (7) becomes
Since we are interested in studying the Poincaré polynomials of the Schubert variety S, we are going to take p = r + 1. Bearing in mind that i q = k −q + 1 (in particular, i p = i r+1 = k −r = i) and c = l − j, if we set s = p − q = r + 1 − q, we have (from left to right, numerators first) and the previous equality becomes (2d pq = 2(p − q)(c + 1 − q) = 2s(c + s − r)) Remark 4.4. From the previous corollary we get an explicit inductive algorithm for the computation of Poincaré polynomials of the intersection cohomology of Special Schubert varieties, which is described by the following equality, where we put g pq = t 2dpq f pq in order to simplify the notation:  0 g r+1,r g r+1,r−1 g r+1,r−2 . . . g r+1,1 0 0 g r,r−1 g r,r−2 . . . g r,1 0 0 0 g r−1,r−2 . . . g r−1,1 . .

Appendix: a symbolic point of view
In this Appendix, we consider the global polynomial identity of Theorem 4.2 from a symbolic point of view. Recall that the requests over the integers i, j, k, j, l and the values r := k − i, c := l − j are 0 < i < k ≤ j < l and 0 < r < c < k, or, equivalently, (8) 0 < i < j and 0 < r < c < r + i ≤ j.
Nevertheless, if we also assume P α = 0 for every α < 0, the polynomial identity of Theorem 4.2 symbolically makes sense, i.e. the denominators do not vanish, under the following weaker assumptions: However, in the few further cases r = 0 or c = r + i or i = 0 or i = j, which we have under the assumptions (9), the polynomial identity trivially holds. For the remaining case c = r, we obtain some experimental evidences verifying the polynomial identity for the 4-uples (i, j, c, r), where c = r varies in [2,20], i in [1,20] and j in [r + i, 40], by direct computations performed in CoCoA5 (see [1]). In the following, we consider the assumptions (8) (except for some special cases that will be highlighted), and give a proof of that identity by a mere algebraic manipulation when 2 = min{k − i, k − c}, which is the first case with a significant geometric meaning in the context of this paper.
By direct computations, we also verified that the polynomial identity of Theorem 4.2 holds for all the 4-uples (i, j, c, r) where i varies in [1,20], r in [2,20], j in [r + i, 40] and c in [r + 1, r + i − 1]. At http://wpage.unina.it/cioffifr/PolynomialIdentity, some CoCoA5 functions that perform such computations are available. 5.1. Case 2 = k − i ≤ k − c. With the same notation of Theorem 4.2 and recalling that k = i + 2 and ℓ = j + c, the global polynomial identity becomes: where only the parameters i, j, c appear. Note that formula (10) does make sense for every j ≥ i + 2 and c ≥ 2. Let Applying the fact that Hence, letting formula (10) holds if and only if F (i, j, c) = 1. Observe that F (i, j, c) does make sense for every c ≥ 2 and for every positive integers i, j. In the following, we repeatedly apply the equality t 2α h β = h α+β − h α−1 , which holds for every α, β ≥ 0.
We now show that F (i, j, Although we should consider i ≥ c ≥ 3 due to (8), symbolically we easily obtain Hence, by induction, we can assume F (i − 1, j, 3) = 1 and prove that So, letting Recall that we are assuming k − i ≤ k − c, hence i ≥ c, and we know that F (i, j, 3) = 1. So, our thesis now is F (i, j, c) − F (i, j, c − 1) = 0. Assuming c > 3, we compute So, letting Being Being Note that, if j = i or j = i + 1, then Q 0 + Q 1 = 0 = Q 2 . So, we now consider the less obvious case j > i + 1. Observe that We are done, because: In this case we have r + i = k = c + 2, ℓ = j + c, c = r + i − 2, and the global polynomial identity becomes: P j P j+r−2 P i P j−i P r P j−2 = P r+i−2 P r P i−2 P r+j P r+i P j−i + +t 2(i−1) P 2 P r+i−2 P r+j−1 P r−1 P i−1 P r+i P j−i−1 + t 4i P 2 P r+i−2 P r+j−2 P 2 P r−2 P i P r+i P j−i−2 where only the parameters i, j, r appear. Note that formula (11) does make sense for every 2 ≤ r and 2 ≤ i ≤ j − 2. Let K := P j P j+r−2 P i P j−i P r P j−2 , E := P r+i−2 P r P i−2 P r+j P r+i P j−i , E 1 := t 2(i−1) P 2 P r+i−2 P r+j−1 P r−1 P i−1 P r+i P j−i−1 , E 2 := t 4i P 2 P r+i−2 P r+j−2 P 2 P r−2 P i P r+i P j−i−2 .