CLASSIFICATION OF DING’S SCHUBERT VARIETIES: FINER ROOK EQUIVALENCE

MIKE DEVELIN, JEREMY L. MARTIN, AND VICTOR REINER

Abstract. K. Ding studied a class of Schubert varieties X_{λ} in type A partial flag manifolds, indexed by integer partitions λ and in bijection with dominant permutations. He observed that the Schubert cell structure of X_{λ} is indexed by maximal rook placements on the Ferrers board B_{λ}, and that the integral cohomology groups H^{∗}(X_{λ}; ℤ), H^{∗}(X_{μ}; ℤ) are additively isomorphic exactly when the Ferrers boards B_{λ}, B_{μ} satisfy the combinatorial condition of rook-equivalence.

We classify the varieties X_{λ} up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

1. Introduction

The goal of this paper is to classify up to isomorphism a certain class of Schubert varieties within partial flag manifolds of type A. Although this is partly motivated as a first step toward the isomorphism classification of all Schubert varieties, we choose here to explain instead our original motivation, stemming from rook theory in combinatorics.

A board B is a subset of the squares on an N × N chessboard, and a k-rook placement on B is a subset of k squares in B, no two in a single row or column. Kaplansky and Riordan [9] considered the problem of when two boards B, B′ are rook-equivalent, that is, when for each k ≥ 0, the number R_{k}(B) of k-rook placements is the same as R_{k}(B′).

Foata and Schützenberger [4] solved the problem for the well-behaved subclass of Ferrers boards B_{λ}; these are the usual Ferrers diagrams associated to partitions^{1}

λ = (0 ≤ λ_{1} ≤ … ≤ λ_{n})  (1)

having all squares left-justified in their row, with λ_{1} squares in the bottom row, λ_{2} in the next, etc. They showed that each rook-equivalence class of Ferrers boards has a unique representative which is a strict partition, i.e., satisfying λ_{i} < λ_{i+1}. Goldman, Joichi and White [8] re-proved this result by showing that B_{λ} and B_{μ} are rook-equivalent if and only if the multisets of integers {λ_{i} − i}_{i=1}^{n} and {μ_{i} − i}_{i=1}^{n} coincide.

Garsia and Remmel [6] defined q-rook polynomials R_{k}(B_{λ}, q) that q-count the k-rook placements on B_{λ} by a certain “inversion” statistic generalizing inversions of permutations. They also showed that the problem of q-rook equivalence is the same as that of rook equivalence. When λ_{i} ≥ i for each i, this can be deduced from a product formula for R_{n}(B_{λ}, q) that counts placements of n rooks: up to a factor of q it is

∏_{i=1}^{n}[λ_{i} − i + 1]_{q}  (2)

where [m]_{q} := \frac{q^{m}−1}{q−1} = 1 + q + q^{2} + ⋯ + q^{m−1}.

Date: August 24, 2004.

2000 Mathematics Subject Classification. Primary 14M15; Secondary 05E05.

Key words and phrases. Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence.

This work was completed while the first author was visiting the Univeristy of Minnesota. First author supported by the American Institute of Mathematics. Second author partially supported by an NSF Postdoctoral Fellowship. Third author partially supported by NSF grant DMS–0245379.

1NB: we are writing our partitions with the parts in weakly increasing order, contrary to usual combinatorial conventions, but more convenient in this setting.

Figure 1. A decomposable partition λ. The unshaded regions are λ^{(1)} and λ^{(2)}.

K. Ding [2, 3] interpreted this product as the Poincaré series for a certain algebraic variety X_{λ} which he called a partition variety. Fix a standard complete flag of subspaces

0 ⊂ ℂ^{1} ⊂ ⋯ ℂ^{N−1} ⊂ ℂ^{N}

and define

X_{λ} := {flags 0 ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n} ⊂ ℂ^{N} : dim_{ℂ} V_{i} = i and V_{i} ⊂ ℂ^{λ_{i}}}.  (3)

The set X_{λ} may be endowed with the structure of a smooth complex projective variety, and (although not stated explicitly in [2]) is in fact a smooth Schubert variety inside the partial flag manifold X_{N^{n}}, where N^{n} denotes the rectangular board with n rows and N columns. As we shall explain below, the Schubert varieties arising in this way are (in the notation of [5, §10.2]) those of the form X_{w}, where w is a 312-avoiding permutation. Equivalently, the fundamental cohomology class [X_{w}] is represented by a Schubert polynomial indexed by a dominant or 132-avoiding permutation. (See [5] for a reference on Schubert varieties, and [10] for a detailed treatment of Schubert polynomials.) Ding observed that the Schubert cell structure inherited by X_{λ} has cells indexed by n-rook placements on B_{λ}, and with the dimension of the cell governed by Garsia and Remmel’s inversion statistic. Since these cells are all even-dimensional, their (co)homology is free abelian, occurring only in even dimension, and the Poincaré series of X_{λ} is given by the q-rook polynomial formula (2). From this Ding concluded [3] that two partition varieties X_{λ}, X_{μ} have additively isomorphic (co)homology groups if and only if B_{λ} and B_{μ} are rook-equivalent.

It is natural to ask when two such Ding partition varieties X_{λ}, X_{μ} have isomorphic (graded) cohomology rings, or even when they are isomorphic as varieties. The main result of this paper is that the answers to both questions are the same. We make use of recent results of Gasharov and the third author [7], giving simple explicit cohomology ring presentations^{2} for a more general class of Schubert varieties in partial flag manifolds (those defined by a conjunction of inclusion conditions of the forms ℂ^{j} ⊂ V_{i} and V_{i} ⊂ ℂ^{j}).

To state our main result, we first note one trivial source of isomorphisms among the partition varieties X_{λ}. We assume throughout that λ_{i} ≥ i for every i, for otherwise X_{λ} = ∅. However, if λ_{k} = k for some k, then the condition V_{k} ⊂ ℂ^{k} with dim_{ℂ} V_{k} = k forces V_{k} = ℂ^{k}, so that X_{λ} is isomorphic to X_{λ^{(1)}} × X_{λ^{(2)}}, where

\begin{aligned}λ^{(1)} & = (λ_{1}, …, λ_{k−1}), \\ λ^{(2)} & = (λ_{k+1} − k, …, λ_{n} − k). \end{aligned}

Here if k = n, so that λ_{n} = n, there is no partition λ^{(2)} and we simply note that X_{λ} ≅ X_{λ^{(1)}}.

Say that λ is decomposable if this occurs (i.e., if λ_{k} = k for some k), and indecomposable otherwise. For example, the partition λ = (5, 5, 5, 6, 6, 6, 8, 9) shown in Figure 1 is decomposable since λ_{6} = 6. In this case, one has λ^{(1)} = (5, 5, 5, 6, 6) and λ^{(2)} = (2, 3), as shown in the figure.

2It is amusing that these cohomology ring presentations for Schubert varieties are often derived for the purposes of enumerative geometry (Schubert calculus), but are used here for a different classical topological purpose, namely distinguishing non-homeomorphic spaces.

Iterating this, one can decompose λ into a multiset of indecomposable partitions {λ^{(i)}}_{i=1}^{r}, which we will call its indecomposable components, such that

X_{λ} ≅ X_{λ^{(1)}} × ⋯ × X_{λ^{(r)}}.  (4)

Our main result is that the Schubert varieties X_{λ} are determined up to isomorphism by these multisets of indecomposable components. It should be compared with the result of Goldman, Joichi and White [8], which can now be rephrased: the varieties X_{λ} are determined up to additive (co-)homology isomorphism by the multisets of numbers {λ_{i} − i}.

Theorem 1.1. The following are equivalent for two partitions, λ = (λ_{1}, …, λ_{m}) and μ = (μ_{1}, …, μ_{m′}): (i) The multisets of indecomposable components, {λ^{(i)}}_{i=1}^{r} and {μ^{(i)}}_{i=1}^{r′}, are identical. (ii) There is an isomorphism X_{λ} ≅ X_{μ} of varieties. (iii) There is a graded isomorphism of integer cohomology rings H^{∗}(X_{λ}; ℤ) ≅ H^{∗}(X_{μ}; ℤ).

The implications (i) ⟹ (ii) ⟹ (iii) are clear; the hard part is to show that (iii) implies (i). It turns out that the key to this implication lies in understanding the nilpotence orders of cohomology elements f ∈ H^{2}(X_{λ}; ℤ); that is, the least k for which f^{k} = 0.

In Section 2, we review some of Ding’s results, and re-prove somewhat more directly the presentation for H^{∗}(X_{λ}; ℤ) from [7]. The three sections that follow are the technical heart of the paper, categorizing elements in H^{2}(X^{λ}; ℤ) of minimal nilpotence order. We begin in Section 3 by setting up some Gröbner basis machinery that we shall use throughout (for a general reference on Gröbner basis theory, see [1]). Section 4 deals with nilpotents in the cohomology of the complete flag variety (that is, when λ is a square Ferrers board) and Section 5 treats the case of arbitrary X_{λ}. Using these tools, we prove in Section 6 that an indecomposable partition λ may be recovered from the structure of H^{∗}(X_{λ}; ℤ) as a graded ℤ-algebra. Finally, in Section 7, we show that in the general case, H^{∗}(X_{λ}; ℤ) has an essentially unique decomposition as a tensor product of graded ℤ-algebras, whose factors correspond to the indecomposable components of the partition λ.

It is curious that this unique tensor decomposition fails if instead of the integer cohomology ring H^{∗}(X_{λ}; ℤ) one takes cohomology with coefficients in a ring where 2 is invertible; see Remark 7.6 below.

2. Review of X_{λ} and the presentation of H^{∗}(X_{λ}; ℤ)

For the sake of completeness, and also to collect facts for future use, we begin by re-proving some of Ding’s results from [2], and re-derive somewhat more directly the presentation given in [7] for the cohomology ring of X_{λ}. Throughout this paper, all cohomology groups and rings are taken with integer coefficients unless otherwise specified. We begin by identifying the Schubert varieties that arise as Ding’s varieties X_{λ}. (See [5, §10.6] for more information on Schubert varieties, and [10] for a detailed treatment of Schubert polynomials.)

Let 𝔖_{N} be the symmetric group of permutations of {1, …, N}, and let 𝔖_{{n+1,n+2,...,N}} be the subgroup of permutations fixing {1, …, n} pointwise. Consider the partial flag variety

X_{N^{n}} = {flags 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{n} ⊂ ℂ^{N} : dim V_{i} = i}.

Let w = w_{1} … w_{n} ∈ 𝔖_{N} be a permutation which is a maximum-length representative for its coset in 𝔖_{N}/𝔖_{{n+1,n+2,...,N}}. The corresponding Schubert variety X_{w} ⊂ X_{N^{n}} is defined to be

X_{w} = {flags 0 ⊂ V_{1} ⊂ ⋯ ⊂ V_{n} ⊂ ℂ^{N} : dim V_{i} = i, dim V_{i} ∩ ℂ^{j} ≥ #{k ≤ i : w_{k} ≤ j}}.

Let λ be a partition of the form (1), and let N = λ_{n}. It is easy to check that Ding’s variety X_{λ} coincides with the Schubert variety X_{w} ⊂ X_{N^{n}}, where w is the unique permutation given by the recursive rule

w_{i} = max ({1, …, λ_{i}} \ {w_{1}, …, w_{i−1}}) .

Note that if n = N, then w corresponds to the maximal rook placement on the Ferrers board B_{λ} given by the following algorithm: let i increase from 1 to n, and for each i, place a rook in row i and column w_{i}, where w_{i} is the rightmost square in row i whose column does not already contain a rook. For instance, if λ = (2, 4, 4, 5, 5), then w = 24351 ∈ 𝔖_{5}. (If n < N, then we must first augment λ with N −n additional rows of length λ_{n}.) It is not hard to verify that the permutations w obtained in this way are exactly those which are 312-avoiding; that is, there do not exist i, j, k for which i < j < k and w(i) > w(k) > w(j). Equivalently,

Figure 2. A partition λ and the subpartition ν (shaded) such that H^{∗}(X_{λ}) / ⟨x_{1}⟩ = H^{∗}(X_{ν}).

the cohomology class [X_{w}] ∈ H^{∗}(X_{N^{n}}) is represented by a Schubert polynomial which is a single monomial, namely the Schubert polynomial indexed by the dominant (or 132-avoiding) permutation w_{0}w, where w_{0} is the unique permutation of maximal length. (We thank Ezra Miller for discussions clarifying these points.)

Because X_{λ} is a Schubert variety, it comes equipped with a Schubert cell decomposition, having cells in only even real dimensions. As observed by Ding, this has important consequences:

Theorem 2.1 (Ding [2]). The integral cohomology ring H^{∗}(X_{λ}; ℤ) is free abelian (that is, it has no torsion), is nonzero only in even homological degrees, and has Poincaré series

Poin(X_{λ}, q) := ∑_{i≥0} q^{i} rank_{ℤ} H^{2i}(X_{λ}; ℤ) = ∏_{i=1}^{n}[λ_{i} − i + 1]_{q}.

Proof. The cohomology is free abelian and concentrated in even degrees because the Schubert cell decomposition for the Schubert variety X_{λ} has cells only in even dimensions.

For the assertion about the Poincaré series, we will induct on n. The map

\begin{matrix}X_{λ} & → & ℙ(ℂ^{λ_{1}}) ≅ ℙ_{ℂ}^{λ_{1}−1} \\ \{V_{i}\}_{i=1}^{n} & ↦ & V_{1} \end{matrix}

is an (algebraic) fiber bundle, with fiber isomorphic to X_{ν}, where

ν = (ν_{1}, …, ν_{n−1}) = (λ_{2} − 1, …, λ_{n} − 1)

is the partition obtained by removing the first row and column from λ (see Figure 2). The Leray-Serre spectral sequence is particularly simple in this situation, because both base and fiber are simply-connected (again due to the Schubert cell decomposition) and have homology concentrated in even dimension. This causes the spectral sequence to degenerate at the E^{1}-page, yielding

Poin(X_{λ}, q) = Poin(X_{ν}, q) · Poin(ℙ_{ℂ}^{λ_{1}−1}, q).

The assertion about Poin(X_{λ}, q) now follows by induction on n, using the fact that [m]_{q} = Poin(ℙ_{ℂ}^{m−1}, q). □

We now set about deriving the presentation for H^{∗}(X_{λ}). To this end, we recall Borel’s picture for the cohomology of the complete flag manifold GL_{N}(ℂ)/B = X_{N^{N}} and the partial flag manifold X_{N^{n}}; see [5, Chapter 10, §3, §6]. We will use the following notation for symmetric functions in various sets of variables. For integers 1 ≤ i ≤ j ≤ N and m ≥ 0, define the m^{th} elementary and complete homogeneous symmetric functions, respectively, by

e_{m}(i, j) := e_{m}(x_{i}, x_{i+1}, …, x_{j}),
h_{m}(i, j) := h_{m}(x_{i}, x_{i+1}, …, x_{j}),

where

e_{m}(N) := e_{m}(1, N) = e_{m}(x_{1}, …, x_{N}) = ∑_{1≤i_{1}<···<i_{m}≤N} x_{i_{1}} ⋯ x_{i_{m}},
h_{m}(N) := h_{m}(1, N) = h_{m}(x_{1}, …, x_{N}) = ∑_{1≤i_{1}≤···≤i_{m}≤N} x_{i_{1}} ⋯ x_{i_{m}}.

According to Borel’s picture, H^{∗}(X_{N^{N}}) ≅ ℤ[x_{1}, …, x_{N}]/J, where J = ⟨e_{1}(N), …, e_{N}(N)⟩ is the ideal generated by all symmetric functions of positive degree, and where x_{i} represents the negative of c_{1}(L_{i}), the first Chern class of the line bundle L_{i} on GL_{N}(ℂ)/B whose fiber over the flag {V_{i}}_{i=0}^{N} is V_{i}/V_{i−1}.

Furthermore, the surjection X_{N^{N}} → X_{N^{n}} which forgets the subspaces of dimension greater than n in a complete flag induces a map H^{∗}(X_{N^{n}}) → H^{∗}(X_{N^{N}}) which turns out to be injective, and the image of H^{∗}(X_{N^{n}}) is identified with the invariant subring H^{∗}(X_{N^{N}})^{𝔖_{{n+1,n+2,...,N}}}. This invariant subring may be presented as S/J′, where

S = ℤ[x_{1}, …, x_{N}]^{𝔖_{{n+1,n+2,...,N}}} = ℤ[x_{1}, …, x_{n}, e_{1}(n + 1, N), …, e_{N−n}(n + 1, N)]

and J′ = ⟨e_{1}(N), …, e_{N}(N)⟩ is the ideal of S with the same generators as J.

The relations in the ideals J and J′ induce further relations among various symmetric functions, which we record here for future use.

Proposition 2.2 (cf. [5, p. 163, eqn. (4)]). For every m ∈ {1, 2, …, N} and j ≥ 0, one has

h_{j}(m) ≡ (−1)^{j}e_{j}(m + 1, N) (mod J).

Proof.

∏_{i=1}^{m}(1 + x_{i}t) ∏_{i=m+1}^{N}(1 + x_{i}t) = ∏_{i=1}^{N}(1 + x_{i}t) = ∑_{j=0}^{n} e_{j}(N)t^{j} ≡ 1 (mod J).

Hence

∑_{j=0}^{∞}(−1)^{j}h_{j}(m)t^{j} = ∏_{i=1}^{m}(1 + x_{i}t)^{−1} ≡ ∏_{i=m+1}^{N}(1 + x_{i}t) = ∑_{j=0}^{N−m} e_{j}(m + 1, N)t^{j} (mod J).

Now comparing coefficients of powers of t^{j} yields the desired equality. □

We now give the general presentation for the integral cohomology of X_{λ} (as pointed out in [7, Remark 3.3]).

Theorem 2.3. Let λ be a partition with 1 ≤ λ_{1} ≤ ⋯ ≤ λ_{n} = N and λ_{i} ≥ i for all i. Let

R^{λ} := ℤ[x_{1}, …, x_{n}]/I_{λ}

where I_{λ} := ⟨h_{λ_{i}−i+1}(i) : 1 ≤ i ≤ n⟩.

Then there is a (grade-doubling) ring isomorphism

R^{λ} → H^{∗}(X_{λ}; ℤ)

sending x_{i} to −c_{1}(L_{i}). Here L_{i} is the same line bundle as above, but restricted to X_{λ} from the partial flag manifold X^{N^{n}}.

Proof. The obvious inclusion X_{λ} ↪ X_{N^{n}} induces a map H^{∗}(X_{N^{n}}) → H^{∗}(X_{λ}). This ring map is surjective, because X_{λ} inherits from X_{N^{n}} a decomposition into Schubert cells, and the dual cocycles to these (evendimensional) cells additively generate the cohomology in each case.

There are further relations on the Chern classes x_{i} in H^{∗}(X_{λ}) due to the conditions V_{i} ⊂ ℂ^{λ_{i}}. Specifically, the bundle ℂ^{N}/V_{i} on X_{λ} will have the same Chern classes as the direct sum ℂ^{N}/ℂ^{λ_{i}} ⊕ ℂ^{λ_{i}}/V_{i}, in which ℂ^{N}/ℂ^{λ_{i}} is a trivial bundle. Thus when restricted to X_{λ}, the bundle ℂ^{N}/V_{i} will have the same Chern classes as the bundle ℂ^{λ_{i}}/V_{i} of rank λ_{i} − i. Hence its Chern classes c_{m} = ±e_{m}(i + 1, N) for m > λ_{i} − i inside H^{∗}(X_{λ}) must vanish. Consequently, we have a surjection of rings

ℤ[x_{1}, …, x_{n}, e_{1}(n + 1, N), e_{2}(n + 1, N), …, e_{N−n}(n + 1, N)]/J_{λ} → H^{∗}(X_{λ})  (5)

where

J_{λ} := J′ + ⟨e_{j}(i + 1, N) : 1 ≤ i ≤ n and j > λ_{i} − i⟩ .

We now manipulate the quotient ring ℤ[x_{1}, …, x_{N}]^{𝔖_{{n+1,n+2,...,N}}}/J_{λ} on the left of (5). We use Proposition 2.2 to draw two conclusions:

(i) Applying Proposition 2.2 with m = n shows that H^{∗}(X_{N^{n}}) and H^{∗}(X_{λ}) are generated as algebras by x_{1}, …, x_{n}, since their generators of the form e_{i}(n+1, N) can be expressed modulo J′ as (symmetric) polynomials in x_{1}, …, x_{n}.

(ii) Applying it with m = i for 1 ≤ i ≤ n shows that h_{λ_{i}−i+1}(i) = 0 in H^{∗}(X_{λ}), because for each

j ≥ λ_{i} − i, h_{j}(i) is congruent modulo J′ to ±e_{j}(i + 1, N). Consequently, there is a surjection of rings

ℤ[x_{1}, …, x_{n}]/ ⟨h_{λ_{i}−i+1}(i) : 1 ≤ i ≤ n⟩ → H^{∗}(X_{λ}).  (6)

On the other hand, the set

{h_{λ_{i}−i+1}(i) : 1 ≤ i ≤ n}

is a Gröbner basis for I_{λ} with respect to the lexicographic term order on ℤ[x_{1}, …, x_{n}] given by x_{1} < ⋯ < x_{n}. Indeed, the initial term of h_{λ_{i}−i+1}(i) is x_{i}^{λ_{i}−i+1}, so these generators have pairwise relatively prime, monic initial terms. Consequently, the quotient ring on the left of (6) is a free ℤ-module of rank ∏_{i=1}^{n}(λ_{i}−i+1), with ℤ-basis given by the standard monomials (those divisible by none of the initial terms), namely {x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} : a_{i} ≤ λ_{i} − i}. Since Theorem 2.1 implies that H^{∗}(X_{λ}) is a free ℤ-module of the same rank, the surjection (6) must be an isomorphism. □

For example, if λ is the partition shown in Figure 1, then the Gröbner basis for I_{λ} is

h_{5}(1), h_{4}(2), h_{3}(3), h_{3}(4), h_{2}(5), h_{1}(6), h_{2}(7), h_{2}(8).

The previous proof shows that I_{λ} is the elimination ideal

I_{λ} = ℤ[x_{1}, …, x_{n}] ∩ J_{λ}.

This observation has some useful corollaries, which can also be proved by direct combinatorial/algebraic arguments avoiding any use of geometry. The first corollary is the algebraic manifestation of the (surjective) map R^{λ} → R^{μ} induced by the inclusion of Schubert varieties X_{λ} → X_{μ}.

Corollary 2.4. Let λ and μ be partitions, both with n nonzero rows, such that λ ⊃ μ.

Then I_{λ} ⊂ I_{μ}, and consequently, R^{λ} is a quotient of R^{μ}.

Proof. By definition of J_{λ}, one has J_{λ} ⊂ J_{μ} in this situation. □

Corollary 2.5. If λ_{i} = λ_{i+1} = ⋯ = λ_{j} for some i < j then the ideal I_{λ} is invariant under permutations of the variables x_{i}, x_{i+1}, …, x_{j}.

Proof. It suffices to show that J_{λ} has this same invariance. Note that the generators for J_{λ} of the form

e_{m}(i′ + 1, N) for i ≤ i′ < j and m > λ_{i′} − i′

are all redundant, as they lie in the ideal generated by {e_{m}(j + 1, N) : m > λ_{j} − j}. The latter generators, and all other generators of J_{λ}, are symmetric in x_{i}, x_{i+1}, …, x_{j}. □

3. Two reduced Gröbner bases

This section examines the Gröbner bases for I_{λ} for two extreme cases of indecomposable partitions. In both cases, one can describe the (unique) reduced Gröbner basis, which will be used in an essential way later in the paper. We assume some familiarity with “Gröbner basics” on the reader’s part; a good reference for this topic is [1].

We begin with some notation regarding Gröbner reduction. Since the generators {h_{λ_{i}−i+1}(i) : 1 ≤ i ≤ n} form a Gröbner basis for I_{λ} with respect to a lexicographic monomial ordering in which x_{1} < … x_{n}, we can compute in the quotient R^{λ} by reducing polynomials modulo this Gröbner basis. For a polynomial f ∈ ℤ[x_{1}, …, x_{n}], we will denote by \overline{f} this standard form of f. That is, \overline{f} is the unique ℤ-linear combination of standard monomials {x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} : a_{i} ≤ λ_{i} − i} which is congruent to f modulo I_{λ}. Given a standard

Figure 3. An indecomposable partition λ and its core subpartition μ (shaded).

monomial M, we denote by [M]\overline{f} the coefficient of M in \overline{f}. (This is well-defined, because the standard monomials form a basis for ℤ[x_{1}, …, x_{n}]/I_{λ} as a free ℤ-module.)

Let λ = (λ_{1} ≤ ⋯ ≤ λ_{n}) and for some fixed m ≤ n, let μ = (λ_{1}, …, λ_{m}). Then the fact that we are using a lexicographic order to perform reductions has the following easy consequence (see also [1, §3.1]), which will be used frequently. It can be viewed as an algebraic consequence of the fibration X_{λ} → X_{μ} that forgets the subspaces of dimension greater than m in a flag, which happens to induce an injective map H^{∗}(X_{μ}) → H^{∗}(X_{λ}).

Proposition 3.1. Let λ and μ be related as above. Suppose that f in ℤ[x_{1}, …, x_{n}] lies in some subalgebra ℤ[x_{1}, …, x_{m}], where m ≤ n.

Then the images of f in R^{λ} and R^{μ} have the same standard form \overline{f}.

Our first extreme case arises when λ is an indecomposable partition with λ_{i} = p, and μ ⊂ λ is the smallest indecomposable partition having μ_{i} = p, namely μ = (2, 3, …, i − 1, i, p).

Proposition 3.2. Let μ = (2, 3, …, i − 1, i, p). With respect to lexicographic order on ℤ[x_{1}, …, x_{m}] with x_{1} < ⋯ < x_{m}, the ideal I_{μ} has reduced Gröbner basis

{x_{1}h_{1}(1), x_{2}h_{1}(2), …, x_{i−1}h_{1}(i − 1), x_{i}^{p−i+1} + x_{i}^{p−i}h_{1}(i − 1)}.  (7)

Proof. It is easy to see that the elements of (7) form a reduced Gröbner basis with respect to the lexicographic order for whatever ideal they generate. We observe that this ideal may also be presented as

⟨h_{2}(1), h_{2}(2), …, h_{2}(i − 1), x_{i}^{p−i+1} + x_{i}^{p−i}h_{1}(i − 1)⟩ .

We will show that this ideal is exactly I_{μ}. By Theorem 2.3,

I_{μ} = ⟨h_{2}(1), h_{2}(2), …, h_{2}(i − 1), h_{p−i+1}(i)⟩ ,

so it remains only to show that h_{p−i+1}(i) and x_{i}^{p−i+1} + x_{i}^{p−i}h_{1}(i − 1) are congruent modulo the ideal ⟨h_{2}(1), h_{2}(2), …, h_{2}(i − 1)⟩. Since h_{p−i+1}(i) = ∑_{j=1}^{p−i+1} x_{i}^{j}h_{p−i−j+1}(i − 1), this congruence is immediate from the fact that

h_{m}(ℓ) ∈ ⟨h_{2}(1), h_{2}(2), …, h_{2}(ℓ)⟩ for m ≥ 2,

which is easily proven by double induction on m and ℓ via the identity h_{m}(ℓ) = x_{ℓ}h_{m−1}(ℓ) + h_{m}(ℓ − 1). □

Our second extreme case arises when λ is an indecomposable partition with n rows. Let k = λ_{1}, and let μ be the smallest indecomposable partition with n rows and μ_{1} = k. That is,

\begin{matrix}μ_{1} = μ_{2} = ⋯ = μ_{k−1} = k, \\ μ_{i} = i + 1 for k ≤ i ≤ n. \end{matrix}  (8)

Then μ is a subpartition^{3} of λ, which we will call the core of λ. For example, the core of λ = (4, 4, 6, 6, 8, 10) is the partition μ = (4, 4, 4, 5, 6, 7) (see Figure 3).

3For the purposes of this paper, the statement “μ is a subpartition of λ” means that μ_{i} ≤ λ_{i} for all rows μ_{i} of μ. Equivalently, the Ferrers diagram of μ is contained inside that of λ, when both are left- and bottom-justified.

Proposition 3.3. For k < n, let λ be a partition which is its own core.

Then the polynomials

\begin{aligned}G_{1} & = h_{k}(1), & G_{2} & = h_{k−1}(2), & …, & G_{k−1} & = h_{2}(k − 1), \\ G_{k} & = x_{k}h_{1}(k), & G_{k+1} & = x_{k+1}h_{1}(k + 1), & …, & G_{n} & = x_{n}h_{1}(n) \end{aligned}  (9)

form a reduced Gröbner basis for I_{λ} under the reverse lexicographic term order given by x_{1} < x_{2} < … < x_{n}.

Proof. The initial terms of the G_{i}’s are (in order) x_{1}^{k}, x_{2}^{k−1}, …, x_{k−1}^{2}, x_{k}^{2}, …, x_{n}^{2}. It is evident that no initial term divides any term of any other G_{i}. Therefore, they are a reduced Gröbner basis for the ideal that they generate.

We claim that for every r ∈ {k, k + 1, …, n},

⟨G_{1}, …, G_{r}⟩ = ⟨h_{k}(1), h_{k−1}(2), …, h_{2}(k − 1), h_{2}(k), …, h_{2}(r)⟩ .

The claim is trivial for r = k. For r > k, it follows from induction and the observation that h_{2}(r)−h_{2}(r−1) = x_{r}h_{1}(r) = G_{r}. In particular, the equality for r = n gives ⟨G_{1}, …, G_{r}⟩ = I_{λ}. □

The form of this reduced Gröbner basis has the following consequence, which we will exploit later.

Corollary 3.4. (“Stickiness”) Let λ be an indecomposable partition which is its own core, and k := λ_{1}. Let M be a monomial in x_{1}, …, x_{n}.

Then:

(1) If k ≤ i ≤ n and M is divisible by x_{i}, then so is \overline{M}.

(2) If M is not divisible by any of the variables x_{k}, …, x_{n}, then neither is \overline{M}.

Proof. (1) is immediate from the previous discussion. For (2), the only Gröbner basis elements that can be used in the reduction of M are G_{1}, …, G_{k−1}, so the reduction process cannot introduce a monomial divisible by any of x_{k}, …, x_{n}. □

One useful consequence of “stickiness” is the following.

Corollary 3.5. Let λ be an indecomposable partition which is its own core, and k := λ_{1}. Let f = ∑_{i=1}^{n} a_{i}x_{i} be an element of the degree-one graded piece R_{1}^{λ} of R^{λ}. Decompose f as f = g + h, where

g = ∑_{i=1}^{k−1} a_{i}x_{i}, h = ∑_{i=k}^{n} a_{i}x_{i}.

If f^{m} = 0 in R^{λ} for some positive integer m, then g^{m} = 0 in R^{λ}.

Proof. Note that f^{m} = g^{m} + p, where p is some polynomial divisible by a_{k}x_{k} + ⋯ + a_{n}x_{n}. Passing to the standard forms, we find that 0 = \overline{g^{m}} + \overline{p}. By Corollary 3.4, no monomial in \overline{g^{m}} is divisible by a sticky variable (that is, one of x_{k}, …, x_{n}), but every monomial in \overline{p} is divisible by a sticky variable. Therefore \overline{g^{m}} = 0 (= \overline{p}). □

4. Nilpotence of linear forms in the cohomology of G/B

The main result of this section, Theorem 4.1, concerns the nilpotence orders of degree-1 elements in the graded ring H^{∗}(G/B). This result may be of independent interest, and it would be nice to have a geometric explanation for it.

Recall that H^{∗}(G/B) = R^{n^{n}} ≅ ℤ[x_{1}, …, x_{n}]/J, where

J = ⟨e_{i}(n) : 1 ≤ i ≤ n⟩ = I_{n^{n}} = ⟨h_{n−i+1}(i) : 1 ≤ i ≤ n⟩ .  (10)

We digress to discuss graded ℤ-algebras and nilpotence. A standard graded ℤ-algebra is a ring R with a ℤ-module direct sum decomposition R = ⨁_{d≥0} R_{d} in which each R_{d} is a free ℤ-module, R_{d} · R_{e} ⊂ R_{d+e} and R is generated over the subalgebra R_{0} = ℤ by R_{1}. Let R be a ring and f ∈ R a nilpotent element (that is, some power of f is zero). The nilpotence order of f is defined as the smallest integer k such that f^{k} = 0; we will sometimes say that f is k-nilpotent. (So f has nilpotence order 1 if and only if f = 0.)

By Theorem 2.3, R^{λ} = ℤ[x_{1}, …, x_{n}]/I_{λ} is a standard graded ℤ-algebra, with R_{1}^{λ} ≅ H^{2}(X_{λ}; ℤ). Furthermore, every element of R_{1}^{λ} is nilpotent, since R^{λ} has finite rank as a ℤ-module. The nilpotence order of these linear forms will be our main tool in distinguishing the rings R^{λ}. In this section, we study the case that λ = n^{n}; we treat the general case in Section 5.

Note that the images of the variables x_{i} in R^{n^{n}} satisfy x_{i}^{n} = 0. Indeed, by Corollary 2.5, it is sufficient to prove that x_{1}^{n} = 0, which follows from (10) since I_{n^{n}} contains the element h_{n−1+1}(1) = h_{n}(1) = x_{1}^{n}. In fact, more is true:

Theorem 4.1. Let f ∈ H^{2}(G/B) ≅ (R^{n^{n}})_{1}.

Then f has nilpotence order greater than or equal to n, with equality if and only if f is congruent, modulo J, to a scalar multiple of one of the variables x_{1}, …, x_{n}.

We first show that n is the minimal nilpotence order achieved by any linear form.

Proposition 4.2. Let f ∈ R_{1}^{n^{n}} be a linear form. If f^{n−1} = 0, then f = 0.

Proof. Let f̂ be a preimage of f under the quotient map ℤ[x_{1}, …, x_{n}] ↠ R^{n^{n}}. Then f^{n−1} = 0 means f̂^{n−1} ∈ J. By degree considerations, this means that f̂^{n−1} belongs to the ideal

I := ⟨e_{i}(n) : 1 ≤ i ≤ n − 1⟩ ⊂ ℤ[x_{1}, …, x_{n}].  (11)

Thus it suffices to show that I is a radical ideal, since then f̂ ∈ I and f = 0 in R^{n^{n}}. We will show something slightly stronger: that the ideal I′ := ⟨e_{i}(n) : 1 ≤ i ≤ n − 1⟩ ⊂ ℂ[x_{1}, …, x_{n}] is radical. Indeed, any nonzero nilpotent in ℤ[x_{1}, …, x_{n}]/I would give rise to a nonzero nilpotent in ℂ[x_{1}, …, x_{n}]/I′.

Let ζ be a primitive n^{th} root of unity. We claim that I′ is the vanishing ideal I(V) for the variety V ⊂ ℂ^{n}, defined as the union of all lines whose slope vector is any permutation of (1, ζ, …, ζ^{n−1}). Note that there are exactly (n − 1)! such lines, because two such slope vectors that differ by multiplication by a root of unity give rise to the same line. Equating coefficients of powers of t in the equation

t^{n} − 1 = ∏_{i=1}^{n}(t − ζ^{i}) = ∑_{i=0}^{n} e_{i}(1, ζ, …, ζ^{n−1})t^{i}

shows that I′ ⊂ I(V). For the reverse inclusion, note that e_{1}(n), …, e_{n}(n) is a regular sequence in ℂ[x_{1}, …, x_{n}], and therefore cuts out scheme-theoretically a complete intersection of Krull dimension 1, that is, a set of curves with various multiplicities. By Bézout’s Theorem, the sum of the degrees of those curves, counted with multiplicities, must be

deg(e_{1}(n)) · deg(e_{2}(n)) ⋯ deg(e_{n}(n)) = 1 · 2 ⋯ (n − 1) = (n − 1)!

But this complete intersection contains at least (n − 1)! lines in V, each of degree 1. Therefore it contains no other curves, and each line occurs with multiplicity 1; that is, I′ = I(V). □

To complete the proof of Theorem 4.1, we must show that the scalar multiples of the variables x_{i} are the only n-nilpotent linear forms in R^{n^{n}}. In what follows, we regard a linear form f = ∑_{i=1}^{n} a_{i}x_{i} as a ℂ-linear functional, mapping v = (v_{1}, …, v_{n}) ∈ ℂ^{n} to ∑_{i=1}^{n} a_{i}v_{i}.

Lemma 4.3. Let f = ∑_{i=1}^{n} a_{i}x_{i}, with a_{i} ∈ ℂ, and let α ∈ ℂ^{∗} be a nonzero constant. Suppose that f(v)^{n} = α for all v ∈ ℂ^{n} whose coordinates are permutations of the distinct n^{th} roots of unity.

Then f ∈ ℂx_{i} + ℂe_{1}(n) for some i.

Proof. Let ζ be a primitive n^{th} root of unity. Let the symmetric group 𝔖_{n} act on ℂ^{n} by permuting coordinates, and for a permutation σ ∈ 𝔖_{n}, abbreviate f(σ(1, ζ, …, ζ^{n−1})) by f(σ). Replacing f with f/α, we may assume that f(σ)^{n} = 1 for all σ ∈ 𝔖_{n}. That f has the desired form is equivalent to the statement that at least n − 1 of the coefficients a_{1}, …, a_{n} are equal. This is trivial if n = 1 or n = 2, and can be checked by direct calculation if n = 3. Therefore, suppose n ≥ 4. By transitivity, it suffices to show that if two coefficients a_{i} are different, then the other n − 2 are mutually equal.

Suppose that a_{1} ≠ a_{2}. Choose i ≠ j ∈ [n] so as to maximize |ζ^{i} − ζ^{j}|, and let σ ∈ 𝔖_{n} such that σ(1) = i and σ(2) = j. Then f(σ) and f((12) ∘ σ) are both n^{th} roots of unity, and

f(σ) − f((12) ∘ σ) = (a_{1} − a_{2})(ζ^{i} − ζ^{j}).  (12)

Taking the magnitude of both sides, the choice of i and j implies that |a_{1} − a_{2}| ≤ 1. On the other hand, if we choose i′ ≠ j′ ∈ [n] to minimize |ζ^{i′} − ζ^{j′}|, the same argument implies that |a_{1} − a_{2}| ≥ 1. We conclude that |a_{1} − a_{2}| = 1.

Note that ζ^{i} and ζ^{j} are the only n^{th} roots of unity whose difference is ζ^{i} − ζ^{j}. (This may be seen most easily by plotting the n^{th} roots of unity in the complex plane, and observing that no two of the line segments joining two maximally distant roots are parallel.) Therefore, the equation (12) implies that the values f(σ) and f((12) ∘ σ) do not depend on σ(3), …, σ(n). Hence a_{3} = … = a_{n} as desired. □

Proposition 4.4. Let f ∈ R_{1}^{n^{n}} be a linear form such that f^{n} = 0.

Then f ∈ ℤx_{i} for some i.

Proof. Let f̂ be a preimage of f under the quotient map ℤ[x_{1}, …, x_{n}] ↠ R^{n^{n}}; that is, f̂^{n} ∈ J. By degree considerations, there is a constant α ∈ ℤ such that f̂^{n} ≡ αe_{n}(n) modulo I. As in the proof of Proposition 4.2, the ideal I vanishes on all vectors v whose coordinates are a permutation of the distinct n^{th} roots of unity. Therefore f̂^{n}(v) = αe_{n}(n)(v) = (−1)^{n−1}α for all such vectors v. By Lemma 4.3, there is some i such that f̂ ∈ ℂx_{i} + ℂe_{1}(n). As f̂ ∈ ℤ[x_{1}, …, x_{n}], this implies f̂ ∈ ℤx_{i} + ℤe_{1}(n). Consequently f ∈ ℤx_{i} in R^{n^{n}}. This completes the proof of the proposition and of Theorem 4.1. □

5. Nilpotence of linear forms in the cohomology of X_{λ}

Throughout this section, λ will be an indecomposable partition. We continue our study of nilpotence orders of linear forms in the graded ℤ-algebra R^{λ} = H^{∗}(X^{λ}). The main result is the following classification of linear forms of minimal nilpotence order, generalizing Theorem 4.1.

Theorem 5.1. Let λ = (0 < λ_{1} ≤ ⋯ ≤ λ_{n}) be an indecomposable partition, and let k := λ_{1}. Then k is the minimal nilpotence order of any linear form in R^{λ}. Moreover, if λ has exactly m parts equal to k (that is, k = λ_{1} = ⋯ = λ_{m} < λ_{m+1}), then the elements of R_{1}^{λ} of nilpotence order exactly k are classified as follows: Case I. Either λ_{k−1} > k, or n < k − 1.

Then the k-nilpotents in R_{1}^{λ} are the multiples of x_{1}, …, x_{m}.

Case II. λ_{k−1} = k (that is, m = k − 1). Subcase IIa. Either λ_{k} > k + 1, or k is odd.

Then the k-nilpotents are x_{1}, …, x_{k−1}, and x_{1} + ⋯ + x_{k−1}.

Subcase IIb. Both λ_{k} = k + 1 and k is even.

Then the k-nilpotents are x_{1}, …, x_{k−1}, x_{1} + ⋯ + x_{k−1}, and x_{1} + ⋯ + x_{k−1} + 2x_{k}.

By way of motivation for the rather technical matter of this section, we explain how the classification of nilpotents will be used in the next two sections to recover a partition from its cohomology ring. Theorem 5.1 implies immediately that λ_{1} is an isomorphism invariant of R^{λ}. Moreover, by the presentation of Theorem 2.3, the quotient ring R^{λ} / ⟨x_{1}⟩ may be identified with the ring R^{ν}, where ν = (λ_{2} − 1, λ_{3} − 1 …, λ_{n} − 1) is the partition obtained by removing the first row and column from λ (see Figure 2). However, it is really necessary to describe R^{ν} as a quotient R^{λ} / ⟨f⟩, where f is some linear form identified intrinsically from the structure of R^{λ} as a standard graded ℤ-algebra, that is, in a way that does not depend on the presentation. The classification of nilpotents in Theorem 5.1 is the tool that allows this. It turns out that we will require almost all, but not quite all of the last assertion in the theorem, so we only prove the parts that will be used. (The arguments we omit are very similar to those that we include.)

In the first part of this section, culminating in Proposition 5.4, we prove the first assertion of Theorem 5.1, namely that k = λ_{1} is the minimal nilpotence order of any linear form in R^{λ}. We begin with a weaker statement: that no linear form in the first k − 1 variables has nilpotence order less than k.

Lemma 5.2. Let λ be indecomposable with k := λ_{1}. Let f = ∑_{i=1}^{k−1} a_{i}x_{i} ∈ R_{1}^{λ}; that is, f is supported only on the first k − 1 variables. Then, in R^{λ},

(a) f^{k−1} = 0 if and only if f = 0, and

(b) if f^{k} = 0, then f is a scalar multiple of one of the following: x_{1}, …, x_{k−1}, x_{1} + ⋯ + x_{k−1}.

Proof. By Proposition 3.1 and the hypothesis that f is supported only on the first k − 1 variables, we may assume without loss of generality that n ≤ k − 1. By Corollary 2.4, we may decrease the part sizes of λ (if necessary), so as to assume that λ = k^{n}. But then using Proposition 3.1 again, we can re-introduce parts λ_{n+1}, λ_{n+2}, …, λ_{k} all of size k, and work in the ring R^{k^{k}} ≅ GL_{k}(ℂ)/B, where assertion (a) follows from Theorem 4.1.

In fact, assertion (b) also follows from Theorem 4.1. The degree-1 graded piece of I_{k^{k}} is generated by e_{1}(k), so the elements of R_{1}^{k^{k}} listed above are the only ones that are congruent modulo I_{k^{k}} to a scalar multiple of a variable x_{i} (here we use the fact that x_{1} + ⋯ + x_{k−1} = e_{1}(k) − x_{k}). □

An immediate consequence of Lemma 5.2 is that every linear form of nilpotence order λ_{1} − 1 must be supported on at least one of the variables x_{k}, …, x_{n}. This is where the concept of “stickiness” introduced in Corollary 3.4 first comes into play.

Proposition 5.3. Let λ be indecomposable with k := λ_{1}, and let f ∈ R_{1}^{λ}. Then f^{k−1} = 0 if and only if f = 0 in R^{λ}.

Proof. Assume f ≠ 0 ∈ R_{1}^{λ}, but f^{k−1} = 0 in R^{λ}. By Lemma 5.2(a), we may assume n ≥ k. By Proposition 3.1, we may assume without loss of generality that λ is its own core.

Writing f = g+h, where g = a_{1}x_{1}+⋯+a_{k−1}x_{k−1} and h = a_{k}x_{k} +⋯+a_{n}x_{n}, it follows from Corollary 3.5 that g^{k−1} = 0. Hence g = 0 by Lemma 5.2. That is, f = h. If f is not supported on x_{n} (that is, a_{n} = 0), then we may replace λ with the partition obtained by removing the nth (largest) row. Repeating this as many times as necessary, we may assume without loss of generality that a_{n} ≠ 0.

Now let M be any monomial in the variables x_{1}, …, x_{k−1}. Note that

[x_{n}M] \overline{f^{k−1}} = [x_{n}M] \overline{(a_{n}x_{n})^{k−1}}  (13)

because the variables x_{k}, …, x_{n−1} are sticky (Corollary 3.4). Reducing (a_{n}x_{n})^{k−1} using the Gröbner basis element G_{n} of (9), we find that

\begin{aligned}(a_{n}x_{n})^{k−1} & = −a_{n}x_{n}^{k−2}(x_{1} + ⋯ + x_{n−1}) \\ & = a_{n}^{2}x_{n}^{k−3}(x_{1} + ⋯ + x_{n−1})^{2} \\ & = … \\ & = αx_{n}(x_{1} + ⋯ + x_{n−1})^{k−2}, \end{aligned}  (14)

where α = (−1)^{k−2}a_{n}^{k−2} ≠ 0. Combining this with (13) yields

[x_{n}M] \overline{f^{k−1}} = α [x_{n}M] \overline{x_{n}(x_{1} + ⋯ + x_{n−1})^{k−2}}
= α [x_{n}M] \overline{x_{n}(x_{1} + ⋯ + x_{k−1})^{k−2}}  (15a)
= α [M] \overline{(x_{1} + ⋯ + x_{k−1})^{k−2}}  (15b)

where (15a) follows from stickiness, and (15b) from the fact that only G_{1}, …, G_{k−1} are used in reducing (15a).

The polynomial x_{1} + ⋯ + x_{k−1} is nonzero in R^{λ} since λ is indecomposable. Thus Lemma 5.2 implies that (x_{1} + ⋯ + x_{k−1})^{k−2} ≠ 0 as well, and so there exists some monomial M in the variables x_{1}, …, x_{k−1} for which [M]\overline{(x_{1} + ⋯ + x_{k−1})^{k−2}} ≠ 0. Note that x_{n}M is also a standard monomial for I_{λ}. Therefore [x_{n}M]\overline{f^{k−1}} ≠ 0, a contradiction. □

Proposition 5.4. When λ is indecomposable, the number k = λ_{1} is an isomorphism invariant of R^{λ} as a graded ring: namely, it is the minimum nilpotence order achieved by any linear form.

Proof. Proposition 5.3 states that no nonzero linear form can have nilpotence order strictly less than k = λ_{1}. On the other hand, x_{1} has nilpotence order at most k, because x_{1}^{k} = h_{k}(1) ∈ I_{λ}. □

In the second part of this section, we show that the various linear forms mentioned in Theorem 5.1 are the only possible k-nilpotents in R^{λ}. We begin by determining the nilpotence order of each variable.

Proposition 5.5. When λ is indecomposable, the variable x_{i} is λ_{i}-nilpotent in R^{λ}.

Proof. Let p = λ_{i}. First, we show that x_{i}^{p} = 0 in R^{λ}. Let κ be the partition given by

κ := (\underbrace{p, …, p}_{i times}, λ_{i+1}, λ_{i+2}, …, λ_{n}).

Then λ is a subpartition of κ, so R^{λ} is a quotient of R^{κ} by Lemma 2.4. It suffices to show that x_{i}^{p} = 0 in R^{κ}, which follows from Corollary 2.5 since x_{1}^{p} ∈ I_{κ}.

It remains to show that x_{i}^{p−1} ≠ 0 in R^{λ}. By Proposition 3.1 and Corollary 2.4, it suffices to show that x_{i}^{p−1} ≠ 0 in R^{μ}, where μ is the subpartition of λ given by

μ := (2, 3, …, i − 1, i, p).

Note that μ is indecomposable, and that R^{μ} has a reduced Gröbner basis given by (7). A Gröbner reduction similar to (14), using the Gröbner basis element x_{i}^{p−i+1} + x_{i}^{p−i}h_{1}(i − 1) yields the equation

x_{i}^{p−1} ≡ (−1)^{i−1}x_{i}^{p−i}h_{1}(i − 1)^{i−1} (mod I_{μ}).

Since further reductions modulo I_{μ} can only involve the other generators h_{2}(1), h_{2}(2), …, h_{2}(i − 1), we may conclude that x_{i}^{p−1} ≠ 0 in R^{μ}, provided that h_{1}(i − 1)^{i−1} ≠ 0 in R^{(2,3,··· ,i−1,i)}. Using the fact that h_{1}(i − 1) = e_{1}(i − 1), this follows from the following more general assertion: for any m ≥ 1 and i ≥ 1,

e_{1}(i − 1)^{m} = e_{m}(i − 1) ≠ 0  in R^{(2,3,··· ,i−1,i)}.  (16)

This is trivially true for i ≤ 2. For i > 2, we prove it by induction on i:

e_{1}(i − 1)^{m} = (x_{i−1} + e_{1}(i − 2))^{m}

= ∑_{j=0}^{m} \binom{m}{j}x_{i−1}^{j}e_{1}(i − 2)^{m−j}

= e_{1}(i − 2)^{m} + ∑_{j=1}^{m} \binom{m}{j}x_{i−1}^{j}e_{1}(i − 2)^{m−j}

≡ e_{1}(i − 2)^{m} + ∑_{j=1}^{m} \binom{m}{j}(−1)^{j−1}x_{i−1}e_{1}(i − 2)^{m−1} (mod I^{(2,3,··· ,i−1,i)}).

This last expression follows from using x_{i−1}h_{1}(i − 1) = x_{i−1}^{2} + x_{1}h_{1}(i − 2) = x_{i−1}^{2} + x_{1}h_{1}(i − 2) to perform repeated Gröbner reduction on each summand. By induction, e_{1}(i − 2)^{m} = e_{m}(i − 2), so we obtain

e_{1}(i − 1)^{m} = e_{m}(i − 2) + x_{i−1}e_{m−1}(i − 2) ∑_{j=1}^{m} \binom{m}{j}(−1)^{j−1}
= e_{m}(i − 2) + x_{i−1}e_{m−1}(i − 2)
= e_{m}(i − 1),

establishing (16) as desired. □

Proposition 5.6. Let f = ∑_{i=1}^{n} a_{i}x_{i} ∈ R^{λ}. Suppose that f^{k} = 0.

Then f is a scalar multiple of one of the following:

\begin{matrix}x_{1}, x_{2}, …, x_{k−1}, \\ x_{1} + ⋯ + x_{k−1}, \\ x_{1} + ⋯ + x_{k−1} + 2x_{k}. \end{matrix}  (17)

The last case can occur only if k is even.

Proof. By Corollary 2.4, we may replace λ with its core. Let g = ∑_{i=1}^{k−1} a_{i}x_{i} be the part of f in the non-sticky variables. Then g^{k} = 0 by Corollary 3.5. By Lemma 5.2(b), g is either zero or of the form αx_{i} for some i ∈ {1, 2, …, k − 1}, or α(x_{1} + ⋯ + x_{k−1}), where α is a nonzero scalar. Without loss of generality, we may assume that α = 1.

If f = g then we are done. Otherwise, we must show that f is a scalar multiple of x_{1} + ⋯ + x_{k−1} + 2x_{k}, and k is even. By Proposition 3.1, we may assume without loss of generality that f involves the variable x_{n} with non-zero coefficient; that is,

f = g + h + ax_{n},

where a := a_{n} ≠ 0 and h is a linear form in the variables x_{k}, …, x_{n−1}. We consider in turn each of the three possibilities: namely, g = 0, g = x_{i}, or g = x_{1} + ⋯ + x_{k−1}.

Case 1: g = 0.

We will rule out this case by deriving a contradiction from the assumption that f^{k} = 0 in R^{λ}. Taking the further quotient of R^{λ} by the variables x_{k}, …, x_{n−1}, one obtains a ring isomorphic to R^{μ}, where

μ = ( \underbrace{k, …, k}_{k−1 times}, k + 1)

is an indecomposable partition, with k parts, equal to its own core. If f^{k} = 0 in R^{λ}, then (ax_{k})^{k} = a^{k}x_{k}^{k} = 0 in R^{μ}. So x_{k}^{k} = 0 in R^{μ} (because a ≠ 0). But this contradicts Corollary 5.5, since μ_{k} = k + 1.

Case 2: g = x_{i}, where i ∈ {1, 2, …, k − 1}.

Assume that k ≥ 3 (the case k = 2 falls under Case 3 below). As in Case 1, we wish to reach a contradiction. Consider the quotient ring

S := R^{λ}/ ⟨x_{k}, …, x_{n−1}, x_{1} + x_{2} + ⋯ + x_{k−1} + x_{n}⟩ ,

which is isomorphic to R^{k^{k}}. Let f̃ = x_{i} − a(x_{1} + ⋯ + x_{k−1}) be the image of f in S; then f̃^{k} = 0. By Theorem 4.1, f̃ must be a scalar multiple of some variable. This is possible only if k = 3 and a = 1; that is, f is a scalar multiple of either x_{1} + x_{3} or x_{2} + x_{3}. All that remains is to check that neither (x_{1} + x_{3})^{3} nor (x_{2} + x_{3})^{3} belongs to the ideal I_{3^{3}} = ⟨h_{3}(1), h_{2}(2), h_{2}(3)⟩; this is a routine calculation. Thus f^{k} ≠ 0 in all cases, a contradiction. Case 2 is therefore ruled out.

Case 3: g = x_{1} + … + x_{k−1}.

Let M be any standard monomial for I_{λ} of degree k−1 in the non-sticky variables x_{1}, …, x_{k−1}; then x_{n}M is also standard. Using stickiness of the variables x_{k}, …, x_{n−1} and the fact that G_{n} = x_{n}(x_{1} +⋯+x_{n}) ∈ I_{λ}, we have for every such monomial

[x_{n}M]f^{k} = [x_{n}M](g + ax_{n})^{k} = [x_{n}M] ∑_{i=0}^{k} \binom{k}{i}a^{i}x_{n}^{i}g^{k−i} = [x_{n}M] ∑_{i=1}^{k} \binom{k}{i}a^{i}x_{n}^{i}g^{k−i}

= [x_{n}M] ∑_{i=1}^{k} \binom{k}{i}a^{i}g^{k−i}(−1)^{i−1}x_{n}(x_{1} + ⋯ + x_{n−1})^{i−1}

= [x_{n}M] ∑_{i=1}^{k} \binom{k}{i}a^{i}g^{k−i}(−1)^{i−1}x_{n}(x_{1} + ⋯ + x_{k−1})^{i−1}

= [M] ∑_{i=1}^{k} \binom{k}{i}a^{i}g^{k−1}(−1)^{i−1} = (∑_{i=1}^{k} \binom{k}{i}a^{i}(−1)^{i−1}) [M]g^{k−1}

= (1 − (1 − a)^{k}) [M]g^{k−1}.

This last expression must be zero since f^{k} = 0 in R^{λ}. On the other hand, g^{k−1} ≠ 0 in R^{λ}, so there is at least one such monomial M in x_{1}, …, x_{k−1} for which [M]g^{k−1} ≠ 0. It follows that 1 − (1 − a)^{k} = 0. Since a ≠ 0, the only possibility is that k is even and a = 2. If n = k, then we are done; we need to rule out the case n > k.

Figure 4. The subpartition μ of Proposition 5.7 (shaded).

Suppose that n > k. Replacing x_{n} with x_{k} in the above calculation, we find that the coefficient a_{k} is either 0 or 2. Bearing in mind that g + x_{k} = x_{1} + ⋯ + x_{k−1} + x_{k} = h_{1}(k), we pass to the quotient ring

T := R^{λ} / ⟨x_{k+1}, x_{k+2}, …, x_{n−1}, g + x_{k}⟩
≅ ℤ[x_{1}, …, x_{k}, x_{n}] / ⟨h_{k}(1), h_{k−1}(2), …, h_{2}(k − 1), x_{k}(g + x_{k}), x_{n}(g + x_{k} + x_{n}), g + x_{k}⟩
≅ ℤ[x_{1}, …, x_{k}, x_{n}] / ⟨h_{k}(1), h_{k−1}(2), …, h_{2}(k − 1), g + x_{k}, x_{n}^{2}⟩
≅ R^{k^{k}} [x_{n}] / ⟨x_{n}^{2}⟩ .

Note that since f equals either g + 2x_{n} or g + 2x_{k} + 2x_{n}, and x_{k} = −g in T, the image p of f in T is of the form p = ±g + 2x_{n}. Since x_{n}^{2} and g^{k} are both zero in T, we have

p^{k} = ∑_{j=0}^{k} \binom{k}{j}(2x_{n})^{j}(±g)^{k−j} = ±2kx_{n}g^{k−1}.

But g^{k−1} ≠ 0 in R^{k^{k}} by Theorem 4.1, so x_{n}g^{k−1} ≠ 0 in T. Hence p^{k} ≠ 0 in T, which implies that f^{k} ≠ 0 in R^{λ}, as desired. □

We now know that every k-nilpotent linear form in R^{λ} is, up to scalar multiplication, one of the linear forms (17). However, if λ is not its own core, then we must consider the possibility that one or more of these linear forms actually has nilpotence order strictly greater than k. We examine each candidate in turn; Proposition 5.5 immediately takes care of the possible nilpotents x_{1}, …, x_{k−1}.

Proposition 5.7. Let λ be indecomposable with n ≥ k − 1 parts and k = λ_{1}. Let g = x_{1} + ⋯ + x_{k−1} ∈ R^{λ}. Then g^{k} = 0 if and only if λ_{1} = ⋯ = λ_{k−1} = k.

Proof. By Proposition 3.1, we may assume that n = k − 1. Suppose that λ_{1} = ⋯ = λ_{k−1} = k. Then

R^{λ} = R^{k^{k−1}} = ℤ[x_{1}, …, x_{k−1}] / ⟨h_{k}(1), h_{k−1}(2), …, h_{2}(k − 1)⟩
≅ ℤ[x_{1}, …, x_{k−1}, x_{k}] / ⟨h_{k}(1), h_{k−1}(2), …, h_{2}(k − 1), h_{1}(k)⟩
= R^{k^{k}},

and g = −x_{k} in R^{k^{k}}, so g^{k} = 0 follows from Theorem 4.1.

Conversely, suppose that λ_{k−1} > k. We will show that g^{k} ≠ 0. Let μ be the subpartition of λ given by

μ = (\underbrace{k − 1, …, k − 1}_{k−2 times}, k + 1)

(see Figure 4). By Corollary 2.4, it will suffice to show that g^{k} ≠ 0 in R^{μ}. We may rewrite the presentation of R^{μ} as

R^{μ} = ℤ[x_{1}, …, x_{k−1}] / ⟨h_{k−1}(1), h_{k−2}(2), …, h_{2}(k − 2), h_{3}(k − 1)⟩
= ℤ[x_{1}, …, x_{k−1}] / ⟨h_{k−1}(1), h_{k−2}(2), …, h_{2}(k − 2), x_{k−1}^{3} + x_{k−1}^{2}h_{1}(k − 2)⟩ ,

using the fact that

h_{3}(k − 1) = x_{k−1}^{3} + x_{k−1}^{2}h_{1}(k − 2) + x_{k−1}h_{2}(k − 2) + h_{3}(k − 2)
= x_{k−1}^{3} + x_{k−1}^{2}h_{1}(k − 2) + x_{k−1}h_{2}(k − 2) + h_{3}(k − 3) + x_{k−2}h_{2}(k − 2),

Therefore x_{k−1}^{j} ≡ (−1)^{j}x_{k−1}^{2}h^{j−2} for all j ≥ 3. Letting h = h_{1}(k −2) = x_{1} +⋯+x_{k−2}, so that g = h+x_{k−1}, we have in R^{μ}

g^{k} = h^{k} + \binom{k}{1}h^{k−1}x_{k−1} + \binom{k}{2}h^{k−2}x_{k−1}^{2} + ∑_{j=3}^{k} \binom{k}{j}h^{k−j}(−1)^{j}x_{k−1}^{2}h^{j−2}

= h^{k} + kh^{k−1}x_{k−1} + h^{k−2}x_{k−1}^{2} ∑_{j=2}^{k}(−1)^{j}\binom{k}{j}
= h^{k} + kh^{k−1}x_{k−1} + (k − 1)h^{k−2}x_{k−1}^{2}.

No further Gröbner reduction is possible, so g^{k} is zero if and only if h^{k}, kh^{k−1}, and (k − 1)h^{k−2} are all zero. But k > 1, and h^{k−2} ≠ 0 by Proposition 5.3. We conclude that g^{k} ≠ 0 in R^{μ} as desired. □

For the remaining assertions of Theorem 5.1, we are left only to consider the potentially k-nilpotent linear form g = x_{1} + … + x_{k−1} + 2x_{k}. Rather than determining exactly when g is k-nilpotent as in the theorem (which can be done by an argument similar to Proposition 5.7), we content ourselves with checking directly the case k = 2, since this is all we need for the present study. Here g = x_{1} + 2x_{2}, and by Proposition 3.1) we may work in the ring

R^{(2,λ_{2})} = ℤ[x_{1}, x_{2}]/ ⟨x_{1}^{2}, h_{λ_{2}−1}(2)⟩ .

Then it is easily seen that g^{2} = x_{1}^{2} + 4x_{1}x_{2} + 4x^{2} is zero in this ring if and only if λ_{2} = 3.

6. The indecomposable case

We now use the results of the previous section to prove that an indecomposable partition is determined uniquely by the cohomology ring of the corresponding Schubert variety.

Theorem 6.1. Every indecomposable partition λ may be recovered from the structure of the ring R^{λ} as a graded ℤ-algebra. In particular, if λ and μ are different indecomposable partitions, then R^{λ} and R^{μ} are not isomorphic.

Proof. We induct on n, the number of parts of λ. Since λ is indecomposable, n is the rank of R_{1}^{λ} as a free ℤ-module. By Theorem 5.1, the smallest part k := λ_{1} is the minimal nilpotence order of any member of R_{1}^{λ}. Moreover, as mentioned at the beginning of Section 5, R^{λ}/ ⟨x_{1}⟩ ≅ R^{ν}, where ν is obtained from λ by deleting the first row and column (see Figure 2). By induction, it suffices to show that we can describe R^{ν} up to isomorphism in a way that is independent of the presentation.

We proceed by examining the same two cases as in Theorem 5.1; however, we subdivide Case II slightly differently into subcases.

Case I. λ_{k−1} > k or n < k − 1.

Let m be the greatest index such that λ_{m} = k. Then Theorem 5.1 tells us that the k-nilpotent linear forms in (R^{λ})_{1} are (up to ℤ-multiples) x_{1}, …, x_{m}. Consequently, up to sign, these are exactly the primitive k-nilpotents, that is, those k-nilpotents f which can only be expressed as a scalar multiple αg for another k-nilpotent g and α ∈ ℤ if α = ±1.

By Corollary 2.5, one has R^{λ}/ ⟨x_{i}⟩ ≅ R^{λ}/ ⟨x_{1}⟩ (≅ R^{ν}) for every i ∈ {1, 2, …, m}, and hence R^{ν} may be identified intrinsically as the quotient of R^{λ} by an arbitrary primitive k-nilpotent linear form.

Case II. λ_{k−1} = k.

Then the primitive k-nilpotents are (up to sign) x_{1}, …, x_{k−1}, x_{1} + ⋯ + x_{k_{1}}, and if k is even, possibly also x_{1} + ⋯ + x_{k−1} + 2x_{k}.

Subcase IIA. k > 2.

If k is odd, then the “extraneous” primitive k-nilpotent x_{1} + ⋯ + x_{k−1} + 2x_{k} is absent. If k is even, then x_{1} + ⋯ + x_{k−1} + 2x_{k} is distinguished intrinsically as the unique primitive k-nilpotent which is ℤ-linearly independent of all the others.

Thus, in all cases when k > 2, we can intrinsically identify the primitive k-nilpotents x_{1}, …, x_{k−1}, x_{1} + ⋯ + x_{k−1}, up to sign. By Corollary 2.5, the first k − 1 forms on this list all have R^{λ}/ ⟨x_{i}⟩ ≅ R^{λ}/ ⟨x_{1}⟩ ≅ R^{ν}. Hence R^{ν} can be identified intrinsically by “majority rule”: it is the ℤ-algebra that occurs (up to isomorphism) as the quotient of R^{λ} by at least k − 1 of the k different primitive k-nilpotent linear forms (other than the one, namely x_{1} + ⋯ + x_{k−1} + 2x_{k}, that is linearly independent from the rest, as above). Note that the fact that k − 1 out of k is a well-defined “majority” uses the assumption that k > 2.

Subcase IIB. k = 2.

If λ_{2} > 3, then x_{1} is the unique primitive k-nilpotent up to sign, so it is distinguished intrinsically, as is R^{ν} ≅ R/ ⟨x_{1}⟩.

If λ_{2} = 3, then there are two primitive k-nilpotents up to sign, namely x_{1} and x_{1} + 2x_{2}. We claim that the graded ℤ-algebra map ω : R^{λ} → R^{λ} defined by

ω(x_{1}) = x_{1} + 2x_{2},  ω(x_{2}) = −x_{2},  ω(x_{i}) = x_{i} for 3 ≤ i ≤ n

is an automorphism of R^{λ} interchanging x_{1} with x_{1} + 2x_{2}. Indeed, it is a routine calculation to check that ω lifts to an automorphism of ℤ[x_{1}, …, x_{n}], and that ω(I_{λ}) = I_{λ}. In particular, R^{ν} ≅ R^{λ}/ ⟨x_{1}⟩ ≅ R^{λ}/ ⟨x_{1} + 2x_{2}⟩ may again be described up to isomorphism as the quotient of R^{λ} by an arbitrary primitive k-nilpotent linear form. □

7. The decomposable case

We now consider the case that λ is decomposable, with indecomposable components λ^{(1)}, λ^{(2)}, …, λ^{(r)}. In this case, X_{λ} ≅ X_{λ^{(1)}} × ⋯ × X_{λ^{(r)}}. Since each X^{λ^{(i)}} has no torsion in its (co-)homology by Theorem 2.1, the Künneth formula [11, §61] implies a tensor decomposition for the associated cohomology rings:

H^{∗}(X_{λ}; ℤ) ≅ ⨂_{i=1}^{r} H^{∗}(X_{λ^{(i)}}; ℤ).  (18)

Together with the uniqueness result for indecomposable partitions (Theorem 6.1), it would seem that we are done. However, there is one remaining technical point: to verify that the partitions λ^{(i)} can be read off intrinsically from the structure of H^{∗}(X_{λ}) as a graded ℤ-algebra, we must check that the tensor decomposition (18) is unique.

To do this, we make further use of the facts about nilpotence established in Section 5. But first we must make precise the notion of tensor decomposition, and point out how it interacts with order of nilpotence.

For R a standard graded ℤ-algebra, a tensor decomposition is an isomorphism of graded ℤ-algebras R ≅ T^{(1)} ⊗ ⋯ ⊗ T^{(r)} in which each T^{(i)} is a standard graded ℤ-algebra. Note that any such decomposition is completely determined by the associated direct sum decomposition of free ℤ-modules R_{1} ≅ ⨁_{i=1}^{r} T_{1}^{(i)}, since T^{(i)} is then the subalgebra of R generated by the direct summand T_{1}^{(i)} of R_{1}. Say that a tensor decomposition of R is nontrivial if T^{(i)} ≠ ℤ for all i. Say R is tensor-indecomposable if it is not ℤ itself, and has no nontrivial tensor decomposition.

Lemma 7.1. Suppose that R = T^{(1)} ⊗ ⋯ ⊗ T^{(r)}. Let x ∈ R_{1}; that is,

x = x_{1} ⊗ 1 ⊗ ⋯ ⊗ 1 + 1 ⊗ x_{2} ⊗ 1 ⋯ ⊗ 1 + … + 1 ⊗ ⋯ ⊗ 1 ⊗ x_{r}

where x_{i} ∈ T_{1}^{(i)}. Let k_{i} be the nilpotence order of x_{i}. (Recall that k_{i} = 1 if and only if x_{i} = 0.) Then the nilpotence order of x is

c = k_{1} + k_{2} + ⋯ + k_{r} − r + 1.

Proof. By the pigeonhole principle, each term of the multinomial expansion of x^{c} is divisible by x_{i}^{k_{i}} for some i; therefore, x^{c} = 0 in R. For the same reason, all but one term of the multinomial expansion of x^{c−1} vanishes; the exception is

\binom{c}{k_{1} − 1, …, k_{n} − 1} x_{1}^{k_{1}−1} ⊗ x_{2}^{k_{2}−1} ⊗ ⋯ ⊗ x_{n}^{k_{n}−1},

which is nonzero, since it is nonzero in each tensor factor. □

This calculation has immediate useful consequences.

Corollary 7.2. Let R be a standard graded ℤ-algebra with a nontrivial tensor decomposition R = ⨂_{i=1}^{r} T^{(i)}. Then any linear form f ∈ R_{1} that achieves the minimal nilpotence among all elements in R_{1} must lie in T^{(i)} for some i.

Combining Lemma 7.1 with Proposition 5.5 yields the following.

Corollary 7.3. Let λ be a partition with indecomposable components {λ^{(j)}}_{j=1}^{r}. If λ_{i} corresponds to λ_{k}^{(j)} in this decomposition, then x_{i} is λ_{k}^{(j)}-nilpotent in R^{λ}.

For example, if λ is the decomposable partition shown in Figure 1, then λ_{1}, …, λ_{5} correspond to the rows of λ^{(1)}, and λ_{7}, λ_{8} to the rows of λ^{(2)}. Thus the variables x_{1}, …, x_{5} have nilpotence orders 5, 5, 5, 6, 6, respectively, in R^{λ} (and in R^{λ^{(1)}}), and x_{7}, x_{8} have nilpotence orders 2 and 3, respectively. (Note that these seven variables are a ℤ-basis for R_{1}^{λ}; x_{6} ≡ −(x_{1} + ⋯ + x_{5}) does not correspond to a variable in the presentation for R^{λ^{(1)}}.)

Proposition 7.4. Let λ be an indecomposable partition. Then the ring R^{λ} is tensor-indecomposable.

Proof. Let n denote the number of parts in λ, and k = λ_{1} its smallest part. We proceed by induction on n.

If n = 1, then clearly R^{λ} = ℤ[x_{1}]/ ⟨x_{1}^{k}⟩ is indecomposable. Otherwise, suppose that R^{λ} = T^{(1)} ⊗ T^{(2)} is a nontrivial tensor decomposition; we will obtain a contradiction.

By Proposition 5.4, x_{1} is a nilpotent of minimal order, and hence by Corollary 7.2, without loss of generality, x_{1} ∈ T^{(1)}. Then R^{λ}/ ⟨x_{1}⟩ = T^{(1)}/ ⟨x_{1}⟩ ⊗ T^{(2)}. On the other hand, R^{λ}/ ⟨x_{1}⟩ ≅ R^{ν}, where ν is the partition obtained from λ by removing the first row and column. Since λ is indecomposable, so is ν. By the inductive hypothesis, the decomposition T^{(1)}/ ⟨x_{1}⟩ ⊗ T^{(2)} must be trivial; that is, T^{(1)}/ ⟨x_{1}⟩ ≅ ℤ, and T^{(1)} must be generated by x_{1} as a ℤ-algebra, i.e., T^{(1)} = ℤ[x_{1}]/ ⟨x_{1}^{k}⟩. Therefore, exactly one member of the set

L = {x_{2} + αx_{1} : α ∈ ℤ}

belongs to T_{1}^{(2)}. Let ℓ be the nilpotence order of that one form; then all other elements of L have nilpotence order k+ℓ−1 > ℓ by Lemma 7.1. Let m = λ_{2}; note that m ≥ 3 since λ is indecomposable. By Proposition 3.1 we can work in the algebra R^{(λ_{1},λ_{2})} = R^{(k,m)}, namely the quotient of ℤ[x_{1}, x_{2}] by the ideal

⟨G_{1} = x_{1}^{k}, G_{2} = x_{2}^{m−1} + x_{2}^{m−2}x_{1} + … + x_{2}^{m−k}x_{1}^{k−1}⟩ .

Let α ∈ ℤ be arbitrary. We will show that no linear form x_{2} + αx_{1} has nilpotence order strictly less than m. Indeed,

(x_{2} + αx_{1})^{m−1} = ∑_{j=0}^{m−1} \binom{m − 1}{j}x_{2}^{j}α^{m−j−1}x_{1}^{m−j−1}

= ( ∑_{j=0}^{m−2} \binom{m − 1}{j}x_{2}^{j}α^{m−j−1}x_{1}^{m−j−1}) + G_{2} − ∑_{j=0}^{m−2} x_{2}^{j}x_{1}^{m−j−1}

= G_{2} + ∑_{j=0}^{m−2} (\binom{m − 1}{j}α^{m−j−1} − 1) x_{2}^{j}x_{1}^{m−j−1}

≡ ∑_{j=m−k}^{m−2} (\binom{m − 1}{j}α^{m−j−1} − 1) x_{2}^{j}x_{1}^{m−j−1}  mod ⟨G_{1}, G_{2}⟩ .

This last expression is exactly the standard form of (x_{2} + αx_{1})^{m−1}. For j = m − 2, the summand is ((m−1)α−1)x_{2}^{m−2}x_{1}; since m > 3 and α is an integer, the coefficient is nonzero. Therefore (x_{2}+αx_{1})^{m−1} ≠ 0.

On the other hand, x_{2}^{m} = 0 in R^{λ} by Proposition 5.5. Therefore x_{2} must be the unique element of L with minimal nilpotence order m = ℓ, and every other element of L must have nilpotence order k + m − 1. But there are no standard monomials in x_{1}, x_{2} of degree greater than (k −1)+(m−2) = k +m−3, which implies that every element of L has nilpotence order k + m − 2 or less. This contradiction completes the proof. □

We now establish the key fact of the decomposable case, that these decompositions are actually unique.

Lemma 7.5. The ring R^{λ} has a unique tensor decomposition into tensor-indecomposables. Specifically, if λ has indecomposable components λ^{(1)}, λ^{(2)}, …, λ^{(r)}, then

R^{λ} = R^{λ^{(1)}} ⊗ ⋯ ⊗ R^{λ^{(r)}}

is the unique tensor decomposition of R^{λ}, up to permuting the factors.

Proof. The existence is immediate, since each R^{λ^{(i)}} is tensor-indecomposable by Lemma 7.4. For uniqueness, we proceed by induction on the number of rows of λ. If λ has only one row, the statement is trivial.

Suppose that R^{λ} = ⊗_{i=1}^{s}T^{(i)} is a tensor decomposition with each T^{(i)} tensor-indecomposable, so that

⨂_{i=1}^{s} T^{(i)} = R^{λ} = ⨂_{j=1}^{r} R^{λ^{(j)}},  (19a)

⨁_{i=1}^{s} T_{1}^{(i)} = R_{1}^{λ} = ⨁_{j=1}^{r} R_{1}^{λ^{(j)}}.  (19b)

Let k be the minimal nilpotence order of any element of R_{1}^{λ}. Then k = min{λ_{1}^{(j)} : 1 ≤ j ≤ r} by Corollary 7.3. Without loss of generality, we may re-index so that k = λ_{1}^{(1)}; then x_{1} is a linear form of nilpotence order k. By Corollary 7.2, x_{1} must belong to one of the T^{(i)}, say T^{(1)}. Let ν, ν^{(1)} be the partitions obtained by removing the left column and bottom row of λ, λ^{(1)}, respectively. Then

T^{(1)}/⟨x_{1}⟩ ⊗ ⨂_{i=2}^{s} T^{(i)} = R^{λ}/⟨x_{1}⟩ = R^{ν^{(1)}} ⊗ ⨂_{j=2}^{r} R^{λ^{(j)}},  (20a)

T_{1}^{(1)}/ℤx_{1} ⊕ ⨁_{i=2}^{s} T_{1}^{(i)} = R_{1}^{λ}/ℤx_{1} = R_{1}^{ν^{(1)}} ⊕ ⨁_{j=2}^{r} R_{1}^{λ^{(j)}}.  (20b)

By induction, the rightmost expression in (20a) is the unique tensor decomposition of R^{ν} into tensorindecomposables (possibly with a superfluous factor R^{ν^{(1)}}= ℤ if λ^{(1)} has only one part). Thus the rightmost expression in (20b) is unique—clearly not as a direct sum decomposition of R_{1}^{λ}/ℤx_{1} as a ℤ-module, but as a direct sum decomposition which induces a tensor decomposition of R^{λ}/ ⟨x_{1}⟩.

Now assume that λ^{(1)} has m rows, so that x_{1}, x_{2}, …, x_{m} generate R^{λ^{(1)}} as a ℤ-subalgebra of R^{λ}. For each ℓ ∈ {2, …, m}, consider the image x̄_{ℓ} of x_{ℓ} in R_{1}^{ν} = R_{1}^{λ}/ℤx_{1}. Since each x̄_{ℓ} belongs to the direct summand R_{1}^{ν^{(1)}} on the left side of the unique decomposition (20b), it must belong either to T_{1}^{(1)}/ℤx_{1}, or to T_{1}^{(i)} for some i ≥ 2. On the other hand, Corollary 7.3 tells us that x_{ℓ} is λ_{ℓ}^{(1)}-nilpotent in R^{λ}, but x̄_{ℓ} is ν_{ℓ−1}^{(1)}-nilpotent in R^{ν}. That is, the nilpotence order of x_{ℓ} drops by 1 in the quotient by x_{1} (because ν_{ℓ−1}^{(1)} = λ_{ℓ}^{(1)} − 1). If x̄_{ℓ} ∈ T_{1}^{(i)} for some i ≥ 2, then this last observation contradicts Lemma 7.1. Therefore x̄_{ℓ} ∈ T_{1}^{(1)}/ℤx_{1}, from which we conclude that T_{1}^{(1)}/ℤx_{1} ⊇ R_{1}^{ν^{(1)}}.

Consequently, the uniqueness property of the decomposition (20b) implies that

T_{1}^{(1)}/ℤx_{1} = R_{1}^{ν^{(1)}} ⊕ ⨁_{u∈U} R_{1}^{λ^{(ju)}}

for some subset U ⊂ {2, 3, …, r}. Since x_{1} lies in both T^{(1)} and R^{λ^{(1)}}, we conclude that

T_{1}^{(1)} = R_{1}^{λ^{(1)}} ⊕ ⨁_{u∈U} R_{1}^{λ^{(ju)}}

and, since T^{(1)} is a standard graded ℤ-algebra,

T^{(1)} = R^{λ^{(1)}} ⊗ ⨂_{u∈U} R^{λ^{(ju)}}.

But T^{(1)} was assumed to be indecomposable, so this forces U = ∅. Hence T_{1}^{(1)} = R_{1}^{λ^{(1)}} and T_{1}^{(1)}/ℤx_{1} = R_{1}^{ν^{(1)}}. By the uniqueness property of (20b), we must have r = s, and after re-indexing, T_{1}^{(i)} = R_{1}^{λ^{(i)}} for i = 2, 3, …, r. Thus the two tensor decompositions in (19a) are identical. □

The nontrivial implication (iii) ⟹ (i) in the main result, Theorem 1.1, is now immediate from Lemma 7.5 and Theorem 6.1.

Remark 7.6. As we shall now demonstrate, it was essential to study the cohomology of X_{λ} with integer coefficients. If A is a coefficient ring in which 2 is invertible, then Proposition 7.4, Lemma 7.5 and Theorem 1.1 would all fail to hold if “graded ℤ-algebra” was replaced with “graded A-algebras”. That is, Ding’s Schubert varieties are not classified up to isomorphism by their cohomology with A-coefficients. For example, consider the indecomposable partition λ = (2, 3). By completing the square, one has

R^{(2,3)} ⊗_{ℤ} A ≅ A[x_{1}, x_{2}] / ⟨x_{1}^{2}, x_{2}^{2} + x_{1}x_{2} + \frac{1}{4}x_{1}^{2}⟩

= A[x_{1}, x_{2}] / ⟨x_{1}^{2}, (x_{2} + \frac{1}{2}x_{1})^{2}⟩

≅ A[x_{1}] / ⟨x_{1}^{2}⟩ ⊗ A[y] / ⟨y^{2}⟩ .

Thus indecomposable partitions do not lead to tensor-indecomposable graded A-algebras. This also leads to “extra” isomorphisms among the cohomology rings H^{∗}(X_{λ}; A) ≅ R^{λ} ⊗_{ℤ} A. For example, the partition μ = (2, 2, 4) has indecomposable components μ^{(1)} = μ^{(2)} = (2). Since R^{(2)} ≅ ℤ[x]/ ⟨x^{2}⟩, one has

R^{μ} ⊗_{ℤ} A ≅ A[x] / ⟨x^{2}⟩ ⊗_{A} A[x] / ⟨x^{2}⟩ ≅ R^{λ} ⊗_{ℤ} A

even though λ = (2, 3) and μ = (2, 2, 4) do not have the same indecomposable partition components.

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[10] I.G. Macdonald, Notes on Schubert polynomials. Publications du LACIM, Université du Québec à Montréal, 1991.

[11] J.R. Munkres, Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

Mike Develin, American Institute of Mathematics, 360 Portage Ave., Palo Alto, CA 94306-2244, USA

  E-mail address: develin@post.harvard.edu

Jeremy L. Martin, School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

  E-mail address: martin@math.umn.edu

Victor Reiner, School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

  E-mail address: reiner@math.umn.edu