HILBERT FUNCTIONS OF POINTS ON SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIAN

SUDHIR R. GHORPADE AND K. N. RAGHAVAN

Abstract. We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.

1. Introduction

Let G be a semisimple algebraic group over an algebraically closed field k and P be a parabolic subgroup of G. Fix a Borel subgroup B of G and a maximal torus T in G such that T ⊂ B ⊂ P. Now, G/P is projective variety and it has a distinguished class of subvarieties, known as Schubert varieties (in G/P); these are indexed by a set W^{P}, which corresponds to the T -fixed points of G/P for the action given by left multiplication. Given w ∈ W^{P}, we denote by e^{w} the corresponding T -fixed point, and by X_{w} the corresponding Schubert variety. In fact, X_{w} is the closure of the B-orbit Be^{w}, and can be decomposed as the union of ‘smaller’ Borbits Be^{v}, where v ranges over elements of W^{P} satisfying v ≤ w; here, ≤ is a certain partial order, called the Bruhat-Chevalley order.

The study of Schubert varieties, in general, and the singularities of Schubert varieties, in particular, has been an active and vibrant area of research in the past three decades. It may suffice to cite the recent monograph [2] by Billey and Lakshmibai, which surveys many known results and can also be a useful reference for the background material. Among the basic questions, insofar as the singularities of Schubert varieties are concerned, are the following: (1) Which points are singular?

(2) what is the multiplicity at a (singular) point? and (3) what is the Hilbert function (of the tangent cone) at a (singular) point? It may be noted that these questions are in an ascending order of generality since the singular points are those of multiplicity > 1, and the Hilbert function determines the Hilbert polynomial whose (normalized) leading coefficient gives the multiplicity. Also note that, thanks to the B-orbit decomposition, it suffices to only look at the points e^{v} in X_{w} where v ≤ w.

The singular loci of Schubert varieties are fairly well understood, thanks to the works of several mathematicians (see [2] for details). Recursive formulas for the multiplicity and for the Hilbert function in the case of minuscule G/P and also in the case of symplectic Grassmannian were obtained by Lakshmibai and Weyman [18] in 1990. More explicit results seem to be available so far in the special case when G = SL_{n} = SL(V), where V is an n-dimensional vector space over k and P = P_{d} the maximal parabolic subgroup given by those g ∈ SL(V) which stabilize a (fixed) d-dimensional subspace of V. Assume for a moment that we are in this case. Thus G/P is the Grassmannian G_{d}(V) and W^{P} may be viewed as the set I(d, n) of subsets of {1, …, n} of cardinality d. Let ϵ denote the element {1, …, d} of I(d, n). An explicit closed-form formula for the multiplicity m_{ϵ}(w) of X_{w} at e^{ϵ} was given by Lakshmibai and Weyman [18] in their 1990 paper. Recently, an explicit description for the Hilbert function of X_{w} at e^{ϵ} was given by Kreiman and Lakshmibai [16]. Further, in the case of arbitrary v, Kreiman and Lakshmibai formulated two conjectures which give an explicit combinatorial description for (i) the Hilbert function of X_{w} at e^{v} and (ii) the multiplicity m_{ϵ}(w) of X_{w} at e^{v}. Subsequently, both the conjectures were proved by Kodiyalam and Raghavan [13] who also gave a reformulation of the main result in terms of Gröbner basis.

Date: 8 September 2004.

1991 Mathematics Subject Classification. 14M15, 13F50, 13A30.

In this paper, we consider the problem of determining the multiplicity as well as the Hilbert function of Schubert varieties in the symplectic Grassmannian G/P where G is the symplectic group Sp_{2d} and P a maximal parabolic subgroup P_{d} of G. Alternatively, G/P is the Grassmannian of maximal isotropic subspaces. [see Section 2 for details]. For Schubert varieties in symplectic Grassmannians, we give an explicit combinatorial description for the multiplicity, and more generally, the Hilbert function at any e^{v} in X_{w}. In effect, we formulate and prove the two conjectures of Kreiman and Lakshmibai in this case. Moreover, as in [13], we obtain a reformulation of the main result in terms of Gröbner basis. A precise statement of our main result is given in Section 2.

A key ingredient in our proof is the Standard Monomial Theory for symplectic Grassmannians and its Schubert subvarieties, as developed by De Concini [5] and by Lakshmibai, Musili and Seshadri [17]. This allows us to translate the problem from geometry to combinatorics. This translation—although we derive it afresh in Section 3 below for the sake of clarity, completeness, and readability—is already there in Lakshmibai-Weyman [18]. Our main job then is the solution of the resulting combinatorial problem. Roughly speaking, it amounts to showing that the combinatorial constructions of [13] behave well with respect to a certain involution. This is carried out in Sections 4 and 5. Moreover, as in [16] and [13], we obtain an interpretation of the multiplicity as the number of certain nonintersecting lattice paths, and this is described in Section 6.

The interpretation of the multiplicity in terms of nonintersecting lattice paths may be viewed as an analogue of the results of Krattenthaler [14, 15]. To review the latter, we note that in the case of the classical Grassmannian, Rosenthal and Zelevinsky [20] obtained in 1998 a closed-form formula for m_{v}(w) for arbitrary v and w, using the recursive formula of Lakshmibai and Weyman [18]. The lattice path interpretation was used by Krattenthaler [14] to explain the relationship between the Rosenthal–Zelevinsky formula and the Lakshmibai-Weyman formula for m_{ϵ}(w). Further, Krattenthaler [15] also used the lattice path interpretation to also prove the Multiplicity Conjecture of Kreiman and Lakshmibai.

We now attempt to outline a connection of Schubert varieties, in general, and the results of this paper, in particular, to a class of affine varieties, broadly known as determinantal varieties. For the last two decades, the study of determinantal varieties has proceeded almost in parallel to the study of Schubert varieties, but often as an independent pursuit. In particular, explicit formulas for the multiplicity and the Hilbert function for various classes of determinantal varieties have been obtained. The relation between determinantal varieties and Schubert varieties is best explained in the case of Grassmannian G_{d}(V) = SL_{n}/P. We have the well-known Plücker embedding of G_{d}(V) in ℙ(⋀^{d} V) and for each v ∈ W^{P}, there is a basic affine open set 𝔸_{v} containing e^{v}. In case, v = ϵ, the intersections X_{w} ∩ 𝔸_{v} are affine varieties, which are precisely the varieties defined by an ideal ‘cogenerated’ by a minor of a generic matrix; in other words, these are exactly the determinantal varieties studied by Abhyankar [1], and later by Herzog-Trung [11], Conca-Herzog [4], and others (see, e.g., [8, 9]). Moreover, this affine variety is a cone and hence it coincides with the tangent cone to X_{w} at e^{ϵ}. Thus, in retrospect, the multiplicity formula of Lakshmibai-Weyman and the Hilbert function result of Kreiman-Lakshmibai could have been deduced from the work of Abhyankar [1] and others. On the other hand, for arbitrary v, the varieties X_{w} ∩ 𝔸_{v} are not so well understood and as far as we know, there is no analogue in the literature on determinantal varieties of the results in [13]. As remarked in [13], the recent work of Knutson and Miller [12] considers a class of determinantal varieties more general than those cogenerated by a minor, but it is not clear if the varieties X_{w} ∩ 𝔸_{v} belong to this class when v ≠ ϵ.

In a similar vein, considering affine patches of Schubert varieties in the symplectic Grassmannian, when v = ϵ, leads to varieties given by ideals cogenerated by a minor of a generic symmetric matrix. Thus in this special case, the results obtained in this paper could be compared with those of Conca [3] on symmetric determinantal varieties. Likewise, one could take up the case of orthogonal Grassmannians (although we do not do this here) and the results thus obtained ought to be compared with those of Ghorpade and Krattenthaler [10] on pfaffian varieties. In either of these cases, for an arbitrary v, the varieties X_{w} ∩𝔸_{v} do not seem to correspond to any of the known classes of varieties defined by the minors of a generic symmetric matrix or the pfaffians of a generic skew-symmetric matrix.

There is a yet another related, but independent, body of work on degeneracy loci and formulas for the fundamental classes of degeneracy loci of maps of vector bundles give rise to multiplicity formulas for determinantal and pfaffian varieties. For a detailed explanation, we refer to the appendix in [10] and the books of Fulton-Pragacz [7] and Manivel [19]. While some of the results on multiplicity could be deduced from the corresponding results on degeneracy loci, it does not seem likely that the latter impinge on the determination of Hilbert functions.

2. The Theorem

The main results of this paper are stated in Theorem 2.1 and Corollary 2.2 below. But first, we fix some notation and terminology to be used in the rest of this paper and briefly review some preliminary notions and results.

Given any nonnegative integer n, we denote by [n] the set {1, …, n}. The cardinality of a finite set v will be denoted by |v|. Given positive integers r and n with r ≤ n, we denote by I(r, n) the set of all subsets of [n] of cardinality r. An element v of I(r, n) may be written as v = (v_{1}, …, v_{r}) where 1 ≤ v_{1} < … < v_{r} ≤ n and v = {v_{1}, …, v_{r}}. Given any v = (v_{1}, …, v_{r}) and w = (w_{1}, …, w_{r}) in I(r, n), we say v ≤ w if v_{1} ≤ w_{1}, …, v_{r} ≤ w_{r}. Clearly, ≤ defines a partial order on I(r, n).

A positive integer d will be kept fixed throughout this paper. For j ∈ [2d], set j^{∗} := 2d + 1 − j. Let I(d) denote the set of subsets v of [2d] with the property that exactly one of j, j^{∗} belongs to v for every j ∈ [d]. Note that I(d) is a subset of I(d, 2d). In particular, we have the partial order ≤ on I(d) induced from I(d, 2d). We denote by ϵ the element [d] = (1, …, d) of I(d).

Fix a vector space V of dimension 2d over an algebraically closed field of arbitrary characteristic. Fix a non-degenerate skew-symmetric bilinear form ⟨ , ⟩ on V. Fix a basis e_{1}, …, e_{2d} of V such that

⟨e_{i}, e_{j}⟩ = \{ \begin{matrix}1 & if i = j^{∗} and i < j \\ −1 & if i = j^{∗} and i > j \\ 0 & otherwise \end{matrix}

A linear subspace of V is called isotropic if the form ⟨ , ⟩ vanishes identically on it. It is well-known that an isotropic subspace of V has dimension at most d and every isotropic subspace is contained in one of dimension d. Denote by G_{d}(V) the Grassmannian of d-dimensional subspaces of V and by 𝔐_{d}(V) the set d-dimensional isotropic subspaces of V. Then 𝔐_{d}(V) is a closed subvariety of G_{d}(V), and is called the variety of maximal isotropic subspaces or the symplectic Grassmannian.

The group Sp(V) of linear automorphisms of V preserving ⟨ , ⟩ acts transitively on 𝔐_{d}(V)—this follows from Witt’s theorem that an isometry between subspaces can be lifted to one of the whole vector space. So 𝔐_{d}(V) is identified as the quotient of Sp(V) by the stabilizer of any point (for example, the span of e_{1}, …, e_{d}). The elements of Sp(V) that are diagonal with respect to the basis e_{1}, …, e_{2d} form a maximal torus T of Sp(V). Similarly the elements of Sp(V) that are upper triangular with respect to e_{1}, …, e_{2d} form a Borel subgroup B of Sp(V)—a linear transformation is upper triangular with respect to e_{1}, …, e_{2d} if for each j ∈ [2d], the image of e_{j} under the transformation is a linear combination of e_{i} with i ≤ j.

The T -fixed points of 𝔐_{d}(V) are parametrized by I(d): for v = (v_{1}, …, v_{d}) in I(d), the corresponding T -fixed point, denoted by e^{v}, is the span of e_{v_{1}}, …, e_{v_{d}}. These points lie in different B-orbits and the union of the B-orbits of these points is all of 𝔐_{d}(V). A Schubert variety in 𝔐_{d}(V) is by definition the closure of such a B-orbit (with the reduced scheme structure). Schubert varieties are thus indexed by T -fixed points and so in turn by I(d). Given w in I(d), we denote by X_{w} the closure of the B-orbit of the T -fixed point e^{w}. We have the B-orbit decomposition:

X_{w} = ∐_{v≤w} Be^{v}.

We are interested in the local rings of various points on a Schubert variety X_{w}. In view of the above B-orbit decomposition, it is enough to focus attention on the T -fixed points contained in X_{w}, that is, the points e^{v} for v ≤ w.

For the rest of this section, fix elements v, w of I(d) with v ≤ w. Define

ℜ^{v} := {(r, c) ∈ v × [2d] \ v : r ≤ c^{∗}} and 𝔑^{v} := {(r, c) ∈ v × [2d] \ v : r > c} .

We will be considering “multi-sets” on ℜ^{v} and 𝔑^{v}. By a multi-set on a finite set S we mean a collection of elements of S in which repetitions are allowed and kept account of. Multi-sets on S can be thought of as monomials in the variables corresponding to the elements of S. The cardinality of a multi-set is the number of elements in it, counting repetitions, or equivalently, the degree of the corresponding monomial. Intersections and unions of multi-sets are defined in a natural way and these correspond to the GCD and the LCM of monomials.

Given any β_{1} = (r_{1}, c_{1}), β_{2} = (r_{2}, c_{2}) in 𝔑^{v}, we say that β_{1} > β_{2} if r_{1} > r_{2} and c_{1} < c_{2}. A sequence β_{1} > … > β_{t} of elements of 𝔑^{v} is called a v-chain. Given a v-chain β_{1} = (r_{1}, c_{1}) > … > β_{t} = (r_{t}, c_{t}), we define

s_{β_{1}} ⋯ s_{β_{t}}v := ({v_{1}, …, v_{d}} \ {c_{1}, …, c_{t}}) ∪ {r_{1}, …, r_{t}},

and note that this is an element of I(d). In case the v-chain is empty, this element is just v. We say that w dominates the v-chain β_{1} > … > β_{t} if w ≥ s_{β_{1}} ⋯ s_{β_{t}}v.

Let 𝔖 be a monomial on ℜ^{v}. By a v-chain in 𝔖 we mean a sequence β_{1} > … > β_{t} of elements of 𝔖 ∩ 𝔑^{v}. We say that w dominates 𝔖 if w dominates every v-chain in 𝔖.

Let S_{w}^{v} denote the set of w-dominated monomials on ℜ^{v}, and S_{w}^{v}(m) the set of such monomials of degree m.

We can now state our theorem:

Theorem 2.1. Let v, w be elements of I(d) with v ≤ w. Let X_{w} be the Schubert variety corresponding to w, e^{v} the T -fixed point corresponding to v, and R the coordinate ring of the tangent cone to X_{w} at the point e^{v} (that is, the associated graded ring gr_{M}(O_{X_{w},e^{v}}) of the local ring O_{X_{w},e^{v}} of X_{w} at the point e^{v} with respect to its maximal ideal M). Then the dimension as a vector space of the m^{th} graded piece R(m) of R equals the cardinality of S_{w}^{v}(m), where S_{w}^{v}(m) is as defined above.

The proof of the theorem occupies sections 3, 4, and 5. For now let us note the following easy consequence.

Corollary 2.2. With notation as in Theorem 2.1 above, the multiplicity of R equals the number of square-free w-dominated monomials on ℜ^{v} of maximum cardinality.

Proof. The proof of the corresponding corollary in [13] works verbatim here. □

3. Reduction to combinatorics

Let 𝔐_{d}(V) ⊆ G_{d}(V) ↪ ℙ(∧^{d}V) be the Plücker embedding. The homogeneous co-ordinate rings of 𝔐_{d}(V) and its Schubert sub-varieties in this embedding have been described by De Concini [5] and Lakshmibai, Musili, and Seshadri [17]. We will use their results to reduce the proof of Theorem 2.1 to combinatorics. Our primary reference will be [5]—its language and approach suit our purpose well.

For θ in I(d, 2d), let p_{θ} denote the corresponding Plücker co-ordinate. Consider the affine patch 𝔸 of ℙ(∧^{d}V) given by p_{ϵ} = 1. The intersection 𝔸 ∩ G_{d,2d} of this patch with the Grassmannian is an affine space. Indeed the d-plane corresponding to an arbitrary point z of 𝔸 ∩ G_{d,2d} has a basis consisting of column vectors of a matrix of the form

B = ( \begin{matrix}I \\ A \end{matrix} )

where I is the identity matrix of size d × d and A is an arbitrary matrix of size d × d. The association z ↦ A is bijective. The restriction of a Plücker co-ordinate p_{θ} to 𝔸 ∩ G_{d,2d} is given by the determinant of a sub-matrix of size d × d of B, the entries of θ determining the rows to be chosen from B to form the sub-matrix.

As can be readily verified, a point z of 𝔸 ∩ G_{d,2d} belongs to 𝔐_{d}(V) if and only if the corresponding matrix A = (a_{ij}) is symmetric with respect to the anti-diagonal: a_{ij} = a_{j^{∗}i^{∗}}, where the columns and rows of A are numbered 1, …, d and d+1, …, 2d respectively, and k^{∗} := 2d − k + 1 for k ∈ [2d]. For example, if d = 4, then a matrix that is symmetric with respect to the anti-diagonal looks like this:

( \begin{matrix}d & c & b & a \\ g & f & e & b \\ i & h & f & c \\ j & i & g & d \end{matrix} )

Notation 3.1. For u = (u_{1}, …, u_{d}) in I(d, 2d), set u^{∗} := (u_{d}^{∗}, …, u_{1}^{∗}). The association u ↦ u^{∗} is an order reversing involution of I(d, 2d). There is another order reversing involution on I(d, 2d), namely u ↦ [2d] \ u. These two involutions commute with each other. Composing them, we obtain an order preserving involution on I(d, 2d): u ↦ u^{#} := [2d] \ u^{∗}. Note that u ∈ I(d) if and only if u = u^{#}.

Lemma 3.2. Given any θ ∈ I(d), we have p_{θ} = p_{θ^{#}} on 𝔐_{d}(V).

Proof. Since 𝔐_{d}(V) is irreducible and its intersection 𝔸 ∩ 𝔐_{d}(V) with the affine patch is non-empty, it is enough to check that p_{θ} = p_{θ^{#}} on 𝔸 ∩ 𝔐_{d}(V), and this follows from the symmetry property just mentioned of the matrix A. □

The relations p_{θ} = p_{θ^{#}} do not span the space of all linear relations among the p_{θ}—see Example 3.5 below. In order to describe a nice parametrizing set for a basis for the space of linear forms in the homogeneous co-ordinate ring of 𝔐_{d}(V)—in fact for describing bases for spaces of forms of any given degree—we make the following definition.

Definition 3.3.^{1} The ϵ-degree of an element x of I(d) is the cardinality of x \ [d] or equivalently that of [d] \ x. More generally, given any v ∈ I(d), the v-degree of an element x of I(d) is the cardinality of x \ v or equivalently that of v \ x. An ordered pair 𝔴 = (x, y) of elements of I(d) is called an admissible pair if x ≥ y and the ϵ-degrees of x and y are equal. We refer to x and y as the top and the bottom of 𝔴 and write 𝔱𝔬𝔭(𝔴) for x and 𝔟𝔬𝔱(𝔴) for y.

Given any admissible pairs 𝔴 = (x, y) and 𝔴′ = (x′, y′), we say 𝔴 ≥ 𝔴′ if y ≥ x′, that is, if x ≥ y ≥ x′ ≥ y′. An ordered sequence (𝔴_{1}, …, 𝔴_{k}) of admissible pairs is called a standard tableau if 𝔴_{i} ≥ 𝔴_{i+1} for 1 ≤ i < k. We often write 𝔴_{1} ≥ … ≥ 𝔴_{k} to denote the standard tableau (𝔴_{1}, …, 𝔴_{k}). Given any w ∈ I(d), we say that a standard tableau 𝔴_{1} ≥ … ≥ 𝔴_{k} is w-dominated if w ≥ 𝔱𝔬𝔭(𝔴_{1}).

Proposition 3.4. There is a bijective map (x, y) ↦ (θ, τ) from the set of admissible pairs (x, y) onto the set of ordered pairs (θ, τ) of elements of I(d, 2d) satisfying

τ = θ^{#}, |θ ∩ [d]| = |θ^{#} ∩ [d]|, and θ ∩ [d] ≥ θ^{#} ∩ [d].

Proof. Let [d]^{c} := [2d] \ [d] = {d + 1, …, 2d}. Given an admissible pair (x, y), set

θ = (x ∩ [d]) ∪ (y ∩ [d]^{c}) and τ = (y ∩ [d]) ∪ (x ∩ [d]^{c})

That (θ, τ) satisfy the three conditions is readily verified. For the map in the other direction, set

x = (θ ∩ [d]) ∪ (τ ∩ [d]^{c}) and y = (τ ∩ [d]) ∪ (θ ∩ [d]^{c})

It is easy to verify that (x, y) is an admissible pair. □

Example 3.5. The relations p_{θ} = p_{θ^{#}} do not span the linear space of relations although [18, page 198, item (b)] seems to claim just that. For example, let d = 4 and θ_{1} = (1, 2, 7, 8), θ_{2} = (1, 4, 5, 8), θ_{3} = (1, 3, 6, 8). Then θ_{1}^{#} = (3, 4, 5, 6), θ_{2}^{#} = (2, 3, 6, 7), θ_{3}^{#} = (2, 4, 5, 7). From the form displayed above of a matrix that is symmetric with respect to the anti-diagonal, we get p_{θ_{1}} = p_{θ_{1}^{#}} = df − cg, p_{θ_{2}} = p_{θ_{2}^{#}} = cg − bi and p_{θ_{3}} = p_{θ_{3}^{#}} = bi − df, and consequently, p_{θ_{1}} + p_{θ_{2}} + p_{θ_{3}} = 0.

1Our admissible pairs are a special case of the admissible minors of De Concini [5]: we are only considering the case k = r in his notation. Our standard tableaux are his standard symplectic tableaux but here again we are only considering the case k = r.

Again, contrary to what is claimed in [18, item (10), page 199], the association of the previous proposition is not a bijection if the second and third conditions are dropped. In the case d = 4 for example there are 42 admissible pairs (x, y) and 43 pairs (θ, θ^{#}). If we try the procedure in the proof above for recovering (x, y) on an arbitrary (θ, θ^{#}), the two resulting elements may not be comparable. Taking for example θ = (2, 3, 6, 7), θ^{#} = (1, 4, 5, 8), we recover (2, 3, 5, 8) and (1, 4, 6, 7). □

Definition 3.6. Given an admissible pair 𝔴 = (x, y), we define the associated Plücker co-ordinate p_{𝔴} to be p_{θ} where (x, y) ↦ (θ, θ^{#}) by the association of Proposition 3.4. This is well-defined since p_{θ} = p_{θ^{#}} by Lemma 3.2. If x = y we sometimes write p_{x} for p_{(x,x)}. To formal products of admissible pairs we associate the product of the associated Plücker co-ordinates. In particular, to a standard tableau 𝔴_{1} ≥ … ≥ 𝔴_{r} we associate p_{𝔴_{1}} ⋯ p_{𝔴_{r}}. Such monomials associated to standard tableaux are called standard monomials. Given any w in I(d), we say that a standard monomial is w-dominated for if the corresponding standard tableau is w-dominated.

We can now state the main theorem of standard monomial theory for 𝔐_{d}(V) and its Schubert sub-varieties.

Theorem 3.7. (De Concini, Lakshmibai-Musili-Seshadri) Standard monomials p_{𝔴_{1}} ⋯ p_{𝔴_{r}} of degree r form a basis for the space of forms of degree r in the homogeneous co-ordinate ring of 𝔐_{d}(V) in the Plücker embedding. More generally, given any w ∈ I(d), the w-dominated standard monomials of degree r form a basis for the space of forms of degree r in the homogeneous co-ordinate ring of the Schubert subvariety X(w) of 𝔐_{d}(V).

Proof. The linear independence of the w-dominated standard monomials in the homogeneous co-ordinate ring of X_{w} is proved in [5, Lemma 3.5]^{2}. That all standard monomials span the homogeneous co-ordinate ring of 𝔐_{d}(V) (and so also that of X_{w}) is the content of [5, Theorem 2.4]; on the other hand, as is easy to see, p_{ϑ} vanishes on X_{w} unless w ≥ 𝔱𝔬𝔭(ϑ), and so the standard monomials that are not w-dominated vanish on X_{w}. □

From the above theorem we now deduce a basis for the co-ordinate ring for an affine patch of a Schubert sub-variety in 𝔐_{d}(V).

Definition 3.8. Given any v ∈ I(d), we say that a standard tableau 𝔴_{1} ≥ … ≥ 𝔴_{r} is v-compatible if for each 𝔴_{i}, either v ≥ 𝔱𝔬𝔭(𝔴_{i}) or 𝔟𝔬𝔱(𝔴_{i}) ≥ v, and 𝔴_{i} ≠ (v, v). A standard monomial is v-compatible if the corresponding standard tableau is vcompatible. Given v and w in I(d), we denote by SM_{w}^{v} the set of w-dominated v-compatible standard tableaux.

Fix elements v ≤ w of I(d) so that the point e^{v} belongs to the Schubert variety X(w). Let 𝔸_{v} denote the affine patch of ℙ(∧^{d}V) given by p_{v} ≠ 0 and set Y_{w}^{v} := X(w) ∩ 𝔸_{v}. The point e^{v} is the origin of the affine space 𝔸_{v}.

The functions f_{𝔴} = p_{𝔴}/p_{v}, 𝔴 an admissible pair, provide a set of co-ordinate functions on 𝔸_{v}. The co-ordinate ring k[Y_{w}^{v}] of Y_{w}^{v} is a quotient of the polynomial ring k[f_{𝔴}], where k is the underlying field.

2The Schubert varieties in [5] are orbits under the lower triangular Borel subgroup as opposed to our choice of upper triangular here and that is why the domination is reversed in the statement of the Lemma there.

Proposition 3.9. As 𝔴_{1} ≥ … ≥ 𝔴_{t} runs over the set SM_{w}^{v} of w-dominated v-compatible standard tableaux, the elements f_{𝔴_{1}} ⋯ f_{𝔴_{t}} form a basis for the coordinate ring k[Y_{w}^{v}] of the affine patch Y_{w}^{v} = X_{w} ∩ 𝔸^{v} of the Schubert variety X_{w}.

Proof. The proof is similar to the proof of Proposition 3.1 of [13]. First consider any linear dependence relation among the f_{𝔴_{1}} ⋯ f_{𝔴_{t}}. Replacing f_{𝔴} by p_{𝔴} and “homogenizing” by p_{v} yields a linear dependence relation among the w-dominated standard monomials p_{ϑ_{1}} ⋯ p_{ϑ_{s}} restricted to X_{w}, and so the original relation must only have been the trivial one, for by Theorem 3.7 the p_{ϑ_{1}} ⋯ p_{ϑ_{s}} are linearly independent.

To prove that the f_{𝔴_{1}} ⋯ f_{𝔴_{t}} span k[Y_{w}^{v}] as a vector space, we need to look at not only the corresponding statement for the Plücker co-ordinates but also the proof of that statement. What is immediate from the corresponding statement for the Plücker co-ordinates is that f_{ϑ_{1}} ⋯ f_{ϑ_{s}} span k[Y_{w}^{v}] as ϑ_{1} ≥ … ≥ ϑ_{s} varies over w-dominated standard monomials—the problem at hand is to show that the v-compatible ones among these are enough.

To an arbitrary monomial p_{ϑ_{1}} ⋯ p_{ϑ_{s}} in the Plücker co-ordinates, attach the following multi-set of {1, …, 2d}:

𝔱𝔬𝔭(ϑ_{1}) ∪ 𝔟𝔬𝔱(ϑ_{1}) ∪ ⋯ ∪ 𝔱𝔬𝔭(ϑ_{s}) ∪ 𝔟𝔬𝔱(ϑ_{s})

We claim that if p_{φ_{1}} ⋯ p_{φ_{s}} is a standard monomial that occurs with non-zero coefficient in the expression of p_{ϑ_{1}} ⋯ p_{ϑ_{s}} as a linear combination of standard monomials, the multi-set attached to the two monomials are the same. The claim follows from the nature of the relations used in the proof in [5] of the spanning by the standard monomials^{3}. Now let f_{φ_{1}} ⋯ f_{φ_{s}} be an arbitrary monomial. Consider the expression

3There are two types of relations used in the proof: those in equation (1.1) and those in Proposition 1.8 (all numbers as in [5]). To get the theorem about spanning by standard monomials in our situation, we only need to use special cases of these and so let us first specialize.

In (1.1) we take s = s′ and further let these equal our d (our d is De Concini’s r); the latter halves of all minors are 1, …, d for us and so we write only the first half—a minor for us is therefore just an element θ of I(d, 2d); the right side H is 0 in our case by the choice s = d. Equation (1.1) gets used in the following way: whenever we have a ‘bad’ product of two minors, it occurs as a term on the left side of an equation of type (1.1) where the other terms are ‘not so bad’; the right side being 0, this allows replacement of a ‘bad’ product by a linear combination of ‘not so bad’ ones. Observe the following: for each term on the left side, the multi-set of {1, …, 2d} that is the union of the indices of the two minors is constant for all terms.

Let us denote a minor θ by the pair (θ, θ^{#}) rather than by just θ. For each term θτ on the left side of (1.1) let us consider the multi-set θ ∪ θ^{#} ∪ τ ∪ τ^{#}; it follows from the observation in the last line of the previous paragraph that this multi-set is constant for all terms.

We need Proposition 1.8 only in the case where k is d and h_{1}, …, h_{k} is 1, …, d (we will omit writing the h_{1}, …, h_{k}). Let θ is the element of I(d, 2d) that denotes the minor that is denoted (J̃∪ Γ, Ĩ∪ Γ) in [5] (J̃, Ĩ, Γ are subsets of {1, …, d}; Γ does not meet J̃∪ Ĩ; it is allowed that Ĩ∩ J̃ is non-empty). Write M := (Ĩ∩ J̃) ∪ Γ, N := {1, …, d} \ (Ĩ∪ J̃ ∪ Γ), Î:= Ĩ\ J̃, and Ĵ:= J̃ \ Ĩ. Then θ = Î ∪ Ĵ^{⋆} ∪ M ∪ M^{⋆} and θ^{#} == Î ∪ Ĵ^{⋆} ∪ N ∪ N^{∗}. In other words, the elements outside of Î ∪ Ĵ^{⋆} occur in θ ∪ θ^{⋆} with multiplicity 2, those of Î^{⋆} ∪ Ĵ with multiplicity 0, and the rest of {1, …, 2d} with multiplicity 1. Now, the right side of the equation in Proposition 1.8 consists of minors of the form (J̃ ∪ Γ̄, Ĩ ∪ Γ̄), where Γ̄ has the same cardinality as Γ and is contained in the complement of Ĩ∪ J̃∪ Γ. Since Î and Ĵ remain invariant for all minors appearing in the equation, it follows that the multi-set θ ∪ θ^{#} is the same for all terms.

Finally, observe that if (θ, θ^{#}) corresponds to an admissible pair, say (x, y), then θ∪θ^{#} = x∪y— see the proof of Proposition 3.4 above. Thus when we rewrite a monomial in the Plücker coordinates in terms of the minors of [5], use the relations of equation (1.1) and Proposition 1.8 as in the proof of Theorem 2.4 of [5] to express it as a linear combination of “standard symplectic tableaux”, and then translate back to get a linear combination of standard monomials in our

as a linear combination of standard monomials for p_{v}^{h} · p_{φ_{1}} ⋯ p_{φ_{s}} where h is larger than 2ds. It follows from the claim that p_{v} must occur in every monomial on the right. Dividing by p_{v}^{h+s} we get an expression for f_{φ_{1}} ⋯ f_{φ_{s}} as a linear combination of f_{𝔴_{1}} ⋯ f_{𝔴_{t}} where 𝔴_{1} ≥ … ≥ 𝔴_{t} are v-compatible. □

Given an admissible pair 𝔴 = (x, y), define the v-degree of 𝔴 by

v-degree(𝔴) := \frac{1}{2} (|x \ v| + |y \ v|)

An easy calculation using the fact that the ϵ-degrees of x and y are equal gives the following:

v-degree(𝔴) = |θ \ v| = |θ^{#} \ v|,

where (θ, θ^{#}) is the pair associated to 𝔴 = (x, y) by Proposition 3.4 above.

The affine patch G_{d}(V) ∩ 𝔸_{v} of the Grassmannian is an affine space whose coordinate ring can be taken to be the polynomial ring in variables of the form X_{rc} where r and c are numbers between 1 and 2d such that c belongs to v and r does not—this is easy to see and in any case explained in [13, §3]. It is readily seen that the ideal of the closed sub-variety 𝔐_{d}(V) ∩ 𝔸^{v} of G_{d,2d} ∩ 𝔸_{v} is generated by elements of the form X_{rc} − ϵX_{c^{∗}r^{∗}} where ϵ = −1 if either r > d and c^{∗} < d or r < d and c^{∗} > d, and ϵ = 1 otherwise. The affine patch 𝔐_{d}(V) ∩ 𝔸^{v} of 𝔐_{d}(V) is thus also an affine space whose co-ordinate ring can be taken to be the polynomial ring in variables of the form X_{β}, β in ℜ^{v}.Taking d = 5 and v = (3, 4, 6, 9, 10) for example, a general element of 𝔐_{d}(V) ∩ 𝔸_{v} has a basis consisting of column vectors of a matrix of the following form—having a negative sign in front of all the variable entries in the first d rows is just a convenient way of getting the signs right:

( \begin{matrix}−X_{13} & −X_{14} & −X_{16} & −X_{19} & −X_{1,10} \\ −X_{23} & −X_{24} & −X_{26} & −X_{29} & −X_{19} \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ −X_{53} & −X_{54} & −X_{56} & −X_{26} & −X_{16} \\ 0 & 0 & 1 & 0 & 0 \\ X_{73} & X_{74} & X_{54} & X_{24} & X_{14} \\ X_{83} & X_{73} & X_{53} & X_{23} & X_{13} \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

The expression for f_{𝔴} := p_{𝔴}/p_{v} in terms of the X_{β}, β ∈ ℜ^{v}, is obtained by taking the determinant of the sub-matrix of a matrix such as above obtained by choosing the rows given by the entries of θ where 𝔴 ↦ (θ, θ^{#}) is the association in Proposition 3.4. Thus it is a homogeneous polynomial of degree the v-degree of 𝔴. Since^{4} the ideal of the Schubert variety X_{w} in the homogeneous co-ordinate ring of 𝔐_{d}(V) ⊆ ℙ(∧^{d}V) is generated by the p_{𝔴} as 𝔴 = (x, y) varies over all admissible pairs such that w ≱ x, it follows that the ideal of Y_{w}^{v} := X_{w} ∩ 𝔸^{v} in 𝔐_{d}(V) ∩ 𝔸^{v} is generated by the same f_{𝔴}. We are interested in the tangent cone to X_{w} at e^{v} (or what is the same, the tangent cone to Y_{w}^{v} at the origin), and since k[Y_{w}^{v}] is graded, its associated graded ring with respect to the maximal ideal corresponding to the origin is k[Y_{w}^{v}] itself.

language, the multi-set attached to any monomial on the right side is the same as the multi-set attached to the original monomial.

4This is a consequence of Theorem 3.7. It is easy to see that the p_{𝔴} such that w ≱ 𝔱𝔬𝔭(𝔴)

vanish on X_{w}. Since all standard monomials form a basis for the homogeneous co-ordinate ring of 𝔐_{d}(V) ⊆ ℙ(∧^{d}V), it follows that w-dominated standard monomials in admissible pairs span the quotient ring by the ideal generated by such p_{𝔴}. Since such monomials are linearly independent in the homogeneous co-ordinate ring of X_{w}, the desired result follows.

Proposition 3.9 tells us that the graded piece of k[Y_{w}^{v}] is generated as a k-vector space by elements of SM_{w}^{v} of degree m, where the degree of a standard monomial 𝔴_{1} ≥ … ≥ 𝔴_{t} is defined to be the sum of the v-degrees of 𝔴_{1}, …, 𝔴_{t}. To prove Theorem 2.1 it therefore suffices to prove that the set SM_{w}^{v}(m) of w-dominated v-compatible standard monomials of degree m is in bijection with S_{w}^{v}(m).

4. Further reductions

In the last section we saw how Theorem 2.1 follows once it is shown that the combinatorially defined sets SM_{w}^{v}(m) and S_{w}^{v}(m) are in bijection with each other. Now we make some further reductions. After these, it will remain only to show that the combinatorial bijection established in [13, §4] has further structure, and this will be shown in Section 5.

An element v of I(d) remains fixed throughout this section.

Let S^{v} denote the set of monomials in ℜ^{v} and T^{v} the set of monomials in 𝔑^{v}. Let SM^{v,v} denote the set of v-compatible standard monomials that are anti-dominated by v: a standard monomial θ_{1} ≥ … ≥ θ_{t} is anti-dominated by v if 𝔟𝔬𝔱(θ_{t}) ≥ v.

Define the domination map from T^{v} to I(d) by sending a monomial in 𝔑^{v} to the least element that dominates it^{5}. Define the domination map from SM^{v,v} to I(d) by sending θ_{1} ≥ … ≥ θ_{t} to 𝔱𝔬𝔭(θ_{1}). Both these maps take, by definition, the value v on the empty monomial.

The desired bijection follows from the following proposition.

Proposition 4.1. There is a bijection between SM^{v,v} and T^{v} that respects domination and degree.

By the above proposition, we have, for w in I(d) with w ≥ v, a bijection SM_{w}^{v,v}(m) ≅ T_{w}^{v}(m), where SM_{w}^{v,v}(m) is the set of w-dominated elements of SM^{v,v} that have degree m, and similarly T_{w}^{v}(m) is the set of w-dominated elements of T^{v} that have degree m.

Now let U^{v} denote the set of monomials in ℜ^{v} \ 𝔑^{v} (that is, in pairs (r, c) of ℜ^{v} with r < c), and SM_{v}^{v} the set of v-compatible and v-dominated standard monomials. As explained below, the “mirror image” of the bijection SM^{v,v} ≅ T^{v} gives a bijection SM_{v}^{v}(m) ≅ U^{v}(m), where SM_{v}^{v}(m) and U^{v}(m) denote respectively the sets of elements of degree m of SM_{v}^{v} and U^{v}.

Putting these bijections together, we get the desired bijection:

SM_{w}^{v}(m) = ⋃_{j=0}^{m} SM_{w}^{v,v}(j) × SM_{v}^{v}(m − j)

≅ ⋃_{j=0}^{m} T_{w}^{v}(j) × U^{v}(m − j)
= S_{w}^{v}(m).

5The poset I(d) has a largest element, namely (d + 1, …, 2d), and this clearly dominates all monomials. The glb (greatest lower bound) in I(d, 2d) of a set of elements in I(d) belongs to I(d)

for glb(u, x)^{#} = glb(u^{#}, x^{#}) for u, x in I(d, 2d).

Here the first equality is obtained by splitting a v-compatible standard monomial θ_{1} ≥ … ≥ θ_{t} into two parts θ_{1} ≥ … ≥ θ_{p} and θ_{p+1} ≥ … ≥ θ_{t}, where p is the largest integer, 1 ≤ p ≤ t, with 𝔟𝔬𝔱(θ_{p}) ≥ v. The last equality is obtained by writing a monomial of degree m in ℜ^{v} as a product of two monomials, one in 𝔑^{v} and the other in ℜ^{v} \ 𝔑^{v}, the sum of their degrees being m.

Let us now briefly explain how to take the “mirror image”. Recall that j^{∗} := 2d − j + 1 for an integer j, 1 ≤ j ≤ 2d. For u = (u_{1}, …, u_{d}) in I(d), define the dual u^{∗} by u^{∗} := (u_{d}^{∗}, …, u_{1}^{∗}). This dual map on I(d) is an order reversing involution. It induces a bijection SM_{v}^{v} ≅ SM^{v^{∗},v^{∗}} by associating to θ_{1} ≥ … ≥ θ_{t} the element θ_{t}^{∗} ≥ … ≥ θ_{1}^{∗}—for an admissible pair θ = (𝔱𝔬𝔭(θ), 𝔟𝔬𝔱(θ)), θ^{∗} denotes of course the admissible pair (𝔟𝔬𝔱(θ)^{∗}, 𝔱𝔬𝔭(θ)^{∗}). The sum of the v-degrees of θ_{1}, …, θ_{t} equals the sum of the v^{∗}-degrees of θ_{t}^{∗}, …, θ_{1}^{∗}, so that we get a bijection SM_{v}^{v}(m) ≅ SM^{v^{∗},v^{∗}}(m).

For an element (r, c) in 𝔑^{v^{∗}}, define the dual to be the element (c, r). Since v ∈ I(d), we have v^{∗} = [2d] \ v. Thus, it follows that (c, r) belongs to ℜ^{v} \ 𝔑^{v}. This induces a degree preserving bijection T^{v^{∗}} ≅ U^{v}. Putting this together with the bijection of the previous paragraph and the one given by Proposition 4.1 (for v^{∗} in place of v), we have

SM_{v}^{v}(m) ≅ SM^{v^{∗},v^{∗}}(m) ≅ T^{v^{∗}}(m) ≅ U^{v}(m)

The proof of Theorem 2.1 is thus reduced to the the proof of Proposition 4.1.

4.1. Proof of Proposition 4.1. The arguments in Section 3 and in the present section thus far have shown that Theorem 2.1 follows from Proposition 4.1. To prove the proposition, we will exploit the work already done in [13]. More precisely, we will deduce the proposition from its earlier version stated in the paragraphs following Proposition 4.2 of that paper. As alluded to in the introduction, the main ingredient in the deduction is showing that the bijection in the earlier version respects the involution induced on the combinatorial entities by the skew-symmetric form—see Lemma 4.5 and Proposition 5.5 below for the precise statements. We also need Lemma 5.15 which roughly speaking is a symmetry property of domination.

Remark 4.2. Proposition 4.1 and its earlier version are worded exactly alike. The difference lies in the meaning attached to the symbols. To distinguish the two meanings, we use in this section a tilde sign over these symbols to indicate that the symbol in question has the meaning given to it in [13]. However, in Section 5, the notation and terminology of that paper will throughout be in force, and so the will be omitted.

Consider v as an element of I(d, 2d). A standard monomial in I(d, 2d) is a totally ordered sequence θ_{1} ≥ … ≥ θ_{t} of elements of I(d, 2d). Such a monomial is v-compatible if each θ_{j} is comparable to v but no θ_{j} equals v; it is anti-dominated by v if θ_{t} ≥ v.

Notation 4.3. In keeping with Remark 4.2, we have the following:

\begin{matrix} \\ SM & v,v \end{matrix} denotes the set of v-compatible standard monomials in I(d, 2d) antidominated by v. ℜ̃^{v} denotes the set of ordered pairs (r, c) such that r ∈ [2d] \ v and c ∈ v. 𝔑̃^{v} denotes the subset of ℜ̃^{v} consisting of those (r, c) with r > c. T̃^{v} denotes the set of monomials in 𝔑̃^{v}.

Consider the natural injection SM^{v,v} → \begin{matrix} \\ SM & v,v \end{matrix}: θ_{1} ≥ … ≥ θ_{t} is mapped to 𝔱𝔬𝔭(θ_{1}) ≥ 𝔟𝔬𝔱(θ_{1}) ≥ … ≥ 𝔱𝔬𝔭(θ_{t}) ≥ 𝔟𝔬𝔱(θ_{t}), but 𝔟𝔬𝔱(θ_{t}) is omitted if it equals v. This map preserves domination and doubles degree. Composing this with the domination and degree preserving bijection \begin{matrix} \\ SM & v,v \end{matrix} → T̃^{v} of [13, §4], we get an injection of SM^{v,v} into T̃^{v}.

To describe the image of SM^{v,v} in T̃^{v}, we introduce a definition. For α = (r, c) in ℜ̃^{v}, set α^{#} := (c^{∗}, r^{∗}). The map α ↦ α^{#} is an involution on ℜ̃^{v} and on 𝔑̃^{v} Elements of the form (r, r^{∗}) of 𝔑̃^{v} are referred to as belonging to the “diagonal”, and the set of diagonal elements of 𝔑̃^{v} is denoted 𝔡^{v}.

Definition 4.4. A monomial 𝔖 of T̃^{v} is special if (1) 𝔖 = 𝔖^{#} and

(2) the multiplicity of any diagonal element in 𝔖 is even.

Equivalently, 𝔖 is special if there exists 𝔗 in T̃^{v} with 𝔖 = 𝔗 ∪ 𝔗^{#}. The set of special monomials is denoted 𝔈.

The following is the main technical result of the present paper. It is an immediate corollary of Proposition 5.5 below. Here we use it to prove Proposition 4.1.

Lemma 4.5. The image of SM^{v,v} in T̃^{v} is the set 𝔈 of special monomials.

In other words, there is a domination preserving and degree doubling bijective map from SM^{v,v} to 𝔈.

On the other hand, there is a domination preserving and degree halving bijective map from 𝔈 to T^{v}. Given 𝔖 in 𝔈, to get the corresponding element of T^{v}, replace those (r, c) of 𝔖 with r > c^{∗} by (c^{∗}, r^{∗}) and then take the (positive) square root. This clearly halves the degrees. The map in the other direction is obvious: given a monomial of T^{v}, replace each (r, c) by (r, c)(c^{∗}, r^{∗})—in other words, a monomial 𝔓 of T^{v} is mapped to 𝔓 ∪ 𝔓^{#}.

To see that this bijection from 𝔈 to T^{v} preserves domination, let 𝔗 be a special monomial and 𝔓 the corresponding monomial in T^{v}. Let y and z be the images respectively of 𝔗 and 𝔓 under the domination maps. Then y is the first co-ordinate of π(𝔗), where π is the map defined in [13, §4]—see Proposition 4.1 (4) of [13]. Since 𝔗 = 𝔗^{#}, it follows from Proposition 5.6 below that y = y^{#}. This is equivalent to saying that y belongs to I(d).

That y dominates 𝔓 is clear: any v-chain in 𝔓 is also a v-chain in 𝔗. Since y is in I(d), it follows from the definition of the domination map on T^{v} that y ≥ z. That z ≥ y follows from Lemma 5.15 below: z^{#} = z, 𝔓 = 𝔓^{up}, and z dominates 𝔓, so that z dominates 𝔓 ∪ 𝔓^{#} = 𝔗.

Composing the two bijective maps SM^{v,v} → 𝔈 and 𝔈 → T^{v} above gives us a domination and degree preserving bijection SM^{v,v} → T^{v}. Proposition 4.1 is thus proved.

5. Completion of proof

The purpose of this section is to establish the statements used in the proof of Proposition 4.1. The terminology and notation of [13] will throughout be in force.

An element v of I(d, n) remains fixed throughout. The symbol w will denote an arbitrary element of I(d, n) satisfying w ≥ v, 𝔖 an arbitrary monomial in 𝔑^{v}, and 𝔖_{w} the unique subset of 𝔑^{v} defined by Proposition 5.1 below.

Let (r, c) and (R, C) be elements of 𝔑^{v}. They are comparable if either they are equal or (R, C) > (r, c) (which means that R > r and C < c) or (r, c) > (R, C). We say that (R, C) dominates (r, c) if R ≥ r and C ≤ c.

5.1. Recall.

Proposition 5.1. To each element w of I(d, n) satisfying w ≥ v there is associated a unique subset 𝔖_{w} = {(r_{1}, c_{1}), …, (r_{p}, c_{p})} of 𝔑^{v} satisfying conditions A–C below.

A. r_{i} ≠ r_{j} and c_{i} ≠ c_{j} for i ≠ j.

B. If r_{i} < r_{j} then either c_{j} < c_{i} or r_{i} < c_{j}.
C. w = (v \ {c_{1}, …, c_{p}}) ∪ {r_{1}, …, r_{p}}.

Furthermore, 𝔖_{w} also satisfies conditions D and E below.

D. The v-degree of w equals the cardinality p of 𝔖_{w}. E. w is the smallest element of I(d, n) to dominate 𝔖_{w}.

Proof. The existence and uniqueness of 𝔖_{w} is the substance of Proposition 4.3 of [13]. That D holds is obvious. For E, see Lemma 5.5 of [13]. □

Remark 5.2. If 𝔖 is a subset of 𝔑^{v} that satisfies condition A of Proposition 5.1, the equation in item C can be taken to be the definition of an element w of I(d, n). If 𝔖 satisfies B also, then 𝔖 = 𝔖_{w} for this w. Let x be an element of I(d, n) and 𝔖 a subset of 𝔖_{x}. Then 𝔖 evidently satisfies A and B. Therefore 𝔖 = 𝔖_{w} for some w. It follows from E that x ≥ w.

Lemma 5.3. Let 𝔅 and ℭ be distinct blocks of a monomial 𝔖. Then 𝔅 ∪ 𝔅′ and ℭ ∪ ℭ′ are disjoint.

Proof. For 𝔅 and ℭ both belonging to the same partition 𝔖_{j} of 𝔖, this follows from the proof of Lemma 4.10 of [13]—see the last line of that proof. Now suppose that 𝔅 ⊆ 𝔖_{i} and ℭ ⊆ 𝔖_{j}, and that (r, c) belongs to both 𝔅 ∪ 𝔅′ and ℭ ∪ ℭ′. Assume without loss of generality that i < j. From the definition of ℭ′, it follows that there exists an element (r′, c) in ℭ with r′ ≥ r. From the definition of the partitions 𝔖_{k}, it follows that there exists (a, b) in 𝔖_{i} such that (a, b) > (r′, c). Now (a, b) > (r, c) and both of them belong to 𝔖_{i} ∪ 𝔖′_{i}, a contradiction to Lemma 4.10 of [13]. □

Corollary 5.4. Given (r, c) in 𝔖 ∪ 𝔖′, there exists a unique block 𝔅 of 𝔖 such that (r, c) belongs to 𝔅 ∪ 𝔅′.□

5.2. Statement of the main proposition. From now on we will assume that n = 2d and that v = v^{#} (equivalently that v belongs to I(d)). The map π of [13, §4.2] is defined only on non-empty monomials. We extend the definition by setting π(𝔖) = (v, 𝔖) if 𝔖 is empty. Similarly we extend the definition of the map ϕ of [13, §4.4] by setting ϕ(v, 𝔗) = 𝔗 if 𝔗 is empty.

The following proposition has as an immediate corollary Lemma 4.5 which was the main ingredient in the proof of Proposition 4.1. Its proof will be given in Subsections 5.4 and 5.5 after some preliminaries in Subsection 5.3.

Proposition 5.5. Let 𝔖 be a special monomial. Set π(𝔖) = (t, 𝔖′) and π(𝔖′) = (u, 𝔗). Then

(1) t ≥ u ≥ v, t^{#} = t, u^{#} = u, and the (1, …, d)-degrees of t and u are equal.

(2) 𝔗 is special and u dominates 𝔗. Conversely, if t and u are elements of I(d, 2d) and 𝔗 a monomial such that the two conditions above are satisfied, then ϕ(t, ϕ(u, 𝔗)) is special.

5.3. Behavior under the operation #. In this subsection we investigate the behavior of the combinatorial constructions of [13, §4] under the operation #.

If β is j-deep in 𝔖, then β^{#} is j-deep in 𝔖^{#}. Thus the partitions 𝔖_{j} of 𝔖 are respected by the hash operation: (𝔖_{j})^{#} = (𝔖^{#})_{j}. It is also easy, given the definitions, to verify the following:

• If 𝔅 is a block of 𝔖_{j}, then 𝔅^{#} is a block of 𝔖_{j}^{#}.

• w(𝔅^{#}) = w(𝔅)^{#}.
• (𝔅^{#})′ = (𝔅′)^{#}.

These observations amount to a proof of the following proposition.

Proposition 5.6. The map π respects #: more precisely, if π(𝔖) = (w, 𝔖′), then π(𝔖^{#}) = (w^{#}, 𝔖′^{#}).

Let us now show that the map ϕ also respects #. For this we need

Proposition 5.7. The association w ↦ 𝔖_{w} respects #: (𝔖_{w})^{#} = 𝔖_{w^{#}}. In particular, w^{#} = w if and only if (𝔖_{w})^{#} = 𝔖_{w}.

Proof. The result follows from Proposition 5.1: (𝔖_{w})^{#} satisfies conditions A, B, and C of that proposition with w replaced by w^{#} in condition C. □

Let 𝔗 be a monomial in 𝔑^{v} and w an element of I(d, 2d) such that w ≥ v and w dominates 𝔗. The following are easily verified from the definitions:

• (𝔖_{w}^{j})^{#} = 𝔖_{w^{#}}^{j} and so (w^{j})^{#} = (w^{#})^{j}.

• (𝔗_{j}^{w})^{#} = (𝔗^{#})_{j}^{w^{#}}.

• (𝔓_{β})^{#} is the piece of 𝔗^{#} corresponding to β^{#}, for β an element of 𝔖_{w}.

• ((𝔗_{j}^{w})^{⋆})^{#} = ((𝔗^{#})_{j}^{w^{#}})^{⋆}.

These observations amount to a proof of the following proposition.

Proposition 5.8. The map ϕ respects #: more precisely, if ϕ(w, 𝔗) = 𝔖 then ϕ(w^{#}, 𝔗^{#}) = 𝔖^{#}.

Proposition 5.9. Suppose that w = w^{#} (equivalently by (3) 𝔖_{w} = 𝔖_{w}^{#}). Then for an element (r, c) of 𝔖_{w}, either (a) c < r ≤ d, or (b) d < c < r, or (c) r^{⋆} = c ≤ d < r. In particular, the ϵ-degree of w equals the number of elements of 𝔖_{w} belonging to the diagonal.

Proof. Suppose that c ≤ d and r > d. Since 𝔖_{w} = 𝔖_{w}^{#}, (c^{∗}, r^{∗}) belongs to 𝔖_{w}. If r < c^{∗}, then r^{∗} > c, and the presence of both (r, c) and (c^{∗}, r^{∗}) in 𝔖_{w} is a violation of condition E of Proposition 5.1, a contradiction. If c^{∗} < r, then c > r^{∗}, and again there is a similar contradiction. Thus c = r^{∗}. □

Proposition 5.10. Let 𝔅 be a block of a monomial 𝔖 satisfying 𝔖 = 𝔖^{#}.

(A) The following are equivalent: (1) 𝔅 = 𝔅^{#}.

(2) w(𝔅) lies in the diagonal.

(3) 𝔅 ∪ 𝔅′ meets the diagonal. (B) Suppose that the conditions in (A) above are met. Then there is a unique

element of the diagonal in 𝔅 ∪ 𝔅′ and the multiplicities of that element in 𝔅 and 𝔅′ differ by 1.

Proof. (A): As noted in the proof of Proposition 5.6, 𝔅^{#} is a block of 𝔖^{#} = 𝔖, w(𝔅^{#}) = w(𝔅)^{#}, and (𝔅^{#})′ = (𝔅′)^{#}.

(1) ⇒ (2): w(𝔅)^{#} = w(𝔅^{#}) = w(𝔅), which means that w(𝔅) is on the diagonal.

(2) ⇒ (1): We have w(𝔅^{#}) = w(𝔅)^{#} = w(𝔅). If 𝔅 and 𝔅^{#} are distinct blocks, then w(𝔅) and w(𝔅^{#}) cannot share a row or column index— Corollary 4.13 of [13]—let alone being equal.

(3) ⇒ (1): It follows from the hypothesis that 𝔅 ∪ 𝔅′ and 𝔅^{#} ∪ (𝔅^{#})′ meet. By Corollary 5.4, 𝔅 = 𝔅^{#}.

(1) ⇒ (3): Since 𝔅 = 𝔅^{#}, there is an element (r, s) of 𝔅 with r ≤ s^{∗}. Let r be the maximal row index of such an element of 𝔅. A portion of the arrangement of the elements of 𝔅 in ascending order of row and column indices looks like this:

…, (a, b), \underbrace{(r, r^{⋆}), …, (r, r^{⋆})}_{m times}, (b^{⋆}, a^{⋆}), …

where either a < r or b < r^{⋆}. If m > 0 then 𝔅 meets the diagonal and we are done. If m = 0, then a = r, and the portion of 𝔅′ corresponding to the one above of 𝔅 looks like this:

…, (?, b), (a, a^{⋆}), (b^{⋆},?), …

so that (a, a^{⋆}) = (r, r^{⋆}) belongs to 𝔅′. (B): Since elements of the diagonal are comparable but no two distinct elements of 𝔅 ∪ 𝔅′ are (the latter statement is immediate from the definitions; or see Lemma 4.10 of [13]), there is at most one diagonal element in 𝔅 ∪ 𝔅′. Since there is at least one such element by hypothesis, there is a unique such element.

Let (r, r^{⋆}) be the unique diagonal element 𝔅 ∪ 𝔅′. Proceed as in the proof of (1) ⇒ (3) above. If m = 0, then (a, a^{∗}) = (r, r^{∗}) belongs to 𝔅′. Further b < r^{∗} and b^{∗} > r, so that the multiplicity of (r, r^{⋆}) in 𝔅′ is 1 and we are done.

If m > 0, the portion of 𝔅′ corresponding to the one above of 𝔅 looks like this:

…, (?, b), (a, r^{⋆}), \underbrace{(r, r^{⋆}), …, (r, r^{⋆})}_{m − 1 times}, (r, a^{⋆}), (b^{⋆},?), …

The multiplicity of (r, r^{⋆}) in 𝔅′ is therefore either m + 1 or m − 1 depending upon whether or not a equals r. □

Corollary 5.11. Let 𝔖 = 𝔖^{#} and (r, r^{⋆}) be an element of the diagonal belonging to 𝔖 ∪ 𝔖′. Then the multiplicities of (r, r^{⋆}) in 𝔖 and 𝔖′ differ by 1.

Proof. By Corollary 5.4, there is a unique block 𝔅 of 𝔖 such that (r, r^{⋆}) belongs to 𝔅 ∪ 𝔅′. The multiplicities of (r, r^{⋆}) in 𝔖 and 𝔖′ equal respectively those in 𝔅 and 𝔅′, and these differ by 1 by the proposition above. □

Definition 5.12. For α = (r, c) in 𝔑^{v}, set

α^{up} := \{ \begin{matrix}α & if r ≤ c^{⋆} \\ α^{#} = (c^{⋆}, r^{⋆}) & if r > c^{∗} \end{matrix} α^{down} := \{ \begin{matrix}α & if r ≥ c^{⋆} \\ α^{#} = (c^{⋆}, r^{⋆}) & if r < c^{∗} \end{matrix}

Proposition 5.13. Let α be an element of 𝔑^{v} and w an element of I(d, 2d) with w ≥ v. If both α^{up} and α^{down} belong to 𝔖_{w} and both of them dominate an element β of 𝔑^{v}, then α^{up} = α^{down}. In particular, if w = w^{#} and α in 𝔖_{w} dominates an element on the diagonal, then α belongs to the diagonal.

Proof. Write α^{up} = (R, C) and β = (r, c). Then C ≤ R^{∗} ≤ c < r ≤ R ≤ C^{∗}—here C ≤ R^{∗} and R ≤ C^{∗} because α^{up} = (R, C), R^{∗} ≤ c because α^{down} = (C^{∗}, R^{∗}) dominates β, c < r because β belongs to 𝔑^{v}, and r ≤ R because α^{up} dominates β. Unless R = C^{∗} the presence of both α^{up} and α^{down} in 𝔖_{w} leads to a violation of condition B of Proposition 5.1 by 𝔖_{w}, a contradiction. □

Corollary 5.14. If α_{1} > … > α_{p} is the v-chain of diagonal elements in 𝔖_{w} for an element w = w^{#}, then α_{j} is j-deep but not j + 1-deep in 𝔖_{w}.□

Lemma 5.15. If w^{#} = w and w dominates a monomial 𝔖 with 𝔖 = 𝔖^{up} (that is, α = α^{up} for every α ∈ 𝔖), then w also dominates 𝔖 ∪ 𝔖^{#}.

Proof. Let β_{1} > … > β_{p} be a v-chain in 𝔖 ∪ 𝔖^{#}. We need to show that this chain is dominated by a v-chain in 𝔖_{w}. We may assume without loss of generality that β_{p}^{up} = β_{p}. If β_{j}^{up} = β_{j} for all j, then the desired result follows immediately from the hypothesis. Otherwise, let k ∈ [p] be the largest such that β_{k}^{down} = β_{k}. Choose v-chain α_{1} > … > α_{p} in 𝔖_{w} such that it dominates β_{1}^{up} > … > β_{p}^{up}. Replacing α_{1} > … > α_{p} by α_{1}^{up} > … > α_{p}^{up}, we may assume that α_{j}^{up} = α_{j}. It follows from Proposition 5.13 that α_{j} lives on the diagonal for 1 ≤ j ≤ k: α_{k}^{down} > β_{k}^{down} = β_{k} > β_{k+1} and α_{k}^{up} > β_{k}^{up} > β_{k+1}^{up} = β_{k+1} implies α_{k} belongs to the diagonal; and α_{k−1} > α_{k} implies that α_{k−1} belongs to the diagonal etc. Thus α_{j} dominates β_{j} for all j ∈ [p]. □

5.4. Proof of the first half of Proposition 5.5. By Proposition 5.6 above, π respects the # operation. Hence it follows that t^{#} = t, (𝔖′)^{#} = 𝔖′, u^{#} = u and (𝔗)^{#} = 𝔗. Also, it follows from 1, 3 and 4 of Proposition 4.1 of [13] that t ≥ u ≥ v, and that u dominates 𝔗. Thus it remains only to prove the following:

(a) 𝔖_{t} meets the diagonal in as many points as 𝔖_{u}—this will imply, by Proposition 5.9, that the ϵ-degrees of t and u are equal and hence that (t, u) is an admissible pair.

(b) 𝔗 is special.

We first prove (a). If (r, r^{⋆}) and (s, s^{⋆}) are elements of the diagonal 𝔡^{v}, then either (r, r^{⋆}) > (s, s^{⋆}) or the other way around. The elements of 𝔖_{u} ∩ 𝔡^{v} therefore form a v-chain. Let β_{1} > … > β_{p} be all the elements of 𝔖_{u} ∩ 𝔡^{v}. By Proposition 5.1 (E), t dominates 𝔖_{u}. By Lemma 4.5 of [13], there exists a v-chain α_{1} > … > α_{p} in 𝔖_{t} that dominates β_{1} > … > β_{p}. By Proposition 5.13, the α_{j} belong to 𝔡^{v}.

To complete the proof of (a), it remains to be seen that the cardinality of 𝔖_{u} ∩𝔡^{v} is not less than that of 𝔖_{t}∩𝔡^{v}. We will show in fact that 𝔖′∩𝔡^{v} has at least as many distinct elements as the cardinality of 𝔖_{t}∩𝔡^{v}. The blocks containing these elements will then clearly be distinct: elements of the diagonal are comparable but no two distinct elements of a block are. Each such block ℭ satisfies ℭ^{#} = ℭ, for blocks are disjoint and ℭ meets the diagonal. If ℭ and ℭ_{1} are distinct blocks, then w(ℭ) and w(ℭ_{1}) are distinct—Corollary 4.13 of [13]—so there are distinct contributions to 𝔖_{u} from these blocks. Finally, these contributions all lie on the diagonal by Proposition 5.10.

Let β be an element of 𝔖_{t} ∩ 𝔡^{v}. Let 𝔅 be the block of 𝔖 with w(𝔅) = β. By Proposition 5.10, 𝔅 ∪ 𝔅′ meets 𝔡^{v}. If 𝔅 meets 𝔡^{v} at say β′, then, since 𝔖 satisfies condition 2 of Definition 4.4, it follows from the definition of 𝔅′ that 𝔅′ also contains β′. Thus we get an element β′ in 𝔖′ ∩ 𝔡^{v} for every β in 𝔖_{t} ∩ 𝔡^{v}. The association β ↦ β′ is injective, for the association β ↦ 𝔅 is injective—see the reason given in the last paragraph—and so is 𝔅 ↦ β′ by Lemma 5.3. This finishes the proof of (a).

For use in the proof of (b), let us record a corollary of the proof of (a):

Corollary 5.16. A block 𝔅 of 𝔖′ meets the diagonal if and only if 𝔅 = 𝔅^{#}.

Proof. If 𝔅 meets the diagonal, then 𝔅^{#} = 𝔅. The number of 𝔅 such that 𝔅^{#} = 𝔅 is the cardinality of 𝔖_{u} ∩ 𝔡^{v}. The number of 𝔅 that meet the diagonal is the cardinality of 𝔖′ ∩ 𝔡^{v}. But we have seen that the cardinalities of 𝔖_{u} ∩ 𝔡^{v}, 𝔖′ ∩ 𝔡^{v}, and 𝔖_{t} ∩ 𝔡^{v} are equal. □

We now turn to the proof of (b). Suppose that (r, r^{⋆}) belongs to 𝔗. Let 𝔅 be the block of 𝔖′ such that (r, r^{⋆}) belongs to 𝔅′. By Proposition 5.10 and the corollary above, 𝔅 meets the diagonal. Since elements of 𝔅∪𝔅′ are incomparable— see Lemma 4.10 of [13]—it follows that (r, r^{⋆}) belongs to 𝔅 and so to 𝔖′. By Corollary 5.11, the multiplicities of (r, r^{⋆}) in 𝔖 and 𝔖′′ = 𝔗 differ by 2 or 0 or −2.

5.5. Proof of the converse in Proposition 5.5. Set 𝔐 = ϕ(u, 𝔗) and 𝔑 = ϕ(t, 𝔐). Since ϕ respects #—see Proposition 5.8—it follows that 𝔐^{#} = 𝔐 and 𝔑^{#} = 𝔑. The only thing that takes some work is to show that 𝔑 satisfies condition 2 of Definition 4.4.

By Proposition 5.10, each diagonal element of 𝔑 belongs to a block 𝔅 of 𝔑 with w(𝔅) on the diagonal. It therefore suffices to show that the multiplicity of a diagonal element in any such block of 𝔑 is even (possibly 0).

Let α_{1} > … > α_{p} and β_{1} > … > β_{p} be respectively the elements 𝔖_{t} ∩ 𝔡^{v} and 𝔖_{u} ∩ 𝔡^{v}. Let 𝔓_{1}, …, 𝔓_{p} be the pieces of 𝔗 with respect to u corresponding respectively to β_{1}, …, β_{p}. Then 𝔓_{j}^{⋆} = 𝔅 is a block of 𝔐 with w(𝔓_{j}^{⋆}) = 𝔅_{j} and 𝔓_{j} = 𝔅′—see the claim at the beginning of §4.5 of [13].

Since 𝔗 is special, it follows from Proposition 5.10 that 𝔓_{j}^{⋆} meets the diagonal with (positive) odd multiplicity. Denoting the diagonal element of 𝔓_{j}^{⋆} by δ_{j}, we have δ_{1} > … > δ_{p}—see Lemma 4.21 of [13]. By Proposition 5.13, the piece of 𝔐 with respect to t to which δ_{j} belongs can only correspond to one of the α_{i}.

No two distinct elements of a piece are comparable, so two different δ cannot belong to the piece corresponding to the same α. Further, it follows from the the definition of the way we break a monomial into pieces that if i < j and δ_{i}, δ_{j} belong respectively to the pieces corresponding to α_{k} and α_{l}, then k < l. Thus δ_{j} belongs to the piece corresponding to α_{j}.

Let 𝔅_{1}, …, 𝔅_{p} be the blocks of 𝔑 with w(𝔅_{j}) = α_{j}. Then 𝔅′_{j} is the piece of 𝔐 with respect to t corresponding to α_{j}—see the claim at the beginning of [13, §4.5]. By Proposition 5.10, the multiplicity of the diagonal element in 𝔅_{j} is even (possibly 0).□

6. Interpretations

Fix elements v, w in I(d) with v ≤ w. It follows from Corollary 2.2 that the multiplicity of the Schubert variety X_{w} in 𝔐_{d}(V) at the point e^{v} can be interpreted as the cardinality of a certain set of non-intersecting lattice paths. We illustrate this by means of two examples. The justification for the interpretation is the same word-for-word as for the usual Grassmannian [13, §5] and so will be omitted. The interpretation leads immediately to a formula for the multiplicity involving Figure 1. A symmetric tuple of non-intersecting lattice paths as in Example 6.1

binomial determinants. Specializing to the case v = ϵ = (1, …, d), we recover Conca’s result [3, Theorem 3.6].

Example 6.1. Let d = 23.

\begin{matrix}v = (1, 2, 3, 4, 5, 11, 12, 13, 14, 19, 20, 22, 23, 26, 29, 30, 31, 32, 37, 38, 39, 40, 41) \\ w = (4, 5, 9, 10, 14, 17, 18, 21, 23, 25, 27, 28, 31, 32, 34, 35, 36, 39, 40, 41, 44, 45, 46) \end{matrix}

so that

\begin{aligned}𝔖_{w} & = & \{(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), \\ & & (27, 26), (28, 19), (34, 30), (35, 29), (36, 11), (44, 38), (45, 37), (46, 1)\} \end{aligned}

We think of v and w as elements of I(d, 2d) and switch to the notation of [13] as we have done in §5. The grid depicting the points of 𝔑^{v} is shown in Figure 6.1. The solid dots represent the points of 𝔖_{w}, this being the monomial associated to w by Proposition 5.1. From each point β of 𝔖_{w} we draw a vertical line and a horizontal line. Let β(start) and β(finish) denote respectively the points where the vertical line and the horizontal line meet the boundary. For example, β(start) = (15, 11) and β(finish) = (36, 32) for β = (36, 11); for β = (21, 20), β(start) = β(finish) = β.

A lattice path between a pair of such points β(start) and β(finish) is a sequence α_{1}, …, α_{q} of elements of 𝔑^{v} with α_{1} = β(start), α_{q} = β(finish), and for j, 1 ≤ j ≤ q − 1, writing α_{j} = (r, c), α_{j+1} is either (r + 1, c) or (r, c + 1). If β(start) = (r, c) and β(finish) = (R, C), then q = (R − r) + (C − c) + 1.

Write 𝔖_{w} = {β_{1}, …, β_{p}}. By Proposition 5.7, 𝔖_{w}^{#} = 𝔖_{w^{#}} = 𝔖_{w}—that is, 𝔖_{w} is symmetric with respect to the anti-diagonal. Consider the set of all p-tuples of paths (Λ_{1}, …, Λ_{p}), where Λ_{j} is a lattice path between β_{j}(start) and β_{j}(finish) such that Figure 2. All the symmetric tuples of non-intersecting lattice paths as in Example 6.2

• No two Λ_{j} intersect.

• If β_{j} = β_{k}^{#}, then Λ_{j} = Λ_{k}^{#}. A particular such symmetric p-tuple is shown in Figure 6.1. The number of such symmetric p-tuples is the multiplicity of X_{w} at the point e^{v}. □

Example 6.2. Let us draw, in a simple case, the pictures of all possible symmetric tuples of non-intersecting lattice paths as defined in the above example. Let d = 5, v = (1, 2, 3, 6, 7), and w = (3, 5, 7, 9, 10), so that 𝔖_{w} = {(5, 2), (9, 6), (10, 1)}. Figure 2 shows all the symmetric 3-tuples of non-intersecting lattice paths. There are 10 of them and thus the multiplicity in this case is 10. □

6.1. The Gröbner basis interpretation. Here we interpret Theorem 2.1 in terms of Gröbner basis—see Proposition 6.3 below. The special case of the proposition when v is the identity coset (that is, v = (1, …, d)) has been obtained by Conca [3] by different methods.

Fix elements v, w in I(d) with v ≤ w.

Recall from §3 that the ideal (f_{𝔴} = p_{𝔴}/p_{v} | 𝔴 admissible pair, 𝔱𝔬𝔭(𝔴) ≰ w) in the polynomial ring P := k[X_{β} | β ∈ ℜ^{v}] defines the tangent cone to the Schubert variety X_{w} at the point e^{v}. We will identify a subset of these generators f_{𝔴} as being Gröbner basis with respect to certain term orders for the ideal of the tangent cone. Observe that the f_{𝔴} are homogeneous polynomials in the variables X_{β}.

Let 𝔴 = (t, u) be an admissible pair and (θ, θ^{#}) the pair associated to 𝔴 as in Proposition 3.4. Let us call 𝔴 “good” (the scope of this terminology is intended to be limited to this section) if

• v ≤ u and t ≰ w;

• 𝔖_{θ} = 𝔖_{θ}^{up}, that is, r ≤ c^{∗} for (r, c) in 𝔖_{θ}, and the elements of 𝔖_{θ} form a v-chain—here 𝔖_{θ} is the monomial attached to θ as in Proposition 5.1.^{6}

Suppose that 𝔴 is good. Then 𝔖_{θ} occurs up to sign as a term in the expression for f_{𝔴} = f_{θ} as the determinant of a sub-matrix of the matrix of the form on page 9 attached to v—here we are abusing notation and not distinguishing between a monomial in ℜ^{v} and the corresponding monomial in the variables X_{β}, β in ℜ^{v}.

6While it is true that in §5 we are using the notation of [13] where the symbols have a different meaning from what they do in §2–4, the condition that r ≤ c^{∗} for (r, c) in 𝔖_{θ} gives us some license for sloppiness.

Proposition 6.3. Fix elements v ≤ w of I(d). Fix any term order on the monomials in the standard graded polynomial ring P := k[X_{β} | β ∈ ℜ^{v}] such that, for any good admissible pair 𝔴, the initial term of f_{𝔴} is 𝔖_{θ}. Then the f_{𝔴} as 𝔴 varies over all good admissible pairs form a Gröbner basis with respect to this term order for the ideal (f_{𝔴} | 𝔴 admissible pair, 𝔱𝔬𝔭(𝔴) ≰ w) defining the tangent cone to the Schubert variety X_{w} at the point e^{v}.

Proof. Denoting by I the ideal (f_{𝔴} | 𝔴 admissible pair, 𝔱𝔬𝔭(𝔴) ≰ w) defining the tangent cone, by in(f) the initial term of a polynomial f in the fixed term order, and by in(I) the ideal (in(f) | f ∈ I), we clearly have a graded surjection

P/ (in(f_{𝔴}) | 𝔴 good admissible pair, 𝔱𝔬𝔭(𝔴) ≰ w) ↠ P/in(I)

The assertion is that this map is an isomorphism. To prove this, it is enough to show that the Hilbert function of the quotient ring dominates that of the ring on the left.

The Hilbert function of P/in(I) is the same as that of the tangent cone P/I, and so by Theorem 2.1 its value at a positive integer m is the cardinality of the set S_{w}^{v}(m) of w-dominated monomials in ℜ^{v} of degree m. It therefore suffices to observe that a monomial in ℜ^{v} that is not w-dominated belongs to the ideal (in(f_{𝔴}) | 𝔴 good admissible pair, 𝔱𝔬𝔭(𝔴) ≰ w). Given such a monomial, choose a v-chain β_{1} > … > β_{t} in it such that w ≱ s_{β_{1}} ⋯ s_{β_{t}}v. Applying the lemma below to the this v-chain, we get a good admissible pair 𝔴 such that the initial term of f_{𝔴} in the fixed term order is {β_{1}, …, β_{t}}. □

Lemma 6.4. Fix elements v ≤ w of I(d). Let β_{1} > … > β_{r} be a v-chain of elements in 𝔑^{v} such that w ≱ s_{β_{1}} ⋯ s_{β_{r}}v. The element of SM^{v,v} that corresponds under the bijection of Proposition 4.1 to the element {β_{1}, …, β_{r}} of T^{v} consists of a single admissible pair 𝔴 = (t, u) which is good and for which 𝔖_{θ} = {β_{1}, …, β_{r}}.

Proof. We follow the notation of the statement and proof of Proposition 4.1. Set 𝔖 = {β_{1}, …, β_{r}} ∪ {β_{1}^{#}, …, β_{r}^{#}}, π(𝔖) = (t, 𝔖′), and π(𝔖′) = (u, 𝔗). We know from that proposition that (t, u) is an admissible pair and that v ≤ u. Since the bijection SM^{v,v} ≃ T^{v} respects domination, and s_{β_{1}} ⋯ s_{β_{r}}v ≰ w by hypothesis, it follows that t ≰ w. It remains to be seen that 𝔗 is the empty monomial and that 𝔖_{θ} = {β_{1}, …, β_{r}}. These will follow from explicit computation which we now perform.

Set β_{j} = (r_{j}, c_{j}); then β_{j}^{#} = (c_{j}^{∗}, r_{j}^{∗}). Since β_{j} ∈ 𝔑^{v}, we have c_{j} ≤ r_{j}^{∗} and c_{j} < r_{j}, and so {c_{1}, …, c_{t}} ⊆ [d].

Consider the two-step partitioning of 𝔖 into the 𝔖_{j} and the 𝔖_{j} into blocks. It is readily seen that 𝔖_{j} = {β_{j}, β_{j}^{#}}. Let s be the largest integer, 0 ≤ s ≤ r, such that r_{s} ≥ r_{s}^{∗}—set s = 0 if r_{1} < r_{1}^{∗}. In other words, s is the least integer, 0 ≤ s ≤ t, such that {r_{s+1}, …, r_{t}} ⊆ [d]. For 1 ≤ j ≤ s, 𝔖_{j} is a single block. For s < j ≤ t, the block decomposition of 𝔖_{j} is {β_{j}} ∪ {β_{j}^{#}}.

The following expressions for 𝔖′, u, t, and 𝔗 follow readily from the definition of the map π:

𝔖′ = {(r_{1}, r_{1}^{∗}), …, (r_{t}, r_{t}^{∗})} and u = (v ∪ {r_{1}, …, r_{s}}) \ ({r_{1}^{∗}, …, r_{s}^{∗}}) ,

and

t = (v ∪ {c_{1}^{∗}, …, c_{t}^{∗}} ∪ {r_{s+1}, …, r_{t}}) \ ({c_{1}, …, c_{t}} ∪ {r_{s+1}^{∗}, …, r_{t}^{∗}}) ,

and 𝔗 is the empty monomial. Now, upon letting [d]^{c} := {d + 1, …, 2d}, put

θ := (t ∩ [d]) ∪ (u ∩ [d]^{c}) = (v ∪ {r_{1}, …, r_{t}}) \ {c_{1}, …, c_{t}}

and

θ^{#} := (u ∩ [d]) ∪ (t∩[ d]^{c}) = (v ∪ {c_{1}^{∗}, …, c_{t}^{∗}}) \ {r_{1}^{∗}, …, r_{t}^{∗}}.

Immediately from the definition of 𝔖_{θ}, we have

𝔖_{θ} = {(r_{1}, c_{1}), …, (r_{t}, c_{t})} and 𝔖_{θ^{#}} = {(c_{1}^{∗}, r_{1}^{∗}), …, (c_{t}^{∗}, r_{t}^{∗})}.
□

We finish by listing some term orders that satisfy the requirement of Proposition 6.3. Fix notation and terminology as in §15.2 of [6]. Defined below are eight total orders >_{1} through >_{8} on the elements of 𝔑^{v}. Let >_{j} denote also any total order on ℜ^{v} that refines the total order >_{j} on 𝔑^{v}. The homogeneous lexicographic orders induced by >_{1}, >_{2}, >_{7}, >_{8} and the reverse lexicographic orders induced by >_{4}, >_{6} satisfy the requirement of Proposition 6.3. The reverse lexicographic orders induced by >_{3}, >_{5} do not satisfy the requirement.

The total orders >_{1} through >_{8} on 𝔑^{v} are as follows: for (r, c) in 𝔑^{v} and (a, b) in ℜ^{v} \ 𝔑^{v},

(1) (r, c) >_{1} (r′, c′) if either (a) r < r′ or (b) r = r′ and c > c′.

(2) (r, c) >_{2} (r′, c′) if either (a) c > c′ or (b) c = c′ and r < r′.
(3) (r, c) >_{3} (r′, c′) if either (a) r < r′ or (b) r = r′ and c < c′.
(4) (r, c) >_{4} (r′, c′) if either (a) c > c′ or (b) c = c′ and r > r′.
(5) (r, c) >_{5} (r′, c′) if either (a) c < c′ or (b) c = c′ and r < r′.
(6) (r, c) >_{6} (r′, c′) if either (a) r > r′ or (b) r = r′ and c > c′.
(7) (r, c) >_{7} (r′, c′) if either (a) c < c′ or (b) c = c′ and r > r′.
(8) (r, c) >_{8} (r′, c′) if either (a) r > r′ or (b) r = r′ and c < c′.

Acknowledgments

A part of this work was done when the first named author visited the Institute of Mathematical Sciences, Chennai, during March 2003, and Purdue University, West Lafayette, during Spring 2004. He would like to thank both these institutions for their hospitality and support. Also, he gratefully acknowledges the travel support from the Commission on Development and Exchanges of the International Mathematical Union for his visit to Purdue.

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Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India.

E-mail address: srg@math.iitb.ac.in

Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, India.

E-mail address: knr@imsc.ernet.in