ON THE ENUMERATION OF INDEXED MONOMIALS AND THE COMPUTATION OF HILBERT FUNCTIONS OF LADDER

DETERMINANTAL VARIETIES

SUDHIR R. GHORPADE

Abstract. We outline the computation of an explicit formula for the Hilbert function of the ladder determinantal varieties defined by the vanishing of all minors of a fixed size of a rectangular matrix with indeterminate entries such that the indeterminates in these minors are restricted to lie in some ladder shaped region of the rectangular array. Finding such a formula is equivalent to enumerating the set of monomials of a fixed degree such that the support of these monomials is a subset of a ‘ladder’ and satisfies a certain “index condition”. We also describe applications of this formula for estimating the dimension of ladder determinantal varieties.

1. Introduction

Typically, a ladder is a subset of a rectangle which looks as follows.

•  • •  • •

Figure 1

Given a rectangular matrix with indeterminate entries, the algebraic variety defined by all minors of a fixed size within a ladder shaped subset of the rectangle, is usually called a ladder determinantal variety. These varieties are intimately related to Schubert varieties in flag manifolds (cf. [21], [22]). Motivated partly by this connection, the ladder determinantal varieties (and, in fact, more general varieties of this kind) were introduced by Abhyankar [1]. It is known that such varieties are irreducible (cf. [1], [23]), Cohen-Macaulay (cf. [12]) and normal (cf. [5], [19]). Some of these properties can also be deduced using the connection with Schubert varieties from the work of Ramanathan [25] (see also [3], [17], [18], [24] [26]). Using [25] or otherwise (cf. [7], [11]) one knows also that most of these varieties have rational singularities. Other properties such as Gorensteinness as well as the determination of the divisor class group have also been investigated (cf. [5], [6], [20]).

Date: October 1, 1999.

1991 Mathematics Subject Classification. Primary 05A15, 13C40, 13D40, 14M12; Secondary 05A19, 05E10, 14M15.

A part of this work was supported by research grant No. 93-106/RG/MATHS/AS from the Third World Academy of Sciences, Italy. Currently, the author is partially supported by a ‘Career Award’ grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.

We consider in this paper the problem of finding an explicit formula for the Hilbert function of ladder determinantal varieties. More precisely, we consider an m(1) × m(2) matrix X = (X_{ij}) whose entries are independent indeterminates over a field K, and a subset L of the rectangle {(i, j) : 1 ≤ i ≤ m(1), 1 ≤ j ≤ m(2)} such that L is like a (one-sided) ladder as in Fig. 1 or, more generally, a ‘biladder’ or a two-sided ladder (See Section 2 for precise definitions). Let K[L] denote the ring of polynomials in the indeterminates {X_{ij} : (i, j) ∈ L} with coefficients in K. Let I_{p}(L) denote the ideal of K[L] generated by all p × p minors of X in K[L].

From a combinatorial viewpoint, the problem of finding the Hilbert function of I_{p}(L) is equivalent to enumerating a set of monomials in K[L] satisfying a certain ‘index condition’ and of a fixed degree. Indeed, from the work of Abhyankar [1], we know that such “indexed monomials” form a K-basis for the graded components of the residue class ring K[L]/I_{p}(L). In fact, this equivalence is a basic starting point for the arguments in [2], [15], [16] and in this paper as well. The connection with indexed monomials is explained in details in Section 2.

The problem of finding explicitly the Hilbert function of the residue class ring K[L]/I_{p}(L) or of the corresponding projective variety V_{p}(L) was first studied by Kulkarni in his 1985 thesis [15] (see also [16]). There he obtained a nice formula in the first nontrivial case of p = 2. It may be noted that in the degenerate case when L is the entire rectangle [1, m(1)] × [1, m(2)], the ideal I_{p}(L) reduces to the classical determinantal ideal I_{p}(X) that arises frequently in Algebraic geometry and Invariant Theory. In the case of I_{p}(X), the Hilbert function is explicitly known from the work of Abhyankar [1]. In particular, the Hilbert function of K[X]/I_{p}(X) coincides with the Hilbert polynomial of I_{p}(X) for all nonnegative integers; ideals with this property are called hilbertian. For a survey of Abhyankar’s work, see [8] and for a short proof of a formula for the Hilbert function of K[X]/I_{p}(X), see [4] or [9]. Returning to ladders, it was shown in 1989 by Abhyankar and Kulkarni [2] that the ideals I_{p}(L) are also hilbertian for any p > 1 and any biladder L; in fact, this result is applicable to sets more general than biladders, called generalized ladders or saturated sets (see Section 2 for details). Ladder determinantal ideals such as I_{p}(L) were considered from the viewpoint of Gröbner bases and lattice paths by Herzog and Trung [12]. They showed that one can describe the Hilbert function of K[L]/I_{p}(L) in terms of the f-vector of the associated simplicial complex. While this would also prove the Hilbertianness of I_{p}(L), there still remains the problem of finding explicitly the Hilbert function of K[L]/I_{p}(L). To this end, Conca and Herzog [4] conjectured a ‘remarkable formula’ for the Hilbert series (that is, the generating function for the sequence of values of the Hilbert function) in the case of one-sided ladders. Recently, Krattenthaler and Prohaska [14] have established this Conjecture using the so called ‘two-rowed arrays’.

Our main result is an explicit (albeit, complicated!) formula for the Hilbert function of K[L]/I_{p}(L) for any biladder L and any p > 1. This may be viewed as a natural extension of the results of Kulkarni [16] and a refinement of the technique used by Abhyankar and Kulkarni [2] to prove that I_{p}(L) is hilbertian. A detailed proof of this result shall appear in a forthcoming paper [10]. For this paper, we have a modest two-fold aim. First, to outline some of the main ideas involved in the proof and give the statements of the main lemmas and theorems. Second, to describe applications of the Hilbert function formula for determining, or estimating, the dimension of ladder determinantal varieties. It is hoped that this would make [10], which appears to be a rather long and technical paper, a little more accessible.

This paper is organized as follows. The next section sets up some notation and preliminary notions that we shall use. We consider the case of I_{2}(L) in Section 3 and the general case in Section 4. Finally, in Section 5, we give results for estimating the degree of the Hilbert function or equivalently the dimension of ladder determinantal varieties. This last section is disjoint from [10] and thus, unlike in the previous sections, complete proofs are given.

2. Preliminaries

By ℤ, ℕ, and ℕ^{+} we denote the sets of all integers, nonnegative integers, and positive integers respectively. Given any a, b ∈ ℤ, we define the closed and semiclosed integral intervals [a, b], [a, b), (a, b] in the obvious way; for example,

[a, b) = {c ∈ ℤ : a ≤ c < b}.

Fix a pair m = (m(1), m(2)) of positive integers, a field K and an m(1) × m(2) matrix X = (X_{ij}) whose entries are independent indeterminates over K. Given any subset Y of the rectangle

[1, m(1)] × [1, m(2)] = {(i, j) : 1 ≤ i ≤ m(1), 1 ≤ j ≤ m(2)},

let K[Y] denote the polynomial ring in the indeterminates {X_{ij} : (i, j) ∈ Y } with coefficients in K. Given any p ∈ ℕ, we let I_{p}(Y) denote the ideal of K[Y] generated by all p × p minors of X in K[Y].

Given any h ∈ ℕ^{+}, by a ladder generating bisequence (LGB) of length h, we mean a map S : [1, 2] × [0, h] → ℕ such that

1 = S(1, 0) ≤ S(1, 1) < S(1, 2) < ⋯ < S(1, h) = m(1)

and

m(2) = S(2, 0) > S(2, 1) > ⋯ > S(2, h − 1) ≥ S(2, h) = 1.

The positive integer h may be denoted by len(S). We shall find it convenient to also consider the empty bisequence, which we declare to be the unique LGB of length 0. Given any LGB S of length h, we define

L(S) = ⋃_{k=1}^{len(S)}[S(1, k − 1), S(1, k)] × [1, S(2, k − 1)],

and

L(S)^{o} = ⋃_{k=1}^{len(S)}[S(1, k − 1), S(1, k)) × [1, S(2, k − 1)).

We call L(S) to be the ladder corresponding to S and L(S)^{o} to be the interior of L(S). Note that if h > 0, then

L(S)^{o} ⊂_{≠} L(S) ⊆ [1, m(1)] × [1, m(2)].

In case h = 0, we have L(S) = L(S)^{0} = ∅, whereas if h = 1, then we have L(S) = [1, m(1)] × [1, m(2)]. We shall denote by ∂S or by ∂L(S) the boundary of L(S), which is defined by ∂S = L(S) \ L(S)^{o}. Points (S(1, k), S(2, k)), where 1 ≤ k ≤ h − 1, are called the nodes of S or of the ladder L(S) and we denote by N (S) the set of all nodes of S.

Given any LGB’s S′ and S such that len(S) ≠ 0 and L(S′) ⊆ L(S), we define

L(S′, S) = L(S) \ L(S′) and L(S′, S)^{o} = L(S)^{o} \ L(S′).

We call L(S′, S) to be the biladder corresponding to S′ and S while L(S′, S)^{o} to be the interior of L(S′, S). Note that since we allow L(S′) = ∅, a ladder is a special case of a biladder. Pictorially, a ladder looks as in Fig. 1 above and a biladder looks as in Fig. 2 (a) or, more generally, as in Fig. 2 (b) below.

•  •  •  •

Figure 2 (a) Figure 2 (b)

It may be remarked that in these pictures, we adopt the ‘matrix notation’ rather than that of Coordinate Geometry to represent points. Thus in Fig. 1, the bullet on the top left hand corner indicates the point (1, 1) while the other bullets indicate the ‘nodes’ (S(1, k), S(2, k)) , 1 ≤ k < h. In Fig. 2 (a) and Fig. 2 (b), we have only marked the points (1, m(2)) and (m(1), 1) corresponding to h = 0 and h = 1.

Given any biladder L = L(S′, S), we shall denote by Δ(S′, S) the intersection of the boundaries of L(S) and L(S′), and by N (S′, S) the set of common nodes of N (S′) and N (S), that is,

Δ(S′, S) = ∂S′ ∩ ∂S and N (S′, S) = N (S′) ∩ N (S).

It may be noted that Δ(S′, S) = ∂S ∩ L(S′).

Observe that ladders as well as biladders are subsets Y of [1, m(1)] × [1, m(2)] with the property that whenever (i_{1}, i_{2}), (j_{1}, j_{2}) ∈ Y with i_{1} < i_{2} and j_{1} < j_{2}, we have that (i_{1}, j_{2}) ∈ Y and (j_{1}, i_{2}) ∈ Y. Sets Y with this property may be called generalized ladders or saturated sets. Some authors simply refer to them as ladders. It is not difficult to see that if a generalized ladder is ‘connected’, then it must be a biladder.

Given any Y ⊆ [1, m(1)] × [1, m(2)], we let mon(Y) denote the set of all maps of Y → ℕ. Given any θ ∈ mon(Y), we let

supp(θ) = {(i, j) ∈ Y : θ(i, j) ≠ 0}

denote the support of θ and

X^{θ} = ∏_{(i,j)∈Y} X_{ij}^{θ(i,j)}

denote the corresponding monomial in K[Y]. Following Abhyankar [1], we define the index of any subset T ⊆ [1, m(1)] × [1, m(2)] by

ind(T) = max{p ∈ ℕ : ∃ (i_{1}, j_{1}), (i_{2}, j_{2}), …, (i_{p}, j_{p}) in T with
i_{1} < i_{2} < ⋯ < i_{p} and j_{1} < j_{2} < ⋯ < j_{p}}.

For a monomial θ ∈ mon(Y), the index is defined by putting

ind(θ) = ind(supp(θ)).

For every p ∈ ℕ we let

mon(Y, p) = {θ ∈ mon(Y) : ind(θ) ≤ p}

and, restricting attention to monomials of a specified degree, for every p ∈ ℕ and V ∈ ℕ we let

mon(Y, p, V) = {θ ∈ mon(Y, p) : ∑_{y∈Y} θ(y) = V }.

We now recall a basic result of Abhyankar [1, Thm. 20.10] (see also [8, Thm. 6.7]), which was alluded to in the Introduction.

2.1. Theorem. Let Y ⊆ [1, m(1)] × [1, m(2)] be any generalized ladder and let p ∈ ℕ. Given any V ∈ ℕ, the set {X^{θ} : θ ∈ mon(Y, p, V)} forms a free Kbasis of the V -th homogeneous component K[Y]_{V} /I_{p+1}(Y)_{V} of the residue class ring K[Y]/I_{p+1}(Y). Consequently, the Hilbert function of the residue class ring K[Y]/I_{p+1}(Y) or the corresponding projective variety V_{p+1}(Y) is given by

H(V) = |mon(Y, p, V)|, (V ∈ ℕ).

Following Kulkarni [15], we consider the so called radicals and skeletons, which are defined as follows. Fix some Y ⊆ [1, m(1)] × [1, m(2)]. A subset R ⊆ Y is called a radical if ind(R) ≤ 1, and it is called a skeleton if for any two distinct elements (i_{1}, i_{2}) and (j_{1}, j_{2}) of R, we have

either: i_{1} < j_{1} and i_{2} > j_{2} or: i_{1} > j_{1} and i_{2} < j_{2}.

The set of all radicals (resp: skeletons) in Y is denoted by rad(Y) (resp: skel(Y)). Note that skel(Y) ⊆ rad(Y). More generally, given any p ∈ N, we let rad^{p}(Y) denote the set of all R ⊆ Y such that ind(R) ≤ p. Elements of rad^{p}(Y) may be called p-fold radicals. Finally, we set for any p ∈ ℕ and d ∈ ℕ,

rad^{p}(Y, d) = {R ∈ rad^{p}(Y) : |R| = d} and skel(Y, d) = {R ∈ skel(Y) : |R| = d}.

3. Radicals and Skeletons

It is easy to see that the problem of counting the desired set of monomials can be reduced to the problem of enumerating the p-fold radicals of a given size.

3.1. Lemma. Given any Y ⊆ [1, m(1)] × [1, m(2)], p ∈ ℕ and V ∈ N, we have

|mon(Y, p, V)| = ∑_{d≥0} \binom{V − 1}{V − d}|rad^{p}(Y, d)|,

where the summation on the right is essentially finite (that is, all except finitely many summands are zero).

Fix any biladder L = L(S′, S), and let L^{o} denote its interior. Let h = len(S) and h′ = len(S′). Also let

δ_{0} = |Δ(S′, S)| = |∂S′ ∩ ∂S|.

Given any (k, k′) ∈ [1, h] × [1, h′], we let

μ_{L}(k, k′) = card([S(1, k − 1), S(1, k)) ∩ [S′(1, k′ − 1), S′(1, k′)))

and

ν_{L}(k, k′) = card([S(2, k), S(2, k − 1)) ∩ (S′(2, k′), S′(2, k′ − 1)]).

Note that these numbers are completely (and easily) determined by L.

As a preliminary step towards calculating |rad^{p}(L, d)|, we shall restrict our attention to the case of p = 1 so as to determine |rad(L, d)|. To this end, we use techniques similar to [16] except that now instead of ladders we consider the more general biladders, and so one has to be a little more careful, especially since we are allowing overlaps (as in Fig. 2 (b)) of smaller ladder L(S′) with the bigger ladder L(S). As in [15], we reduce the problem to skeletons by constructing two maps

λ : rad(L) → skel(L^{o}) and μ : skel(L^{o}) → rad(L)

such that λ is surjective, μ is injective and moreover, μ is the inverse of the restriction of λ to maximal subsets of rad(L). This leads to the following.

3.2. Theorem. Let M = m(1) + m(2) − 1. Given any d ∈ ℕ, we have

|rad(L, d)| = ∑_{ℓ≥0} \binom{M − δ_{0} − ℓ}{d − ℓ}|skel(L^{o}, ℓ)|,

where the summation on the right is essentially finite (that is, all except finitely many summands are zero).

To describe an explicit formula for the number of skeletons in the interior of a biladder, we need some notation.

Given any ℓ ∈ ℕ, let M_{h,h′}(ℕ, ℓ) denote the set of all h×h′ matrices with integral entries such that the sum of all the entries is ℓ. Note that this is a finite set. Given any α = (α_{kk′}) ∈ M_{h,h′}(ℕ, ℓ) and any (i, j) ∈ [1, h] × [1, h′], we let

σ_{i}(α) = ∑_{k=1}^{i} ∑_{k′=1}^{h′} α_{kk′} and τ_{j}(α) = ∑_{k=1}^{h} ∑_{k′=j}^{h′} α_{kk′}.

Given any ℓ ∈ ℕ and α, β ∈ M_{h,h′}(ℕ, ℓ), we define

σ(β) ≤ σ(α) to mean that σ_{i}(β) ≤ σ_{i}(α) for all i ∈ [1, h]

and

τ(β) ≤ τ(α) to mean that τ_{j}(β) ≤ τ_{j}(α) for all j ∈ [1, h′].

Finally, for any ℓ ∈ ℕ, we define

S(L^{o}, ℓ) = ∑_{\begin{matrix}α, β ∈ M_{h h′} (ℕ, l) \\ σ(β) ≤ σ(α), τ(β) ≤ τ(α) \end{matrix}} ∏_{\begin{matrix}1 ≤ k ≤ h \\ 1 ≤ k′ ≤ h′ \end{matrix}} \binom{μ_{L}(k, k′)}{α_{kk′}}\binom{ν_{L}(k, k′)}{β_{kk′}}.

3.3. Theorem. Given any ℓ ∈ ℕ, we have

|skel(L^{o}, ℓ)| = S(L^{o}, ℓ).

As a consequence of the above results, we obtain the following formula, which may be viewed as an extension of Kulkarni’s formula [16, Thm. 11]

3.4. Theorem. The Hilbert function as well as the Hilbert polynomial of I_{2}(L) in K[L] is given by

F (V) = ∑_{ℓ≥0} \binom{V + M − δ_{0} − 1 − ℓ}{M − δ_{0} − 1}S(L^{o}, ℓ)

where M = m(1) + m(2) − 1 and S(L^{o}, ℓ) is given by the formula above.

4. General Case

As in Section 3, we fix a biladder L = L(S′, S), and let L^{o} denote its interior. Let h, h′, δ_{0} and ν be as defined in Section 3. Also, let M = m(1) + m(2) − 1.

Given any LGB S^{∗}, we shall write S^{∗} ≤ S to mean that L(S^{∗}) ⊆ L(S). Further, given any p ∈ ℕ^{+}, we let D_{p}(L) denote the set of all (p−1)-tuples S = (S_{1}, …, S_{p−1}) of LGB’s such that

len(S_{i}) ≠ 0 and Δ(S_{i−1}, S_{i}) = Δ(S_{i−1}, S), for 1 ≤ i ≤ p − 1,

and

S′ = S_{0} ≤ S_{1} ≤ ⋯ ≤ S_{p−1} ≤ S_{p} = S.

Observe that D_{p}(L) is nonempty since it always contains the (p−1)-tuple (S, …, S).

The following basic result allows us to tackle the general case recursively by applying the results of Section 3. The map Γ mentioned in the theorem below can be described quite explicitly and it yields a decomposition of rad(L), which may be viewed as a refinement of the superskeleton decomposition of [2, Thm. 10].

4.1. Theorem. Let p ∈ ℕ^{+} and L be as above. Then there exists a LGB S^{∗} such that S′ < S^{∗} ≤ S and there exists an injective map

Γ : rad^{p}(L) → rad(L) × rad^{p−1}(L^{∗}),

where L^{∗} denotes the biladder L(S^{∗}, S).

The map Γ is not surjective, in general. But its image can be characterized reasonably well. This leads to the following enumerative result.

4.2. Theorem. Given any d ∈ ℕ and any p ∈ ℕ^{+}, we have

|rad^{p}(L, d)| = ∑_{S^{∗}} ∑_{d_{1}+d_{2}=d} \binom{M − δ_{0} − ν^{∗}}{d_{1} − ν^{∗}}|rad^{p−1}(L^{∗}, d_{2})|,

where the first sum is taken over all LGB’s S^{∗} such that

S′ ≤ S^{∗} ≤ S, len(S^{∗}) ≠ 0 and Δ(S′, S^{∗}) = Δ(S′, S),

and the second sum is over all nonnegative integer pairs (d_{1}, d_{2}) such that d_{1} + d_{2} = d, and where ν^{∗} = |N (S^{∗}) \ N (S) ∩ N (S^{∗})| and L^{∗} denotes the biladder L(S^{∗}, S).

Successive applications of the above result yields the following.

4.3. Theorem. Given any d ∈ ℕ and any p ∈ ℕ^{+}, we have

|rad^{p}(L, d)| = ∑_{S∈D_{p}(L)} ∑_{ℓ≥0} \binom{pM − δ(S) − ν_{1} − ⋯ − ν_{p−1} − ℓ}{d − ν_{1} − ⋯ − ν_{p−1} − ℓ}S(L_{p−1}^{o}, ℓ)

where given any S = (S_{1}, …, S_{p−1}) ∈ D_{p}(L), we have put

δ(S) = ∑_{i=0}^{p−1} δ_{i} , where δ_{i} = |Δ(S_{i}, S)| = |∂S_{i} ∩ ∂S|, for 0 ≤ i ≤ p − 1

(with the convention that S_{0} = S′ and S_{p} = S), and ν_{i} = |N (S_{i}) \ (N (S_{i}) ∩ N (S))| for 1 ≤ i ≤ p − 1, and L_{p−1} = L(S_{p−1}, S).

Given S = (S_{1}, …, S_{p−1}) ∈ D_{p}(L) and u ∈ ℕ, we let

F_{u}(S) = ∑_{ℓ≥0} \binom{ν_{1} + ⋯ + ν_{p−1} + ℓ}{u}S(L_{p−1}^{o}, ℓ),

where ν_{i} and L_{p−1} are as in Theorem 4.3. The main result of [10] is as follows.

4.4. Theorem. Let p ∈ ℕ^{+} and L be as above. The Hilbert function as well as the Hilbert polynomial of K[L]/I_{p+1}(L) is given by

F (V) = ∑_{u≥0} ∑_{S∈D_{p}(L)}(−1)^{u}F_{u}(S)\binom{V + pM − δ(S) − 1 − u}{pM − δ(S) − 1 − u}

where δ(S) is as in Theorem 4.3. In particular, I_{p+1}(L) is a Hilbertian ideal.

4.5. Remarks. 1. The first two theorems in this section may motivate the use of biladders although one may only be interested in (one-sided) ladders. Indeed, even if L were a ladder to begin with, the L^{∗} that one obtains in Theorem 4.1 is necessarily a biladder. Thus it makes sense to have the results of Section 3 in the general case of biladders.

2. The formulae in Theorem 4.3 and Theorem 4.4 are no doubt complicated and perhaps they may seem unworthy of being called ‘explicit’, in view of the rather unwieldy summation over D_{p}(L). Nevertheless, they can be used to deduce some interesting information about the variety associated to I_{p+1}(L). For example, it is shown in the next section that one can derive fairly simple estimates for the degree of the Hilbert polynomial. Also, as Krattenthaler and Prohaska [14, Sec. 7] seem to suggest, it appears unlikely that an elegant and simple formula for the Hilbert function of I_{p+1}(L) can be found.

5. Degree Computations

It is well-known that if H(V) is the Hilbert polynomial of a projective variety, and if

H(V) = \frac{e}{d!}V^{d} + c_{1}V^{d−1} + ⋯ + c_{d} with e, c_{1}, …, c_{d} ∈ ℤ and e ≠ 0,

then the degree d of H(V) is the dimension of that projective variety and the ‘normalized leading coefficient’ e is its order or the multiplicity. Note that the dimension d can also be read off from the Hilbert series; indeed, we have

∑_{V =0}^{∞} H(V)t^{V} = \frac{P(t)}{(1 − t)^{d+1}} , where P(t) ∈ ℤ[t] with P(1) ≠ 0.

Moreover, d + 1 is the (Krull) dimension of the corresponding homogeneous coordinate ring. With this in view, we now attempt to extract some information about the degree as well as the leading coefficient of the Hilbert polynomial F (V) of the ladder determinantal variety V_{p+1}(L). First, we need some notation.

Fix some p ∈ ℕ^{+} and a biladder L = L(S′, S). Let M, δ(S) and F (V) be as defined in Section 4. Further, we let

δ^{∗}(L) = min{δ(S) : S ∈ D_{p}(L)}

and

D_{p}^{∗}(L) = {S ∈ D_{p}(L) : δ(S) = δ^{∗}(L)}.

5.1. Proposition. The degree of the Hilbert polynomial F (V) of V_{p+1}(L) equals

pM − δ^{∗}(L) − 1 = p(m(1) + m(2) − 1) − δ^{∗}(L) − 1

and the normalized leading coefficient equals

∑_{S∈D_{p}^{∗}(L)} |skel(L_{p−1}^{o})| = ∑_{S∈D_{p}^{∗}(L)} ∑_{ℓ≥0} S(L_{p−1}^{o}, ℓ)

where for S = (S_{1}, …, S_{p−1}) ∈ D_{p}(L), by L_{p−1} we denote the biladder L(S_{p−1}, S).

Proof. From Theorem 4.4, we see that F (V) is a sum of terms of the form

(−1)^{u}F_{u}(S)\binom{V + pM − δ(S) − 1 − u}{pM − δ(S) − 1 − u}

where the coefficients (−1)^{u}F_{u}(S) are independent of V. Clearly, the binomial coefficient above is a polynomial in V of degree pM − δ(S) − 1 − u, and this degree is maximum when u = 0 and δ(S) = δ^{∗}(L). The corresponding leading coefficient

\frac{1}{(pM − δ^{∗}(L) − 1)!} ∑_{S∈D_{p}^{∗}(L)} F_{0}(S) = ∑_{S∈D_{p}^{∗}(L)} ∑_{ℓ≥0} S(L_{p−1}^{o}, ℓ)

is clearly positive since D_{p}(L) is nonempty.

It is not difficult to get a simple upper bound for the degree of F (V). To this end, we begin with some elementary observations concerning the boundary of a ladder.

5.2. Proposition. If S^{∗} is a LGB with len(S^{∗}) ≠ 0, then

|∂S^{∗}| = M = m(1) + m(2) − 1 (1)

and moreover, we can write ∂S^{∗} = {P_{1}, P_{2}, …, P_{M}} where

P_{1} = (1, m(2)), P_{M} = (m(1), 1); P_{j} − P_{j−1} = (1, 0) or (0, 1) for 1 < j ≤ M. (2)

Proof. From the definitions of a ladder and its interior, we easily see that

∂S^{∗} = L(S^{∗}) \ L(S^{∗})^{o} = {P_{1}} ∪ L_{1} ∪ L_{2}

where P_{1} = (S^{∗}(1, 0), S^{∗}(2, 0)) = (1, m(2)) and

L_{1} = ∐_{k=1}^{h}(S^{∗}(1, k − 1), S^{∗}(1, k)] × {S^{∗}(2, k − 1)}

and

L_{2} = ∐_{k=1}^{h}{S^{∗}(1, k)} × [S^{∗}(2, k), S^{∗}(2, k − 1)).

This readily implies (1) and (2).

5.3. Lemma. Given any S = (S_{1}, …, S_{p−1}) ∈ D_{p}(L), we have

δ_{i} = |∂S_{i} ∩ ∂S| ≥ min{δ_{0} + 2i, M}, for 1 ≤ i ≤ p − 1, (3)

and consequently, if t = max{i ∈ [1, p] : δ_{0} + 2(i − 1) ≤ M}, then

δ(S) = ∑_{i=0}^{p−1} δ_{i} ≥ tδ_{0} + t(t − 1) + (p − t)M. (4)

Further, if we assume that p < (M − δ_{0} + 1)/2, then we have

δ^{∗}(L) ≥ pδ_{0} + p(p − 1) and dim V_{p+1}(L) ≤ p(m(1) + m(2) − p − δ_{0}) − 1. (5)

Proof. We prove (3) by induction on i. The case of i = 0 being trivial, assume that i ≥ 1 and that δ_{i−1} ≥ min{δ_{0} + 2i − 2, M}. Now since ∂S_{i} ∩ ∂S_{i−1} = ∂S_{i−1} ∩ ∂S, we have ∂S_{i} ∩ ∂S ⊇ ∂S_{i−1} ∩ ∂S, and so δ_{i} ≥ δ_{i−1}. Thus in case δ_{i−1} ≥ M, we have δ_{i} ≥ M ≥ min{δ_{0} + 2i, M}. Suppose δ_{i−1} ≤ M − 1. Then, by (1), |∂S_{i} ∩∂S_{i−1}| ≤ |∂S_{i−1}|−1, and so we can write ∂S_{i} = {P_{1}, P_{2}, …, P_{M}} where P_{j}’s are as in (2), and further, we can find r ∈ [1, M] such that

P_{r} ∉ ∂S_{i−1} but P_{j} ∈ ∂S_{i−1} for 1 ≤ j < r. (6)

Also, let us write

∂S_{i−1} = {Q_{1}, Q_{2}, …, Q_{M}} and ∂S = {R_{1}, R_{2}, …, R_{M}}.

where Q_{j} and R_{j} satisfy the conditions in (2). Using (2) and (6), we see that P_{j} = Q_{j} for 1 ≤ j < r. Moreover, since ∂S_{i} ∩ ∂S_{i−1} = ∂S_{i−1} ∩ ∂S, we have P_{j} = R_{j} for 1 ≤ j < r. Now P_{r} ≠ Q_{r} and P_{r−1} = R_{r−1}, and thus in view of (2), we have that P_{r} ∉ ∂S_{i−1} and further, R_{r} = P_{r} or R_{r} = Q_{r}. But if R_{r} = Q_{r}, then Q_{r} ∈ ∂S_{i−1} ∩ ∂S ⊆ ∂S_{i}, and this forces that Q_{r} = P_{r}, which is a contradiction. Thus R_{r} = P_{r} and so

δ_{i} = |∂S_{i} ∩ ∂S| ≥ |∂S_{i−1} ∩ ∂S| + |{P_{r}}| = δ_{i−1} + 1.

In particular, if δ_{i−1} = M − 1, then δ_{i} ≥ M, and hence (3) holds. Now suppose δ_{i−1} < M − 1. Then we can also find some s ∈ [1, M] such that r < s < M and

P_{s} ∉ ∂S_{i−1} but P_{j} ∈ ∂S_{i−1} for s < j ≤ M.

Arguing as in the case of P_{r}, we obtain that P_{s} = R_{s}, and thus

δ_{i} = |∂S_{i} ∩ ∂S| ≥ |∂S_{i−1} ∩ ∂S| + |{P_{r}, P_{s}}| = δ_{i−1} + 2.

Hence, using the induction hypothesis, we obtain δ_{i} ≥ δ_{0} + 2i. This proves (3). Next, if t is as defined in the Lemma, then by (3), we have

δ(S) = ∑_{i=0}^{p−1} δ_{i} ≥ ∑_{i=0}^{t−1}(δ_{0} + 2i) + ∑_{i=t}^{p−1} M = tδ_{0} + t(t − 1) + (p − t)M.

Thus (4) is proved. To prove (5) consider the quadratic function

q(t) = tδ_{0} + t(t − 1) + (p − t)M.

Its derivative with respect to t equals 2(t − t_{0}), where t_{0} = (M − δ_{0} + 1)/2. Hence q(t) is strictly decreasing for t < t_{0}, and thus if p < t_{0}, then we have q(t) ≥ q(p) for all t ∈ [1, p]. This yields (5).

5.4. Remark. It may be noted that the condition p < (M − δ_{0} + 1)/2 in Lemma 5.3 is not restrictive. Indeed, the intersection ∂S′ ∩ ∂S of the boundaries of L(S) and L(S′) can be split into ‘horizontal overlaps’ and ‘vertical overlaps’ (see Fig. 2 (b)); if δ_{h} is the cardinality of the former and δ_{v} is the cardinality of the latter, then we have δ_{0} = |∂S′ ∩∂S| ≤ δ_{h} +δ_{v}. Moreover, in order that that a (p+1)×(p+1) minor of X has its entries in L = L(S) \ L(S′), the columns should avoid the horizontal overlaps and the rows should avoid the vertical overlaps. Thus p+1 ≤ m(2)−δ_{h} and p+1 ≤ m(1)−δ_{v}. This implies that 2p < m(1)+m(2)−(δ_{h}+δ_{v}) ≤ m(1)+m(2)−δ_{0}, and so p < (M − δ_{0} + 1)/2.

5.5. Corollary. Let L = L(S) be the ladder corresponding to S. Assume that p + 1 ≤ min{m(1), m(2)}. Then dim V_{p+1}(L) ≤ p(m(1) + m(2) − p) − 1.

Proof. Apply Lemma 5.3 with S′ as the empty LGB, and note that in this case δ_{0} = 0 and t = p.

5.6. Remark. The upper bound for dim V_{p+1}(L) in the Corollary above is, in fact, the dimension of the classical determinantal variety defined by the ideal I_{p+1}(X) generated by the (p + 1) × (p + 1) minors of X (see, e.g., [1, Thm. 20.15]). In this particular case of (one-sided) ladders, the same bound also follows from the expression for the Hilbert series in [14, Thm. 2]. However, this bound need not be attained, in general. For instance, in the Example considered in [14, p. 1022], the (projective) dimension is seen to be 103 whereas the value predicted by the above bound as well as by [14, Thm. 2] is 124.

Finally in this section, we illustrate how in some cases the actual value of dim V_{p+1}(L) can be determined.

5.7. Examples. 1. Consider the most trivial case when L = ∅. This corresponds to taking S′ = S. Hence in this case

S = (S_{1}, …, S_{p−1}) ∈ D_{p}(L) ⟺ S_{i} = S ∀i ⟹ ν_{i} = 0 ∀i and δ(S) = pM.

Further, L_{p−1} = ∅, and so S(L_{p−1}^{0}, ℓ) = 1 if ℓ = 0, and 0 otherwise. Hence,

F_{u}(S) = ∑_{ℓ≥0} \binom{l}{u}S(L_{p−1}^{0}, ℓ) = \{ \begin{matrix}1 & if u = 0 \\ 0 & if u > 0. \end{matrix}

Consequently, for any V ∈ ℕ, we have

F (V) = \binom{V + pM − δ(S) − 1}{pM − δ(S) − 1} = \binom{V − 1}{−1} = 0.

Of course, this is to be expected since I_{p+1}(L) is the zero ideal and K[L]/I_{p+1}(L) is the trivial ring.

2. Suppose L is the full rectangle [1, m(1)] × [1, m(2)], which corresponds to the case when S′ is the empty LGB and S is the unique LGB of length 1; also suppose p < min{m(1), m(2)}. More generally, let L be a ladder L(S) (so that S′ is the empty LGB) where S is a LGB of length h > 0 with S(1, 1) ≥ p and S(2, h−1) ≥ p. In this case, by (4), we have

δ^{∗}(L) = min{δ(S) : S ∈ D_{p}(L)} ≥ p(p − 1).

Moreover, if we take S = (S_{1}, …, S_{p−1}), where S_{i}’s are the ‘hook-like’ LGB’s determined by the conditions

len(S_{i}) = 2 and (S_{i}(1, 1), S_{i}(2, 1)) = (i, i), for 1 ≤ i ≤ p − 1,

then it is easy to see that for 1 ≤ i ≤ p − 1, we have

∂S_{i}∩∂S_{i−1} = {(1, m(2)), …, (i−1, m(2)), (m(1), 1), …, (m(1), i−1)} = ∂S_{i−1}∩∂S,

and therefore S ∈ D_{p}(L) and δ_{i} = |∂S_{i} ∩ ∂S| = 2i for 0 ≤ i ≤ p − 1; hence δ(S) = p(p − 1). It follows that δ^{∗}(L) = p(p − 1), and therefore,

dim V_{p+1}(L) = p(m(1) + m(2) − p) − 1.

3. Let L be a ladder with one corner missing, i.e, S′ is the empty LGB and L = L(S), where S is any LGB of length 2. Assume that p < min{m(1), m(2)}. Let (S(1, 1), S(2, 1)) = (a, b). Then we have

(7) δ^{∗}(L) = \{ \begin{matrix}p(p − 1) & if a ≥ p and b ≥ p \\ p(p − 1) + (p − a)(m(2) − b) & if a < p and b ≥ p \\ p(p − 1) + (p − b)(m(1) − a) & if a ≥ p and b < p \\ m(1)(p − b) + m(2)(p − a) + ab − p & if a < p and b < p \end{matrix}

and consequently,

(8) dim V_{p+1}(L) = \{ \begin{matrix}pM − 1 & if a ≥ p and b ≥ p \\ pM − 1 + (a − p)(m(2) − b) & if a < p and b ≥ p \\ pM − 1 + (b − p)(m(1) − a) & if a ≥ p and b < p \\ bm(1) + am(2) − ab & if a < p and b < p. \end{matrix}

To see this, note that when a ≥ p and b ≥ p, the result follows from the preceding example. In the remaining cases, we can argue as in Lemma 5.3 and the preceding example. Thus, for instance, if a < p and b ≥ p, then for any S ∈ D_{p}(L), we must have

δ_{i} ≥ 2i for 0 ≤ i < a and δ_{i} ≥ 2i + (m(2) − b) for a ≤ i ≤ p − 1.

Moreover, there exists a configuration S ∈ D_{p}(L) for which the above inequalities are equalities. Alternatively, we can directly prove (8) in the last three cases from the following simple observations. If a < p and b ≥ p, then V_{p+1}(L) is a cylinder over the determinantal variety V_{p+1}(Y), where Y is the m(1) × b rectangular submatrix of X obtained by taking the first b columns of X. The case when a ≥ p and b < p is similar. Lastly, when a < p and b < p, we have I_{p+1}(L) = (0).

5.8. Remarks. 1. The observation about the dimension of dim V_{p+1}(L) in the second example above may perhaps explain why in Kulkarni’s formula for the Hilbert function of V_{2}(L), the degree is the same as that in the case of V_{2}(X). Indeed, when p = 1, the conditions S(1, 1) ≥ p and S(2, h − 1) ≥ p always hold.

2. In the case L is the full rectangle [1, m(1)] × [1, m(2)], Proposition 5.1 gives a curious formula for the multiplicity of the classical determinantal ideal I_{p+1}(X). It may be interesting to compare this with the more elegant formulae described in [12, p. 17].

References

[1] S. S. Abhyankar, “Enumerative Combinatorics of Young Tableaux”, Marcel Dekker, New York, 1988.

[2] S. S. Abhyankar and D. M. Kulkarni, On hilbertian ideals, Linear Algebra Appl., 116 (1989), 53–79

[3] H. H. Andersen, Schubert varieties and Demazure’s character formula, Invent. Math., 79 (1985), 611–618.

[4] A. Conca and J. Herzog, On the Hilbert function of determinantal rings and their canonical module, Proc. Amer. Math. Soc., 122 (1994), 677–681.

[5] A. Conca, Ladder determinantal rings, J. Pure Appl. Algebra, 98 (1995), 119–134.

[6] A. Conca, Gorenstein ladder determinantal rings, J. London Math. Soc., 54 (1996), 453– 474.

[7] A. Conca and J. Herzog, Ladder determinantal rings have rational singularities, Adv. Math., 132 (1997), 120–147.

[8] S. R. Ghorpade, Abhyankar’s work on Young tableaux and some recent developments, in “Algebraic Geometry and its Applications” (C. Bajaj, Ed.), pp. 235–265, Springer-Verlag, New York, 1994.

[9] S. R. Ghorpade, Young bitableaux, lattice paths and Hilbert functions, J. Statist. Plann. Inference, 54 (1996), 55–66.

[10] S. R. Ghorpade, Hilbert functions of ladder determinantal varieties, forthcoming.

[11] D. Glassbrenner and K. E. Smith, Singularities of certain ladder determinantal varieties, J. Pure Appl. Algebra, 101 (1995), 59–75.

[12] J. Herzog and N. V. Trung, Gröbner bases and multiplicity of determinantal and pfaffian ideals, Adv. Math., 96 (1992), 1–37.

[13] W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Camb. Phil. Soc., 39 (1943), 22–30.

[14] C. Krattenthaler and M. Prohaska, A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns, Trans. Amer. Math. Soc., 351 (1999), 1015–1042.

[15] D. M. Kulkarni, Thesis, Purdue Univ., West Lafayette, USA, 1985.

[16] D. M. Kulkarni, Hilbert polynomial of a certain ladder determinantal ideal, J. Algebraic Combin., 2 (1993), 57–71.

[17] V. B. Mehta and V. Srinivas, Normality of Schubert varieties, Amer. J. Math., 109 (1987), 987–989.

[18] V. B. Mehta and V. Srinivas, A note on Schubert varieties in G/B, Math. Ann., 284 (1989), 1–5.

[19] J. Motwani and M. Sohoni, Normality of ladder determinantal rings, J. Algebra, 186 (1996), 323–337.

[20] J. Motwani and M. Sohoni, Divisor class group of ladder determinantal varieties, J. Algebra, 186 (1996), 338–367.

[21] S. B. Mulay, Determinantal loci and the flag variety, Adv. Math., 74 (1989), 1–30.

[22] C. Musili, Applications of standard monomial theory, in “Proceedings of the Hyderabad Conference on Algebraic Groups”, (S. Ramanan, Ed.), pp. 381–406, Manoj Prakashan, Madras (Distributed outside India by the American Math. Society, Providence), 1991.

[23] H. Narasimhan, Irreducibility of ladder determinantal varieties, J. Algebra, 102 (1986), 162–185.

[24] S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math., 79 (1985), 217–224.

[25] A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math., 80 (1985), 283–294.

[26] C. S. Seshadri, Line bundles on Schubert varieties, in “Vector Bundles on Algebraic Varieties” (Papers presented at the Bombay Colloquium, 1984), pp. 495–528, Oxford Univ. Press, Bombay, 1987.

Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 400076 - INDIA

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