QUANTUM DETERMINANTAL IDEALS

K. R. Goodearl and T. H. Lenagan

Abstract. Quantum determinantal ideals in coordinate algebras of quantum matrices are investigated. The ideal I_{t} generated by all (t + 1) × (t + 1) quantum minors in O_{q}(M_{m,n}(k)) is shown to be a completely prime ideal, that is, O_{q}(M_{m,n}(k))/I_{t} is an integral domain. The corresponding result is then obtained for the multiparameter quantum matrix algebra O_{λ,p}(M_{m,n}(k)). The main idea involved in the proof is the construction of a preferred basis for O_{q}(M_{n}(k)) in terms of certain products of quantum minors, together with rewriting rules for expressing elements of this algebra in terms of the preferred basis.

Introduction

Fix a base field k. The quantized coordinate ring of n × n matrices over k, denoted O_{q}(M_{n}(k)), is a deformation of the classical coordinate ring of n × n matrices, O(M_{n}(k)). As such it is a k-algebra generated by n^{2} indeterminates X_{ij}, for 1 ≤ i, j ≤ n, subject to relations which we recall in (1.1). Here q is a nonzero element of the field k. When q = 1, we recover O(M_{n}(k)), which is the commutative polynomial algebra k[X_{ij}]. The algebra O_{q}(M_{n}(k)) has a distinguished element D_{q}, the quantum determinant, which is a central element. Two important algebras O_{q}(GL_{n}(k)) and O_{q}(SL_{n}(k)) are formed by inverting D_{q} and setting D_{q} = 1, respectively.

The structures of the primitive and prime ideal spectra of the algebras O_{q}(GL_{n}(k)) and O_{q}(SL_{n}(k)) have been investigated recently; see, for example, [2], [6] and [9]. Results obtained in these investigations can be pulled back to partial results about the primitive and prime ideal spectra of O_{q}(M_{n}(k)). However, these techniques give no information about the closed subset of the spectrum determined by D_{q}. In this paper, we begin the study of this portion of the spectrum.

In the classical commutative setting, much attention has been paid to determinantal ideals, that is, the ideals generated by the minors of a given size. In particular, these are special prime ideals of O(M_{n}(k)) containing the determinant. Moreover, there are interesting geometrical and invariant theoretical reasons for the importance of these ideals; see, for example, [4]. In order to put our results into context, it may be useful to review some highlights of the commutative theory.

Let M_{l,m}(k) denote the algebraic variety of l×m matrices over k. For t ≤ n, the general linear group GL_{t}(k) acts on M_{n,t}(k) × M_{t,n}(k) via

g · (A, B) := (Ag^{−1}, gB).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16A40, 16W30, 16D30, 16S15, 13C40, 17B37.

This research was partially supported by National Science Foundation research grant DMS-9622876 and NATO Collaborative Research Grant 960250.

Matrix multiplication yields a map

μ : M_{n,t}(k) × M_{t,n}(k) → M_{n}(k),

the image of which is the set of matrices with rank at most t.

There is an induced map

μ_{∗} : O(M_{n}(k)) → O(M_{n,t}(k) × M_{t,n}(k)) = O(M_{n,t}(k)) ⊗ O(M_{t,n}(k)).

The First Fundamental Theorem of invariant theory identifies the fixed ring of the coordinate ring O(M_{n,t}(k) × M_{t,n}(k)) under the induced action of GL_{t}(k) as precisely the image of μ_{∗}. The Second Fundamental Theorem states that the kernel of μ_{∗} is I_{t}, the ideal generated by the (t + 1) × (t + 1) minors of O(M_{n}(k)), so that the coordinate ring of the variety of n × n matrices of rank at most t is O(M_{n}(k))/I_{t}. As a consequence, since this variety is irreducible, the ideal I_{t} is a prime ideal of O(M_{n}(k)).

Our main result, Theorem 2.5, is a quantum analog of the Second Fundamental Theorem. (We conjecture that there is also a quantum analog of the First Fundamental Theorem, but do not address that problem in the present paper.) If I, J are subsets of {1, …, n} with |I| = |J|, then D(I, J) denotes the quantum minor obtained by evaluating the quantum determinant of the subalgebra of O_{q}(M_{n}(k)) generated by those X_{ij} with i ∈ I and j ∈ J. The ideal I_{t} is then the ideal generated by the (t + 1) × (t + 1) quantum minors of O_{q}(M_{n}(k)). Theorem 2.5 states that O_{q}(M_{n}(k))/I_{t} is an integral domain, for 0 ≤ t ≤ n − 1. The case (t = n − 1) of this result has been proved by Jordan [8] and Levasseur-Stafford [10, p. 182], and the case (t = 1) has recently been obtained by Rigal [14]. The case (t = 0) holds trivially.

The classical commutative appproach to the Second Fundamental Theorem is as follows. By geometrical considerations, the variety of n × n matrices of rank at most t is an irreducible variety, and it is easy to see that the coordinate ring of this variety is O(M_{n}(k))/\sqrt{I_{t}}. Thus the main problem is to show that I_{t} is a radical ideal of O(M_{n}(k)). This is achieved via the notion of algebras with straightening laws; see [3] or [4]. In order to simplify the problem, the algebra O(M_{n}(k)) is replaced temporarily by O(M_{n,2n}(k)). The subalgebra B of O(M_{n,2n}(k)) generated by the maximal minors of O(M_{n,2n}(k)) is the coordinate ring of the Grassmanian of the n-dimensional subspaces of k^{2n}. The products of maximal minors span B, but do not form a basis – the famous Plücker relations generate the relations between the maximal minors. The Plücker relations are used to produce straightening laws leading to a standard basis of B. All this is now specialised by setting the rightmost n × n block of X_{ij}’s equal to the identity matrix. The images of the maximal minors become all of the minors of O(M_{n}(k)), and the standard basis of B induces a standard basis of O(M_{n}(k)) consisting of certain products of minors of O(M_{n}(k)). This establishes that O(M_{n}(k)) is an algebra with a straightening law. The conclusion that I_{t} is radical then follows easily.

The classical approach breaks down completely in the quantum setting. There is no group acting, and setting noncentral elements equal to 0 or 1 produces a homomorphic image that is far too small. The action of the group GL_{t}(k) can be replaced by a coaction of the Hopf algebra O_{q}(GL_{t}(k)). Otherwise, the only thing that survives is the idea of a basis constructed of products of (quantum) minors and straightening laws. However, an added complication appears: as well as straightening laws to deal with linear dependencies, it is also necessary to generate commutation laws to deal with re-ordering products. This latter problem leads us to choose a different ordering of variables than that chosen in the classical case, so that we can give preference to the best approximation to central elements – the normal elements that occur in profusion in the quantum case. As a result, we call our basis a preferred basis rather than a standard basis.

The second main ingredient in the proof is the exploitation of the fact that O_{q}(M_{n}(k)) is a bialgebra. Quantum minors behave well under the comultiplication map Δ, and using this fact we produce an embedding of O_{q}(M_{n}(k))/I_{t} into the algebra O_{q}(M_{n,t}(k)) ⊗ O_{q}(M_{t,n}(k)). This latter algebra is an iterated Ore extension of k and is thus a domain, establishing our theorem.

In the latter part of the paper, we show, using the twisting methods of Artin-Schelter-Tate [1], that our results also hold for multiparameter coordinate rings of quantum matrices.

I. A basis for quantum matrices

This section is devoted to establishing the existence of a basis for O_{q}(M_{n}(k)) which is built from products of quantum minors. This basis is crucial to our calculations with quantum determinantal ideals. A basis of this type was constructed in [7] for a class of quantum matrix superalgebras which includes the O_{q}(M_{n}(k)) for q not a root of unity. Our modification of their construction allows q to be an arbitrary nonzero scalar. For convenience of notation and in applying results from the literature, we work mainly with the quantum coordinate rings of square matrices. At the end of the section, we shall see that our basis theorem readily carries over to the case of O_{q}(M_{m,n}(k)).

The calculations involved in constructing and verifying our basis rely on several general identities concerning products of quantum minors. Although some of these identities are of standard types, they are not available in the literature in precisely the forms we require, and so we derive them from known forms. In order not to disrupt the line of this section, we relegate the discussions of the identities to appendices in sections V and VI.

1.1. Throughout this section, we fix an integer n ≥ 2, a base field k, and a nonzero scalar q ∈ k^{×}. No other restrictions are assumed; in particular, k need not be algebraically closed, and q is allowed to be a root of unity. We work with the one-parameter quantized coordinate ring of n × n matrices over k, namely the algebra A = O_{q}(M_{n}(k)) with generators X_{ij} for i, j = 1, …, n and relations

X_{ij}X_{lj} = qX_{lj}X_{ij}_{X_{ij}X_{im} = qX_{im}X_{ij}}_{X_{im}X_{lj} = X_{lj}X_{im}}_{X_{ij}X_{lm} − X_{lm}X_{ij} = (q − q^{−1})X_{im}X_{lj}} when i < l;

when j < m;
when i < l and j < m;
when i < l and j < m.

As is well known, this algebra is in fact a bialgebra, with comultiplication Δ : A → A ⊗ A and counit ϵ : A → k such that

Δ(X_{ij}) = ∑_{l=1}^{n} X_{il} ⊗ X_{lj}  and  ϵ(X_{ij}) = δ_{ij}

for all i, j.

1.2. We shall need several partial order relations on index sets. Let A, B ⊆ {1, …, n}, not necessarily of the same cardinality. First, we define a “row ordering”, denoted ≤_{r}. To describe this, write A and B in descending order:

A = {a_{1} > a_{2} > ⋯ > a_{α}}  and  B = {b_{1} > b_{2} > ⋯ > b_{β}}.

Define A ≤_{r} B to mean that α ≥ β and a_{i} ≥ b_{i} for i = 1, …, β. For the “column ordering”, ≤_{c}, write A and B in ascending order:

A = {a_{1} < a_{2} < ⋯ < a_{α}}  and  B = {b_{1} < b_{2} < ⋯ < b_{β}}.

Define A ≤_{c} B to mean that α ≥ β and a_{i} ≤ b_{i} for i = 1, …, β.

By an index pair we will mean a pair (I, J) where I, J ⊆ {1, …, n} and |I| = |J|. Order index pairs by (≤_{r}, ≤_{c}), that is, define (I, J) ≤ (I′, J′) if and only if I ≤_{r} I′ and J ≤_{c} J′. For example, when n = 3 the poset of index pairs can be drawn as in Diagram 1.2, where we have abbreviated the descriptions of index sets by eliminating braces and commas.

[figure: (∅,∅); (1,3); (1,2); (2,3); (1,1); (2,2); (3,3); (2,1); (21,23); (3,2); (21,13); (3,1); (31,23); (21,12); (31,13); (32,23); (31,12); (32,13); (32,12); (321,123)] Diagram 1.2

1.3. The basis we construct will be indexed by certain bitableaux (pairs of tableaux) with specifications as below. Recall that, in general, a tableau consists of a Young diagram with entries in each box. We consider only tableaux with entries from {1, …, n} and no repetitions in any row. Allowable bitableaux are pairs (T, T′) where

(a) T and T′ have the same shape;

(b) T has strictly decreasing rows;

(c) T′ has strictly increasing rows. Rows of T or T′ can be identified with subsets of {1, …, n} listed in descending or ascending order. Hence, allowable bitableaux can be labelled in the form

(••)  ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{l} & J_{l} \end{matrix} )

where (I_{1}, J_{1}), …, (I_{l}, J_{l}) are index pairs such that |I_{1}| ≥ |I_{2}| ≥ ⋯ ≥ |I_{l}|. The pair (I_{1}, J_{1}) will be called the top row of (T, T′).

We say that a bitableau (T, T′) is preferred if it is allowable, the columns of T are nonincreasing, and the columns of T′ are nondecreasing. In the format (••) above, (T, T′) is preferred if and only if (I_{1}, J_{1}) ≤ (I_{2}, J_{2}) ≤ ⋯ ≤ (I_{l}, J_{l}).

For induction purposes, we shall also need an ordering on bitableaux. Suppose that

(S, S′) = ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{s} & J_{s} \end{matrix} )  and  (T, T′) = ( \begin{matrix}K_{1} & L_{1} \\ K_{2} & L_{2} \\ ⋮ & ⋮ \\ K_{t} & L_{t} \end{matrix} )

are bitableaux presented in the format (••). Define (S, S′) ≺ (T, T′) if and only if either the shape of S is larger than the shape of T relative to the lexicographic ordering on shapes, that is,

(|I_{1}|, …, |I_{s}|) >_{lex} (|K_{1}|, …, |K_{t}|);

or else the shapes of S and T coincide and

((I_{1}, J_{1}), …, (I_{s}, J_{s})) <_{lex} ((K_{1}, L_{1}), …, (K_{t}, L_{t})),

that is, there is an index l such that (I_{j}, J_{j}) = (K_{j}, L_{j}) for j < l and (I_{l}, J_{l}) < (K_{l}, L_{l}) (relative to the ordering defined in (1.2)).

Note that if (I, J) and (K, L) are index pairs, then (I, J) < (K, L) implies (I J) ≺ (K L) but not vice versa (unless |I| = |K|).

1.4. We adopt the notation of [7], using [•|•] in place of (•|•) to help distinguish quantum minors and products of these from the index pairs and bitableaux labelling them.

For any index pair (I, J), there is a quantum minor D(I, J) ∈ A defined in terms of the X_{ij} for i ∈ I and j ∈ J. One can obtain D(I, J) as the image of the quantum determinant in O_{q}(M_{|I|}(k)) under the natural isomorphism of this algebra with k⟨X_{ij} | i ∈ I, j ∈ J⟩ (cf.

[13, (4.3)]). By convention, D(∅, ∅) = 1. In this section and the next, we shall abbreviate D(I, J) by the symbol [I|J].

For any (allowable) bitableau (T, T′), write

(T, T′) = ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{l} & J_{l} \end{matrix} )

in the format (••) of (1.3). We define [T |T′] = [I_{1}|J_{1}][I_{2}|J_{2}] ⋯ [I_{l}|J_{l}].

1.5. The content of a tableau T is the multiset 1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}} where m_{i} is the number of times i appears in T. The bicontent of a bitableau (T, T′) is the pair of multisets (content(T), content(T′)).

There is a natural ℤ^{n} × ℤ^{n} bigrading on A, under which each X_{ij} has bidegree (ϵ_{i}, ϵ_{j}) where ϵ_{1}, …, ϵ_{n} is the standard basis for ℤ^{n}. Observe that any quantum minor [I|J] is homogeneous of bidegree (χ_{I}, χ_{J}) where χ_{X} stands for the characteristic function of a subset X of {1, …, n}. More generally, if a bitableau (T, T′) has bicontent

(1^{r_{1}}2^{r_{2}} ⋯ n^{r_{n}}, 1^{c_{1}}2^{c_{2}} ⋯ n^{c_{n}}),

then [T |T′] is homogeneous of bidegree (r_{1}, …, r_{n}, c_{1}, …, c_{n}).

1.6. Lemma. Let (R_{1}, C_{1}), …, (R_{l}, C_{l}) be index pairs such that |R_{1}| ≥ |R_{2}| ≥ ⋯ ≥ |R_{j}| but |R_{j}| < |R_{j+1}| for some j. Let g < j be the largest index such that |R_{g}| ≥ |R_{j+1}|, or g = 0 if |R_{1}| < |R_{j+1}|. Then the product

P := [R_{1}|C_{1}][R_{2}|C_{2}] ⋯ [R_{l}|C_{l}]

can be expressed as a linear combination of products [T |T′] of the same bidegree as P, where each (T, T′) is a bitableau of the form

( \begin{matrix}R_{1} & C_{1} \\ ⋮ & ⋮ \\ R_{g} & C_{g} \\ K_{1} & K′_{1} \\ K_{2} & K′_{2} \\ ⋮ & ⋮ \end{matrix} )

with |K_{1}| = |R_{j+1}| and (K_{1}, K′_{1}) ≤ (R_{j+1}, C_{j+1}).

Proof. By induction on j.

In view of Proposition 5.3, [R_{j}|C_{j}][R_{j+1}|C_{j+1}] can be written as a linear combination of products

[K_{1}|K′_{1}][K_{2}|K′_{2}] ⋯ [K_{s}|K′_{s}]

of the same bidegree as [R_{j}|C_{j}][R_{j+1}|C_{j+1}], such that |K_{1}| = |R_{j+1}| and (K_{1}, K′_{1}) ≤ (R_{j+1}, C_{j+1}). (Here we write any X_{ij} occurring in a monomial M′ as a 1 × 1 quantum minor [i|j].) Substituting this linear combination into P, we obtain an expression for P as a linear combination of products

[R_{1}|C_{1}] ⋯ [R_{j−1}|C_{j−1}][K_{1}|K′_{1}][K_{2}|K′_{2}] ⋯ [K_{s}|K′_{s}][R_{j+2}|C_{j+2}] ⋯ [R_{l}|C_{l}]

with the same bidegree as P. After expanding each of [R_{j+2}|C_{j+2}], …, [R_{l}|C_{l}] as a linear combination of monomials, we can express P as a linear combination of products

(†)  [R_{1}|C_{1}] ⋯ [R_{j−1}|C_{j−1}][K_{1}|K′_{1}][K_{2}|K′_{2}] ⋯ [K_{t}|K′_{t}]

of the same bidegree as P, such that |K_{1}| = |R_{j+1}| and (K_{1}, K′_{1}) ≤ (R_{j+1}, C_{j+1}), while |K_{i}| = 1 for i > 1.

If either j = 1 or |R_{j−1}| ≥ |K_{1}|, these products can be written in the form [T |T′] for bitableaux (T, T′) of the desired type, and we are done. If j > 1 and |R_{j−1}| < |K_{1}|, the induction hypothesis applies to each of the products (†); collecting terms, we are again done. □

1.7. Recall the (non-total) ordering ≺ on bitableau defined in (1.3).

Lemma. Let (S, S′) be a bitableau with bicontent γ and top row (I_{1}, J_{1}), and suppose that (S, S′) is not preferred.

(a) (S, S′) is not minimal with respect to ≺ among bitableau with bicontent γ.

(b) [S|S′] can be expressed as a linear combination of products [T |T′] where each (T, T′) is a bitableau with bicontent γ such that (T, T′) ≺ (S, S′). Further, each (T, T′) can be chosen with a top row (X_{1}, Y_{1}) such that either |X_{1}| > |I_{1}| or (X_{1}, Y_{1}) ≤ (I_{1}, J_{1}).

Although part (a) can be obtained as a consequence of part (b), we find it clearer to give an explicit proof of (a).

Proof. Let δ denote the bidegree of [S|S′].

Since (S, S′) is not preferred, it must have at least two rows. Write

(S, S′) = ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{l} & J_{l} \end{matrix} )

in the format (••) of (1.3); then either I_{j} ≰_{r} I_{j+1} or J_{j} ≰_{c} J_{j+1} for some j.

\underline{Case I}: Suppose that I_{j} ≰_{r} I_{j+1} for some j. We may assume that j is minimal with respect to this property, so that I_{1} ≤_{r} I_{2} ≤_{r} ⋯ ≤_{r} I_{j}. Write

I_{j} = {a_{1} > a_{2} > ⋯ > a_{α}}
I_{j+1} = {b_{1} > b_{2} > ⋯ > b_{β}};

then α ≥ β (by the shape of S) but a_{i} < b_{i} for some i ≤ β. We may assume that i is minimal, so that a_{1} ≥ b_{1}, …, a_{i−1} ≥ b_{i−1}. Set

A_{1} = {a_{1} > a_{2} > ⋯ > a_{i−1}}
A_{2} = {b_{i+1} > ⋯ > b_{β}}
K = {b_{1} > ⋯ > b_{i} > a_{i} > ⋯ > a_{α}}.

(a) Since {b_{1}, …, b_{i}} has one more element than A_{1}, there must be an index p ≤ i such that b_{p} ∉ A_{1}. In addition, b_{p} ≥ b_{i} > a_{i} > ⋯ > a_{α}, and so b_{p} ∉ I_{j}. Similarly, there is an index q ≥ i such that a_{q} ∉ I_{j+1}, and b_{p} ≥ b_{i} > a_{i} ≥ a_{q}. Now set

I′_{j} = I_{j} ∪ {b_{p}} \ {a_{q}}
I′_{j+1} = I_{j+1} ∪ {a_{q}} \ {b_{p}}

and observe that I′_{j} and I′_{j+1} have the same cardinalities as I_{j} and I_{j+1}, respectively. Further, I′_{j} ∪ I′_{j+1} = I_{j} ∪ I_{j+1}, and I′_{j} <_{r} I_{j} because b_{p} > a_{q}. Set

(R, R′) = ( \begin{matrix}I_{1} & J_{1} \\ ⋮ & ⋮ \\ I_{j−1} & J_{j−1} \\ I′_{j} & J_{j} \\ I′_{j+1} & J_{j+1} \\ I_{j+2} & J_{j+2} \\ ⋮ & ⋮ \\ I_{l} & J_{l} \end{matrix} )

and note that (R, R′) is a bitableau with the same shape and bicontent as (S, S′). Since I′_{j} <_{r} I_{j}, we also have (I′_{j}, J_{j}) < (I_{j}, J_{j}), and therefore (R, R′) ≺ (S, S′).

(b) The exchange formula (6.2)(b) gives us a relation of the form

(†) ∑_{K=K′⊔K′′} ±q^{•}[A_{1} ⊔ K′|J_{j}][K′′ ⊔ A_{2}|J_{j+1}]

= ∑_{J_{ν}=J′_{ν}⊔J′′_{ν}} ±q^{•}[A_{1}|J′_{j}][K|J′′_{j} ⊔ J′′_{j+1}][A_{2}|J′_{j+1}]

with all terms of the same bidegree. Note that [I_{j}|J_{j}][I_{j+1}|J_{j+1}] occurs on the left hand side of (†) when K′ = {a_{i} > ⋯ > a_{α}} and K′′ = {b_{1} > ⋯ > b_{i}}. In any other term on the left, K′ contains at least one of b_{1}, …, b_{i}, from which we see that A_{1} ⊔ K′ <_{r} I_{j}.

We now have a relation of the form

(*) ∑_{K=K′⊔K′′} ±q^{•}[I_{1}|J_{1}] ⋯ [I_{j−1}|J_{j−1}][A_{1} ⊔ K′|J_{j}][K′′ ⊔ A_{2}|J_{j+1}][I_{j+2}|J_{j+2}] ⋯ [I_{l}|J_{l}]

= ∑_{\begin{matrix}J_{ν}=J′_{ν}⊔J′′_{ν} \\ (ν=j,j+1) \end{matrix}} ±q^{•}[I_{1}|J_{1}] ⋯ [I_{j−1}|J_{j−1}][A_{1}|J′_{j}][K|J′′_{j} ⊔ J′′_{j+1}][A_{2}|J′_{j+1}][I_{j+2}|J_{j+2}] ⋯ [I_{l}|J_{l}]

with all terms of bidegree δ. On the left hand side, a term ±q^{•}[S|S′] occurs once, and all other terms are of the form ±q^{•}[T |S′] with (T, S′) ≺ (S, S′). Moreover, if j > 1 the top row of (T, S′) equals (I_{1}, J_{1}), while if j = 1, the top row of (T, S′) equals (A_{1} ⊔ K′, J_{1}), and in this case (A_{1} ⊔ K′, J_{1}) < (I_{1}, J_{1}) because A_{1} ⊔ K′ <_{r} I_{1}. Hence, to prove part (b) we just need to express the right hand side of (*) in the desired form.

Note that |A_{1}| = i − 1 < α while |K| = α + 1. Let g < j be the largest index such that |I_{g}| ≥ α + 1, or g = 0 if |I_{1}| ≤ α. Applying Lemma 1.6 to each term, we can express the right-hand side of (*) as a linear combination of products [T |T′] of bidegree δ where each (T, T′) is a bitableau of the form

( \begin{matrix}I_{1} & J_{1} \\ ⋮ & ⋮ \\ I_{g} & J_{g} \\ K_{1} & K′_{1} \\ K_{2} & K′_{2} \\ ⋮ & ⋮ \end{matrix} )

with |K_{1}| = α + 1. Since |I_{g+1}| ≤ α, the shape of T is larger than the shape of S, and so (T, T′) ≺ (S, S′). If g ≥ 1, the top row of (T, T′) equals (I_{1}, J_{1}), while if g = 0, the top row of (T, T′) equals (K_{1}, K′_{1}), and in this case |K_{1}| = α + 1 > α ≥ |I_{1}|. This establishes part (b) in Case I.

\underline{Case II}: Suppose that J_{j} ≰_{c} J_{j+1} for some j. This case can be handled in the same manner as Case I, by using (6.2)(a) rather than (6.2)(b). □

1.8. Corollary. Let (S, S′) be a bitableau with bicontent γ and top row (I_{1}, J_{1}). Then [S|S′] can be expressed as a linear combination of products [T |T′] where

(a) (T, T′) is a preferred bitableau with bicontent γ.

(b) (T, T′) has a top row (X_{1}, Y_{1}) such that either |X_{1}| > |I_{1}| or (X_{1}, Y_{1}) ≤ (I_{1}, J_{1}).

Proof. This follows from Lemma 1.7 by induction with respect to ≺. □

Part (b) of this corollary is a weak form of the straightening law for a classical standard basis, and is useful in computing preferred bases for certain ideals of O_{q}(M_{n}(k)).

1.9. Theorem. Let δ = (r_{1}, …, r_{n}, c_{1}, …, c_{n}) ∈ (ℤ^{+})^{n}×(ℤ^{+})^{n}, let V be the homogeneous component of A with bidegree δ, and set γ = (1^{r_{1}}2^{r_{2}} ⋯ n^{r_{n}}, 1^{c_{1}}2^{c_{2}} ⋯ n^{c_{n}}). The products [T |T′], as (T, T′) runs over all preferred bitableaux with bicontent γ, form a basis for V.

Proof. Observe that [S|S′] ∈ V for all bitableau (S, S′) with bicontent γ, and that there are only finitely many such bitableaux. Further, such products [S|S′] include all monomials X_{i_{1}j_{1}}X_{i_{2}j_{2}} ⋯ X_{i_{r}j_{r}} with bidegree δ, and these monomials span V. Hence, Corollary 1.8 implies that V is spanned by the products [T |T′] as (T, T′) runs over all preferred bitableaux with bicontent γ. It remains to show that these products are linearly independent. To see this, it suffices to prove that the number of preferred bitableaux with bicontent γ is equal to the dimension of V.

We may write A as an iterated Ore extension with the variables X_{ij} in the order

X_{nn}, X_{n,n−1}, …, X_{n1}, X_{n−1,n}, X_{n−1,n−1}, …, X_{n−1,1}, …, X_{1n}, X_{1,n−1}, …, X_{11}.

Hence, A has a basis consisting of monomials X_{i_{1}j_{1}}X_{i_{2}j_{2}} ⋯ X_{i_{r}j_{r}} satisfying the conditions (a) i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{r};

(b) j_{l} ≥ j_{l+1} whenever i_{l} = i_{l+1}.

Since X_{im}X_{ij} = q^{−1}X_{ij}X_{im} when m > j, we can reverse any product of generators with the same row index, at the cost of a nonzero scalar coefficient. Hence, A has a basis B consisting of monomials X_{i_{1}j_{1}}X_{i_{2}j_{2}} ⋯ X_{i_{r}j_{r}} such that

(a) i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{r};
(b’) j_{l} ≤ j_{l+1} whenever i_{l} = i_{l+1}.

Note that under conditions (a),(b’), the list i_{1}j_{1}, …, i_{r}j_{r} of double indices is in lexicographic order, \underline{provided} we write our row alphabet in reverse order, i.e., n, n − 1, …, 1, while keeping our column alphabet 1, 2, …, n in the usual order. With this convention, the monomials in B are in bijection with those two-rowed matrices

( \begin{matrix}i_{1} & i_{2} & ⋯ & i_{r} \\ j_{1} & j_{2} & ⋯ & j_{r} \end{matrix} )

having entries from {1, …, n} and columns in lexicographic order. Note that the monomial X_{i_{1}j_{1}}X_{i_{2}j_{2}} ⋯ X_{i_{r}j_{r}} has bidegree δ iff the pair of multisets ({i_{1}, …, i_{r}}, {j_{1}, …, j_{r}}) coincides with γ.

By the Robinson-Schensted-Knuth Theorem [5, p. 40], the two-rowed matrices corresponding to monomials from B with bidegree δ are in bijection with standard bitableaux (Q, P) of bicontent γ. In this result, standard tableaux are required to be nondecreasing on each row and strictly increasing on each column, relative to the total orders given on the two alphabets. Note this means that relative to the usual ordering of integers, Q has nonincreasing rows and strictly decreasing columns. Thus the ordering conditions on (Q, P) hold precisely when the pair (Q^{tr}, P^{tr}) is preferred in our sense (1.3).

Therefore there exists a bijection between the monomials of bidegree δ in B and the preferred bitableaux with bicontent γ. Since the former make a basis for V, we conclude that the number of preferred bitableaux with bicontent γ is precisely dim_{k} V, as required. □

1.10. Corollary. The products [T |T′], as (T, T′) runs over all preferred bitableaux, form a basis for A = O_{q}(M_{n}(k)). □

1.11. The existence of analogous bases for rectangular quantum matrix algebras follows easily from Corollary 1.10. For m < n, we may define O_{q}(M_{m,n}(k)) as the k-subalgebra of O_{q}(M_{n}(k)) generated by the X_{ij} with i ≤ m; the case (m > n) is handled by writing O_{q}(M_{m,n}(k)) as a subalgebra of O_{q}(M_{m}(k)). Note that in the first case, there is a kalgebra retraction π : O_{q}(M_{n}(k)) → O_{q}(M_{m,n}(k)) such that π(X_{ij}) = X_{ij} when i ≤ m and π(X_{ij}) = 0 when i > m.

Corollary. Let m, n be any positive integers, and let B_{m,n} be the set of all products [T |T′] where (T, T′) runs over all preferred bitableaux in which the entries of T lie in {1, …, m} while the entries of T′ lie in {1, …, n}. Then B_{m,n} is a basis for O_{q}(M_{m,n}(k)).

Proof. We prove only the case (m < n); the other case is identical. By Corollary 1.10, the set B_{n,n} is a basis for O_{q}(M_{n}(k)). On one hand, B_{m,n} ⊆ B_{n,n} and so B_{m,n} is linearly independent. On the other hand, π(B_{n,n}) = B_{m,n} ∪ {0}, and therefore B_{m,n} spans O_{q}(M_{m,n}(k)). □

II. One-parameter quantum determinantal ideals

In this section, we prove that quantum determinantal ideals in O_{q}(M_{m,n}(k)) are completely prime. The case of the ideal generated by all 2×2 quantum minors has been proved by Rigal [14], using different methods.

2.1. As in the previous section, we fix n ≥ 2, a field k, a scalar q ∈ k^{×}, and set A = O_{q}(M_{n}(k)). Fix t ∈ {1, …, n−1}, and let I_{t} = I_{q}^{[t]}(M_{n}(k)) denote the ideal of A generated by all (t + 1) × (t + 1) quantum minors. Again, it is convenient to remain with this case until the main result is proved, and to derive the corresponding result for O_{q}(M_{m,n}(k)) as an easy corollary. We proceed by establishing a quantized version of the theorem stating that, in the classical case, I_{t} equals the kernel of the k-algebra homomorphism

μ_{∗} : O(M_{n}(k)) → O(M_{n,t}(k) × M_{t,n}(k)) = O(M_{n,t}(k)) ⊗ O(M_{t,n}(k))

discussed in the introduction.

First, some labels: set

A_{nt} = O_{q}(M_{n,t}(k)) = k⟨X_{ij} | j ≤ t⟩ ⊆ A
A_{tn} = O_{q}(M_{t,n}(k)) = k⟨X_{ij} | i ≤ t⟩ ⊆ A.

For τ = nt or tn, let π_{τ} : A → A_{τ} denote the natural k-algebra retraction. Thus

π_{nt}(X_{ij}) = \{ \begin{matrix}X_{ij} & (j ≤ t) \\ 0 & (j > t); \end{matrix} π_{tn}(X_{ij}) = \{ \begin{matrix}X_{ij} & (i ≤ t) \\ 0 & (i > t). \end{matrix}

The kernels of these homomorphisms are the ideals ⟨X_{ij} | j > t⟩ and ⟨X_{ij} | i > t⟩, respectively. Finally, define the k-algebra homomorphism

θ_{t} : A ⟶^{Δ} A ⊗ A ⟶^{π_{nt}⊗π_{tn}} A_{nt} ⊗ A_{tn},

where Δ denotes the comultiplication on the bialgebra A.

By [12, (1.9)], comultiplication of quantum minors is given by the rule

Δ[I|J] = ∑_{|K|=|I|}[I|K] ⊗ [K|J].

Since all (t + 1) × (t + 1) quantum minors are killed by π_{nt}, we see that I_{t} ⊆ ker θ_{t}. We shall prove equality in Proposition 2.4.

Note that any product [T |T′] for which the shape of T has more than t columns lies in I_{t}. Hence, A/I_{t} is spanned by the images of those products [T |T′] indexed by preferred bitableaux (T, T′) with shapes having at most t columns.

2.2. Consider an allowable bitableau (T, T′). For l = 1, …, n, let ρ_{l}(T) be the number of rows of T of length ≥ l, and set \overline{ρ}(T) = (ρ_{1}(T), ρ_{2}(T), …, ρ_{n}(T)). Let μ(T) and μ′(T) denote the tableaux with the same shape as T and entries as follows: each row of length l is filled 1, 2, …, l in μ(T) and is filled l, l − 1, …, 1 in μ′(T). If (T, T′) is preferred, then (T, μ(T)) and (μ′(T), T′) are preferred bitableaux.

For any homogeneous element x ∈ A, label the bidegree of x as

(\overline{r}(x), \overline{c}(x)) = (r_{1}(x), r_{2}(x), …, r_{n}(x), c_{1}(x), c_{2}(x), …, c_{n}(x)).

Thus with respect to the usual PBW basis of ordered monomials, r_{l}(x) records the number of X_{l?} factors in each monomial in x, and c_{l}(x) the number of X_{?l} factors. For [T |T′] as in the previous paragraph, r_{l}[T |T′] is the number of l’s in T and c_{l}[T |T′] is the number of l’s in T′. Note that c_{l}[T |μ(T)] = r_{l}[μ′(T)|T′] = ρ_{l}(T).

We shall write <_{rlex} for the reverse lexicographic order on n-tuples of integers.

2.3. Lemma. Let (T, T′) be an allowable bitableau with a shape having at most t columns. Then

θ_{t}[T |T′] = [T |μ(T)] ⊗ [μ′(T)|T′] + ∑_{i} X_{i} ⊗ Y_{i}

where the X_{i} and Y_{i} are homogeneous with \overline{c}(X_{i}) = \overline{r}(Y_{i}) >_{rlex} \overline{ρ}(T).

Proof. Write

(T, T′) = ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{s} & J_{s} \end{matrix} )

where the (I_{j}, J_{j}) are index pairs. Then

θ_{t}[I_{j}|J_{j}] = ∑_{\begin{matrix}|K|=|I_{j}| \\ K⊆\{1,...,t\} \end{matrix}}[I_{j}|K] ⊗ [K|J_{j}]

for each j. Hence, θ_{t}[T |T′] is the sum of all possible terms

X_{i} ⊗ Y_{i} = [I_{1}|K_{1}][I_{2}|K_{2}] ⋯ [I_{s}|K_{s}] ⊗ [K_{1}|J_{1}][K_{2}|J_{2}] ⋯ [K_{s}|J_{s}]

where each K_{j} ⊆ {1, …, t} and |K_{j}| = |I_{j}|. Obviously X_{i} and Y_{i} are homogeneous.

Let i = i_{0} label the special case where K_{j} = {1, 2, …, |I_{j}|} for all j. This yields the term X_{i_{0}} ⊗ Y_{i_{0}} = [T |μ(T)] ⊗ [μ′(T)|T′]. Now assume that i ≠ i_{0}. Obviously \overline{c}(X_{i}) = \overline{r}(Y_{i}), so it remains to show that \overline{c}(X_{i}) >_{rlex} \overline{ρ}(T). Note that c_{l}(X_{i}) = ρ_{l}(T) = 0 for l > t.

We claim that if c_{l}(X_{i}) = ρ_{l}(T) for l = n, n − 1, …, h + 1, then K_{j} = {1, 2, …, |I_{j}|} for all j such that |I_{j}| ≥ h. This is vacuously true for h > t. Now suppose that h ≤ t and that K_{j} = {1, 2, …, |I_{j}|} whenever |I_{j}| > h. For l > h, there are ρ_{l}(T) indices j for which |I_{j}| ≥ l, and l ∈ K_{j} for each such j. Since c_{l}(X_{i}) = ρ_{l}(T), this uses up all the available column l’s in X_{i}, and so l ∉ K_{j} for any j with |I_{j}| < l. Thus K_{j} ⊆ {1, 2, …, h} for all j with |I_{j}| ≤ h. In particular K_{j} = {1, 2, …, h} for all j with |I_{j}| = h, verifying the induction step. This establishes the claim.

Since we are in the case i ≠ i_{0}, we cannot have K_{j} = {1, 2, …, |I_{j}|} for all j, and so the claim shows that we cannot have c_{l}(X_{i}) = ρ_{l}(T) for all l. Hence, there is an index g ≥ 1 such that c_{g}(X_{i}) ≠ ρ_{g}(T) while c_{l}(X_{i}) = ρ_{l}(T) for all l > g. By the claim, K_{j} = {1, 2, …, |I_{j}|} for all j such that |I_{j}| ≥ g. Hence, g ∈ K_{j} for all j with |I_{j}| ≥ g, and so c_{g}(X_{i}) ≥ ρ_{g}(T). By our choice of g, we thus must have c_{g}(X_{i}) > ρ_{g}(T). Therefore \overline{c}(X_{i}) >_{rlex} \overline{ρ}(T), as required. □

2.4. Proposition. I_{t} = ker θ_{t}.

Proof. If ker θ_{t} properly contains I_{t}, then ker θ_{t} contains a nonzero element of the form

x = ∑_{i=1}^{m} α_{i}[T_{i}|T′_{i}]

where the α_{i} are nonzero scalars and the (T_{i}, T′_{i}) are distinct preferred bitableaux with shapes having at most t columns. Let \overline{ρ} be the minimum of the n-tuples \overline{ρ}(T_{i}) under reverse lexicographic order. Without loss of generality, there exists m′ such that \overline{ρ}(T_{i}) = \overline{ρ} for i ≤ m′ and \overline{ρ}(T_{i}) >_{rlex} \overline{ρ} for i > m′.

Applying Lemma 2.3 to each θ_{t}[T_{i}|T′_{i}] and collecting terms, we see that

0 = θ_{t}(x) = ∑_{i=1}^{m′} α_{i}[T_{i}|μ(T_{i})] ⊗ [μ′(T_{i})|T′_{i}] + ∑_{j} X_{j} ⊗ Y_{j}

where the X_{j} and Y_{j} are homogeneous with \overline{c}(X_{j}) = \overline{r}(Y_{j}) >_{rlex} \overline{ρ}. Since \overline{c}[T_{i}|μ(T_{i})] = \overline{ρ} for i ≤ m′, all of the X_{j} belong to different homogeneous components than the [T_{i}|μ(T_{i})] for i ≤ m′. Consequently,

∑_{i=1}^{m′} α_{i}[T_{i}|μ(T_{i})] ⊗ [μ′(T_{i})|T′_{i}] = 0.

For 1 ≤ i < j ≤ m′, either T_{i} ≠ T_{j} or T′_{i} ≠ T′_{j}, so (T_{i}, μ(T_{i})) ≠ (T_{j}, μ(T_{j})) or (μ′(T_{i}), T′_{i}) ≠ (μ′(T_{j}), T′_{j}). It thus follows from the linear independence of the preferred products [•|•] in A_{nt} and A_{tn} (Corollary 1.11) that the terms [T_{i}|μ(T_{i})] ⊗ [μ′(T_{i})|T′_{i}] are linearly independent. But then α_{i} = 0 for i = 1, …, m′, contradicting our assumptions. □

2.5. Theorem. A/I_{t} = O_{q}(M_{n}(k))/I_{q}^{[t]}(M_{n}(k)) is an integral domain.

Proof. By Proposition 2.4, A/I_{t} embeds in A_{nt} ⊗ A_{tn}. Now A_{nt} and A_{tn} are iterated Ore extensions of k, with respect to k-algebra automorphisms and k-linear skew derivations. In particular, both of these algebras are domains. Further, A_{nt} ⊗ A_{tn} is an iterated Ore extension of A_{nt}, and so it too is a domain. Therefore A/I_{t} is a domain. □

2.6. Corollary. Let m, n, t be any positive integers such that t < min{m, n}, and let I_{q}^{[t]}(M_{m,n}(k)) be the ideal of O_{q}(M_{m,n}(k)) generated by all (t + 1) × (t + 1) quantum minors. Then O_{q}(M_{m,n}(k))/I_{q}^{[t]}(M_{m,n}(k)) is an integral domain.

Proof. Consider the case (m < n), and put I = I_{q}^{[t]}(M_{m,n}(k)) and J = I_{q}^{[t]}(M_{n}(k)). Obviously I ⊆ J ∩ O_{q}(M_{m,n}(k)). For the reverse inclusion, we use the retraction π : O_{q}(M_{n}(k)) → O_{q}(M_{m,n}(k)) discussed in (1.11). Note that the image of any quantum minor [X|Y] under π is either [X|Y] or 0, and hence π(J) ⊆ I. Since π is the identity on O_{q}(M_{m,n}(k)), it follows that I = J ∩ O_{q}(M_{m,n}(k)). Therefore the corollary follows from Theorem 2.5. □

III. Twisting

Artin, Schelter, and Tate showed in [1] that multiparameter quantum matrix algebras O_{λ,p}(M_{n}(k)) can be obtained from the one-parameter versions by a process of twisting by 2-cocycles. In this section, we recall some details of this process and determine its effect on quantum minors.

3.1. Let k be a field, and let p = (p_{ij}) be a multiplicatively antisymmetric matrix over k^{×}, that is, p_{ii} = 1 and p_{ji} = p_{ij}^{−1} for all i, j. Let λ ∈ k^{×} \{−1}. The algebra O_{λ,p}(M_{n}(k)) is the k-algebra with generators Y_{ij} for i, j = 1, …, n and relations

Y_{lm}Y_{ij} = \{ \begin{matrix}p_{li}p_{jm}Y_{ij}Y_{lm} + (λ − 1)p_{li}Y_{im}Y_{lj} & when l > i and m > j \\ λp_{li}p_{jm}Y_{ij}Y_{lm} & when l > i and m ≤ j \\ p_{jm}Y_{ij}Y_{lm} & when l = i and m > j. \end{matrix}

We shall denote this algebra A_{λ,p} for short.

3.2. The quantum exterior algebra Λ_{p} = Λ_{p}(k^{n}) is the k algebra with generators η_{1}, …, η_{n} and relations

η_{i}^{2} = 0, η_{i}η_{j} = −p_{ij}η_{j}η_{i}

for all i, j. For any subset I ⊆ {1, …, n}, write the elements of I in ascending order, say I = {i_{1} < i_{2} < ⋯ < i_{r}}, and set η_{I} = η_{i_{1}}η_{i_{2}} ⋯ η_{i_{r}}. By convention, η_{∅} = 1. The elements η_{I} form a k-basis for Λ_{p}.

As is well known (and easily checked), there is a k-algebra homomorphism

L_{λ,p} : Λ_{p} ⟶ A_{λ,p} ⊗ Λ_{p}

such that L_{λ,p}(η_{i}) = ∑_{j} Y_{ij} ⊗ η_{j} for all i (cf. [11, Chapter 6, Theorem 3], [1, (11), (12)]). For any nonempty subset I ⊆ {1, …, n}, the image of η_{I} under L_{λ,p} has the form

L_{λ,p}(η_{I}) = ∑_{|J|=|I|} U_{IJ} ⊗ η_{J}

[1, Lemma 1]. The elements U_{IJ} ∈ A_{λ,p} are unique due to the linear independence of the η_{J}. Each U_{IJ} is the quantum minor corresponding to the rows i ∈ I and columns j ∈ J; we shall use the notation

D_{λ,p}(I, J) = U_{IJ}

to indicate the dependence on the parameters λ, p_{ij}. Explicit formulas for D_{λ,p}(I, J) are given in [1, Lemma 1].

3.3. The quantum minors in A_{q} = O_{q}(M_{n}(k)) can be obtained as in (3.2), of course. Since we must consider both settings simultaneously, let us use ξ_{i} for the generators of the quantum exterior algebra in this case. Thus, Λ_{q} = Λ_{q}(k^{n}) is the k-algebra with generators ξ_{1}, …, ξ_{n} and relations

ξ_{i}^{2} = 0 (all i); ξ_{j}ξ_{i} = −qξ_{i}ξ_{j} (i < j).

There is a basis consisting of elements ξ_{I} = ξ_{i_{1}}ξ_{i_{2}} ⋯ ξ_{i_{r}} where I = {i_{1} < i_{2} < ⋯ < i_{r}} runs through all subsets of {1, …, n}.

There is a k-algebra homomorphism L_{q} : Λ_{q} → A_{q} ⊗ Λ_{q} such that L_{q}(ξ_{i}) = ∑_{j} X_{ij} ⊗ ξ_{j} for all i. The quantum minors in A_{q}, which we shall now denote D_{q}(I, J) to indicate the dependence on q, arise in the formulas

L_{q}(ξ_{I}) = ∑_{|J|=|I|} D_{q}(I, J) ⊗ ξ_{J}

(cf. [13, Lemma 4.4.2]; [12, Remark, p. 36]).

3.4. As observed in [1, p. 889], the algebra A_{λ,p} can be obtained as a cocycle twist of A_{q} provided λ = q^{−2} (we take q^{−2} rather than q^{2} to account for the difference q ↔ q^{−1} between [1, (43)] and our choice of relations for A_{q}). We thus carry out the twisting process under the assumption that λ has a square root in k; the general cases of our results require a passage to \overline{k}. Since we must simultaneously work with twists of A_{q}, Λ_{q}, and a subalgebra of A_{q} ⊗ Λ_{q}, it is helpful to give the appropriate cocycle explicitly.

For the remainder of this section, p = (p_{ij}) will be an arbitrary multiplicatively antisymmetric matrix over k^{×}, and we will take λ = q^{−2} for some (fixed) q ∈ k^{×}. Define a map c : ℤ^{n} × ℤ^{n} → k^{×} by the rule

c((a_{1}, …, a_{n}), (b_{1}, …, b_{n})) = ∏_{i>j}(qp_{ji})^{a_{i}b_{j}}.

Then c is a multiplicative bicharacter on ℤ^{n} (that is, c(a + a′, b) = c(a, b)c(a′, b), and similarly in the second variable), and hence also a 2-cocycle. Note that

c(ϵ_{i}, ϵ_{j}) = \{ \begin{matrix}qp_{ji} & (i > j) \\ 1 & (i ≤ j) \end{matrix}

where ϵ_{1}, …, ϵ_{n} denotes the standard basis for ℤ^{n}.

3.5. Recall the ℤ^{n} × ℤ^{n}-bigrading on A_{q} from (1.5). Following [1, Theorem 4], we simultaneously twist A_{q} on the left by c^{−1} and on the right by c. This results in a new algebra, denoted A′_{q}, as follows. As a graded vector space, A_{q} is isomorphic to A′_{q} via an isomorphism a ↦ a′. The multiplication in A′_{q} is given by

a′b′ = c(u_{1}, v_{1})^{−1}c(u_{2}, v_{2})(ab)′

for homogeneous elements a, b ∈ A_{q} of bidegrees (u_{1}, u_{2}) and (v_{1}, v_{2}). In particular,

X′_{ij}X′_{lm} = \{ \begin{matrix}p_{il}p_{mj}(X_{ij}X_{lm})′ & (i > l, j > m) \\ q^{−1}p_{il}(X_{ij}X_{lm})′ & (i > l, j ≤ m) \\ qp_{mj}(X_{ij}X_{lm})′ & (i ≤ l, j > m) \\ (X_{ij}X_{lm})′ & (i ≤ l, j ≤ m). \end{matrix}

Observe that Λ_{q} has a natural ℤ^{n}-grading, where ξ_{i} has degree ϵ_{i}. We twist Λ_{q} by c^{−1} to obtain a new algebra Λ′_{q}. Note that

ξ′_{i}ξ′_{j} = \{ \begin{matrix}q^{−1}p_{ij}(ξ_{i}ξ_{j})′ & (i > j) \\ (ξ_{i}ξ_{j})′ & (i ≤ j). \end{matrix}

3.6. Lemma. There are k-algebra isomorphisms

ϕ : A_{λ,p} ⟶ A′_{q}  and  ψ : Λ_{p} ⟶ Λ′_{q}

such that ϕ(Y_{ij}) = X′_{ij} and ψ(η_{i}) = ξ′_{i} for all i, j.

Proof. The existence of ϕ follows from [1, Theorem 4], and the existence of ψ is proved in the same manner. One first checks that the elements X′_{ij} ∈ A′_{q} and ξ′_{i} ∈ Λ′_{q} satisfy the same relations as the Y_{ij} and the η_{i}. For instance, for i < j we have ξ_{j}ξ_{i} = −qξ_{i}ξ_{j} and so

ξ′_{j}ξ′_{i} = q^{−1}p_{ji}(ξ_{j}ξ_{i})′ = −p_{ji}(ξ_{i}ξ_{j})′ = −p_{ji}ξ′_{i}ξ′_{j}.

Hence, there exist k-algebra homomorphisms ϕ and ψ sending Y_{ij} ↦ X′_{ij} and η_{i} ↦ ξ′_{i}. Since A_{q} has a basis of ordered monomials X_{i_{1}j_{1}} ⋯ X_{i_{t}j_{t}}, and since each (X_{i_{1}j_{1}} ⋯ X_{i_{t}j_{t}})′ is a nonzero scalar multiple of X′_{i_{1}j_{1}} ⋯ X′_{i_{t}j_{t}}, we see that A′_{q} has a basis of ordered monomials X′_{i_{1}j_{1}} ⋯ X′_{i_{t}j_{t}}. In addition, A_{λ,p} has a basis of ordered monomials Y_{i_{1}j_{1}} ⋯ Y_{i_{t}j_{t}}, which ϕ maps to the X′_{i_{1}j_{1}} ⋯ X′_{i_{t}j_{t}}. Therefore ϕ is an isomorphism, and similarly so is ψ. □

3.7. There is a ℤ^{n}-graded subalgebra B_{q} ⊆ A_{q} ⊗ Λ_{q} where

(B_{q})_{u} = ⨁_{v∈ℤ^{n}}(A_{q})_{uv} ⊗ (Λ_{q})_{v}

for u ∈ ℤ^{n}. Using this grading, we twist B_{q} by c^{−1} to obtain a new algebra B′_{q}. Note that there is a vector space embedding B′_{q} → A′_{q} ⊗ Λ′_{q} where (a ⊗ b)′ ↦ a′ ⊗ b′ for a ∈ (A_{q})_{uv} and b ∈ (Λ_{q})_{v}. We shall identify B′_{q} with its image in A′_{q} ⊗ Λ′_{q} via this embedding.

Lemma. Under the above identification, B′_{q} is a k-subalgebra of A′_{q} ⊗ Λ′_{q}.

Proof. It suffices to show that the product of any two homogeneous elements from B′_{q} is the same in both algebras. Given x ∈ (B_{q})_{u_{1}} and y ∈ (B_{q})_{u_{2}}, we can write x = ∑_{i} x_{i} and y = ∑_{j} y_{j} where each x_{i} = a_{i} ⊗ b_{i} ∈ (A_{q})_{u_{1}v_{i}} ⊗ (Λ_{q})_{v_{i}} for some v_{i} ∈ ℤ^{n} and y_{j} = d_{j} ⊗ e_{j} ∈ (A_{q})_{u_{2}w_{j}} ⊗ (Λ_{q})_{w_{j}} for some w_{j} ∈ ℤ^{n}. It is enough to compare the products of any x_{i} with any y_{j}. Hence, there is no loss of generality in assuming that x = a ⊗ b ∈ (A_{q})_{u_{1}v_{1}} ⊗ (Λ_{q})_{v_{1}} and y = d ⊗ e ∈ (A_{q})_{u_{2}v_{2}} ⊗ (Λ_{q})_{v_{2}}.

Under the product in B′_{q}, we have x′y′ = c(u_{1}, u_{2})^{−1}(xy)′. On the other hand, under the product in A′_{q} ⊗ Λ′_{q}, we have

x′y′ = (a′ ⊗ b′)(d′ ⊗ e′) = a′d′ ⊗ b′e′
= [c(u_{1}, u_{2})^{−1}c(v_{1}, v_{2})(ad)′] ⊗ [c(v_{1}, v_{2})^{−1}(be)′]
= c(u_{1}, u_{2})^{−1}(ad)′ ⊗ (be)′ = c(u_{1}, u_{2})^{−1}(xy)′.

Therefore the two products do coincide, as required. □

3.8. Observe that the k-algebra homomorphism L_{q} : Λ_{q} → A_{q} ⊗ Λ_{q} actually maps Λ_{q} to B_{q}. Viewed as a map from Λ_{q} to B_{q}, the homomorphism L_{q} is homogeneous of degree 0 with respect to the ℤ^{n}-gradings on these algebras. Since we have twisted both algebras by the same cocycle, namely c^{−1}, we see that L_{q} induces a k-algebra homomorphism L′_{q} : Λ′_{q} → B′_{q}, where L′_{q}(a′) = L_{q}(a)′ for a ∈ Λ_{q}.

The various k-algebra homomorphisms we have been discussing fit into the following diagram:

[figure: Λ_{p}; L_{λ,p}; ψ; ≅; A_{λ,p} ⊗ Λ_{p}; ≅; ϕ⊗ψ; Λ′_{q}; L′_{q}; B′_{q}; ⊆; A′_{q} ⊗ Λ′_{q}]

This diagram commutes, since

(ϕ ⊗ ψ)L_{λ,p}(η_{i}) = ∑_{j} ϕ(Y_{ij}) ⊗ ψ(η_{j}) = ∑_{j} X′_{ij} ⊗ ξ′_{j}
L′_{q}ψ(η_{i}) = L_{q}(ξ_{i})′ = ∑_{j} X′_{ij} ⊗ ξ′_{j}

for all i.

3.9. Proposition. ϕ(D_{λ,p}(I, J)) = D_{q}(I, J)′ for all I, J.

Proof. We first show that ψ(η_{H}) = ξ′_{H} for all H ⊆ {1, …, n}. This is clear for |H| ≤ 1. If H = {h_{1} < h_{2} < ⋯ < h_{r}} = {h_{1}} ⊔ J for some r ≥ 2, we may assume by induction that ψ(η_{J}) = ξ′_{J}. Hence,

ψ(η_{H}) = ψ(η_{h_{1}}η_{J}) = ξ′_{h_{1}}ξ′_{J} = c(ϵ_{h_{1}}, ϵ_{h_{2}} + ⋯ + ϵ_{h_{r}})^{−1}(ξ_{h_{1}}ξ_{J})′.

But c(ϵ_{h_{1}}, ϵ_{h_{2}} +⋯+ϵ_{h_{r}}) = c(ϵ_{h_{1}}, ϵ_{h_{2}})c(ϵ_{h_{1}}, ϵ_{h_{3}}) ⋯ c(ϵ_{h_{1}}, ϵ_{h_{r}}) = 1 because h_{1} < h_{2} < ⋯ < h_{r}, and so ψ(η_{H}) = (ξ_{h_{1}}ξ_{J})′ = ξ′_{H}. This establishes the induction step for our claim.

Now let I be an arbitrary nonempty subset of {1, …, n}. In view of the commutativity of the diagram in (3.8),

∑_{|J|=|I|} ϕ(D_{λ,p}(I, J)) ⊗ ξ′_{J} = (ϕ ⊗ ψ)L_{λ,p}(η_{I}) = L′_{q}ψ(η_{I})

= L_{q}(ξ_{I})′ = ∑_{|J|=|I|} D_{q}(I, J)′ ⊗ ξ′_{J}.

Since the ξ′_{J} are linearly independent, the proposition follows. □

IV. Multiparameter quantum determinantal ideals

Using the twisting method discussed in the previous section, we extend our main result from quantum determinantal ideals in one-parameter quantum matrix algebras O_{q}(M_{m,n}(k)) to those in multiparameter quantum matrix algebras O_{λ,p}(M_{m,n}(k)).

4.1. Let A_{λ,p} = O_{λ,p}(M_{n}(k)) be an arbitrary multiparameter quantum matrix algebra over an arbitrary base field k, as in (3.1). Fix t ∈ {1, …, n−1}, and let I_{λ,p} = I_{λ,p}^{[t]}(M_{n}(k)) denote the ideal of A_{λ,p} generated by all (t+1)×(t+1) quantum minors, i.e., all D_{λ,p}(I, J) with |I| = |J| = t + 1.

Theorem. O_{λ,p}(M_{n}(k))/I_{λ,p}^{[t]}(M_{n}(k)) is an integral domain.

Proof. First set \overline{A}_{λ,p} = O_{λ,p}(M_{n}(\overline{k})) and \overline{I}_{λ,p} = I_{λ,p}^{[t]}(M_{n}(\overline{k})). We identify \overline{A}_{λ,p} with A_{λ,p} ⊗ \overline{k}. Since the quantum minors in A_{λ,p} and \overline{A}_{λ,p} are the same, \overline{I}_{λ,p} = I_{λ,p} ⊗ \overline{k}. As a result, \overline{A}_{λ,p}/\overline{I}_{λ,p} ≅ (A_{λ,p}/I_{λ,p}) ⊗ \overline{k}; in particular, A_{λ,p}/I_{λ,p} embeds in \overline{A}_{λ,p}/\overline{I}_{λ,p}. Thus it suffices to show that latter algebra is a domain, and hence we may pass to the case where k is algebraically closed.

Now there exists q ∈ k^{×} such that q^{−2} = λ. Let c be the 2-cocycle defined in (3.4), set A_{q} = O_{q}(M_{n}(k)), and twist A_{q} on the left by c^{−1} and on the right by c as in (3.5). In view of Lemma 3.6 and Proposition 3.9, there is a k-algebra isomorphism ϕ : A_{λ,p} → A′_{q} such that ϕ(I_{λ,p}) = I′_{q}, where I_{q} = I_{q}^{[t]}(M_{n}(k)). Thus A_{λ,p}/I_{λ,p} ≅ (A_{q}/I_{q})′, a twist of A_{q}/I_{q}. Since A_{q}/I_{q} is a domain by Theorem 2.5, it only remains to check that the property of being a domain is preserved in the twist (A_{q}/I_{q})′.

We may view A_{q}/I_{q} as graded by ℤ^{2n}, which can be made into a totally ordered group; then (A_{q}/I_{q})′ is graded by the same totally ordered group. To see that the product of any two nonero elements of (A_{q}/I_{q})′ is nonzero, it suffices to show that the product of their highest terms is nonzero. Hence, we just need to show that the product of any two nonzero homogeneous elements a′, b′ ∈ (A_{q}/I_{q})′ is nonzero. But that is clear since a′b′ is a nonzero scalar multiple of (ab)′, while ab ≠ 0 because A_{q}/I_{q} is a domain. Therefore (A_{q}/I_{q})′ is a domain, as required. □

4.2. Just as in Corollary 2.6, the rectangular case follows directly from Theorem 4.1:

Corollary. Let m, n, t be positive integers with t < min{m, n}, and let I_{λ,p}^{[t]}(M_{m,n}(k)) be the ideal of O_{λ,p}(M_{m,n}(k)) generated by all (t + 1) × (t + 1) quantum minors. Then O_{λ,p}(M_{m,n}(k))/I_{λ,p}^{[t]}(M_{m,n}(k)) is an integral domain. □

4.3. The method of proof used above can also be applied to the other results of Sections I and II. In particular, we obtain a basis of products of quantum minors for A_{λ,p} as follows.

Define preferred bitableaux as in (1.3). For any preferred bitableau

(T, T′) = ( \begin{matrix}I_{1} & J_{1} \\ I_{2} & J_{2} \\ ⋮ & ⋮ \\ I_{l} & J_{l} \end{matrix} ) ,

where (I_{1}, J_{1}) ≤ (I_{2}, J_{2}) ≤ ⋯ ≤ (I_{l}, J_{l}) are index pairs, define

[T |T′]_{λ,p} = D_{λ,p}(I_{1}, J_{1})D_{λ,p}(I_{2}, J_{2}) ⋯ D_{λ,p}(I_{l}, J_{l}).

Theorem. The products [T |T′]_{λ,p}, as (T, T′) runs over all preferred bitableaux, form a basis for O_{λ,p}(M_{n}(k)).

Proof. First note that the symbols [T |T′]_{λ,p} stand for the same elements in the algebras A_{λ,p} and \overline{A}_{λ,p} = O_{λ,p}(M_{n}(\overline{k})) = A_{λ,p} ⊗ \overline{k}. If these elements form a \overline{k}-basis for \overline{A}_{λ,p}, then they must also form a k-basis for A_{λ,p}. Hence, there is no loss of generality in assuming that k is algebraically closed.

Now choose q ∈ k^{×} such that q^{−2} = λ, and twist A_{q} as in (3.5). In view of Lemma 3.6 and Proposition 3.9, there is a k-algebra isomorphism ϕ : A_{λ,p} → A′_{q} such that ϕ([T |T′]_{λ,p}) is a nonzero scalar multiple of [T |T′]′ for all preferred bitableaux (T, T′). Since the products [T |T′]′ form a basis for A′_{q} by Corollary 1.10, the theorem follows. □

V. Appendix: Commutation Relations

We derive some commutation relations for quantum minors in O_{q}(M_{n}(k)), expressed using the notation and conventions of (1.1)–(1.4).

5.1. We begin by restating some identities from [13], given there for generators and maximal minors, in a form that applies to minors of arbitrary size. Note the difference between our choice of relations for O_{q}(M_{n}(k)) (see (1.1)) and that in [13, p. 37]. Because of this, we must interchange q and q^{−1} whenever carrying over a formula from [13].

Lemma. Let r, c ∈ {1, …, n} and I, J ⊆ {1, …, n} with |I| = |J| ≥ 1.

(a) If r ∈ I and c ∈ J, then X_{rc}[I|J] = [I|J]X_{rc}.

(b) If r ∈ I and c ∉ J, then

X_{rc}[I|J] − q^{−1}[I|J]X_{rc} = (q^{−1} − q) ∑_{\begin{matrix}j∈J \\ j>c \end{matrix}}(−q)^{−|J∩[c,j]|}[I|J ∪ \{c\} \ \{j\}]X_{rj}.

(c) If r ∉ I and c ∈ J, then

X_{rc}[I|J] − q[I|J]X_{rc} = (q − q^{−1}) ∑_{\begin{matrix}i∈I \\ i<r \end{matrix}}(−q)^{|I∩[i,r]|}[I ∪ \{r\} \ \{i\}|J]X_{ic}.

(d) If r ∉ I and c ∉ J, then

X_{rc}[I|J] − [I|J]X_{rc} = (1 − q^{−2})(∑_{\begin{matrix}i∈I \\ i<r \end{matrix}}(−q)^{|I∩[i,r]|}[I ∪ \{r\} \ \{i\}|J]X_{ic}

− ∑_{\begin{matrix}j∈J \\ j>c \end{matrix}}(−q)^{|J∩[c,j]|}X_{rj}[I|J ∪ \{c\} \ \{j\}]).

Proof. Let t = |I| = |J|.

(a) There is a k-algebra isomorphism

O_{q}(M_{t}(k)) ⟶^{≅} k⟨X_{ij} | i ∈ I, j ∈ J⟩ ⊆ O_{q}(M_{n}(k))

which sends the quantum determinant of O_{q}(M_{t}(k)) to [I|J]. Since the quantum determinant is central in O_{q}(M_{t}(k)), part (a) follows.

(b) Pick r_{0} ∈ \{1, …, n\} \ I. Set \overline{I} = I ∪ \{r_{0}\} and \overline{J} = J ∪ \{c\}, and label the elements of these sets in ascending order, say

\overline{I} = \{i_{1} < i_{2} < ⋯ < i_{t+1}\}  and  \overline{J} = \{j_{1} < j_{2} < ⋯ < j_{t+1}\}.

There exists a k-algebra embedding ϕ : O_{q}(M_{t+1}(k)) → O_{q}(M_{n}(k)) such that ϕ(X_{ab}) = X_{i_{a}j_{b}} for a, b = 1, …, t + 1. Let ρ, γ, σ be the indices such that i_{ρ} = r, j_{γ} = c, and i_{σ} = r_{0}. Then ϕ(X_{ργ}) = X_{rc} and ϕ(A(σ γ)) = [I|J], where A(σ γ) is (in the notation of [13]) the t × t quantum minor in O_{q}(M_{t+1}(k)) obtained by deleting the σ-th row and γ-th column.

By the second part of [13, 4.5.1(2)],

(†)  X_{ργ}A(σ γ) − q^{−1}A(σ γ)X_{ργ} = (q^{−1} − q) ∑_{δ>γ}(−q)^{γ−δ}A(σ δ)X_{ρδ}.

Note that δ − γ = |\overline{J} ∩ (c, j_{δ}]| = |J ∩ [c, j_{δ}]| for δ = γ + 1, …, t + 1. Thus, part (b) results from applying ϕ to (†).

(c)(d) These follow, in the same manner, from the first part of [13, 4.5.1(4)] and the first part of [13, 5.1.2], respectively. □

5.2. Corollary. Let r, c ∈ {1, …, n} and I, J ⊆ {1, …, n} with |I| = |J|. Then the term

Y := X_{rc}[I|J] − q^{δ(c,J)−δ(r,I)}[I|J]X_{rc}

is a linear combination of terms [I′|J′]X_{ij}, with the same bidegree as X_{rc}[I|J], such that |I′| = |I| and (I′, J′) < (I, J).

Proof. We allow the trivial case I = J = ∅ for completeness. Now assume that |I| = |J| ≥ 1. The cases in which r ∈ I or c ∈ J (or both) are clear from the first three parts of Lemma 5.1. Hence, we may assume that r ∉ I and c ∉ J.

If the corollary fails, we may suppose that we have a counterexample in which J is minimal with respect to ≤_{c}. By Lemma 5.1(d), Y is a linear combination of

(i) Terms [I′|J]X_{ic} of the desired form, and

(ii) Terms X_{rj}[I|J ∪ {c} \ {j}] with j ∈ J and j > c. Note that the terms in (ii) have the form X_{rj}[I|J′] with the same bidegree as X_{rc}[I|J], and with J′ <_{c} J. By the minimality of J, each term in (ii) equals [I|J′]X_{rj} plus a linear combination of terms [I′′|J′′]X_{st}, with the same bidegree as X_{rj}[I|J′], such that |I′′| = |I| and (I′′, J′′) < (I, J′) < (I, J). But if these expressions for the terms in (ii) are substituted in our initial expression for Y, we have written Y in the desired form, contradicting the assumption of a counterexample.

Therefore the corollary holds. □

5.3. Proposition. Let R, C, I, J ⊆ {1, …, n} with |R| = |C| and |I| = |J|. If M is any element of O_{q}(M_{n}(k)) of bidegree (χ_{R}, χ_{C}), then the term

Z := M[I|J] − q^{|C∩J|−|R∩I|}[I|J]M

is a linear combination of terms [I′|J′]M′ such that (a) [I′|J′]M′ has the same bidegree as M[I|J];

(b) M′ is a monomial of length |R|;

(c) |I′| = |I| and (I′, J′) < (I, J). In particular, this holds for M = [R|C].

Proof. The proposition holds trivially if either R, C or I, J are empty. Now assume that R, C, I, J are all nonempty. Write M as a linear combination of monomials M_{l} with length |R| and bidegree (χ_{R}, χ_{C}). If each of the terms

Z_{l} := M_{l}[I|J] − q^{|C∩J|−|R∩I|}[I|J]M_{l}

is a linear combination of terms [I′|J′]M′ satisfying (a),(b),(c), then so is Z. Thus, we may assume that M is a monomial.

We now induct on the length of M, namely |R|. The case |R| = 1 is given by Corollary 5.2.

If |R| > 1, write M = X_{rc}N for some r, c and some monomial N of length |R| − 1. Note that N has bidegree (χ_{Q}, χ_{B}) where Q = R \ {r} and B = C \ {c}. By induction,

(1)  N[I|J] = q^{|B∩J|−|Q∩I|}[I|J]N + lin. comb. of terms [I_{1}|J_{1}]N_{1}

such that • [I_{1}|J_{1}]N_{1} has the same bidegree as N[I|J];

• N_{1} is a monomial of length |R| − 1;
• |I_{1}| = |I| and (I_{1}, J_{1}) < (I, J).

Multiplying (1) on the left by X_{rc}, we obtain

(*)  M[I|J] = q^{|B∩J|−|Q∩I|}X_{rc}[I|J]N + lin. comb. of terms X_{rc}[I_{1}|J_{1}]N_{1}.

Next, apply Corollary 5.2 to both X_{rc}[I|J] and X_{rc}[I_{1}|J_{1}]. In the first case,

(2)  X_{rc}[I|J] = q^{δ(c,J)−δ(r,I)}[I|J]X_{rc} + lin. comb. of terms [I_{2}|J_{2}]X_{ij}

such that [I_{2}|J_{2}]X_{ij} has the same bidegree as X_{rc}[I|J], while |I_{2}| = |I| and (I_{2}, J_{2}) < (I, J). Since

|B ∩ J| − |Q ∩ I| + δ(c, J) − δ(r, I) = |C ∩ J| − |R ∩ I|,

it follows that

(†) q^{|B∩J|−|Q∩I|}X_{rc}[I|J]N = q^{|C∩J|−|R∩I|}[I|J]M + lin. comb. of terms [I_{2}|J_{2}]X_{ij}N.

In the second case, for each term X_{rc}[I_{1}|J_{1}] we have an expression of the following type, where we incorporate the [I_{1}|J_{1}]X_{rc} term with the remaining terms:

(3)  X_{rc}[I_{1}|J_{1}] = lin. comb. of terms [I_{3}|J_{3}]X_{st}

such that [I_{3}|J_{3}]X_{st} has the same bidegree as X_{rc}[I_{1}|J_{1}], while |I_{3}| = |I_{1}| = |I| and (I_{3}, J_{3}) ≤ (I_{1}, J_{1}) < (I, J). Consequently,

(‡)  X_{rc}[I_{1}|J_{1}]N_{1} = lin. comb. of terms [I_{3}|J_{3}]X_{st}N_{1}.

Finally, substitute (†) and (‡) in (*), which yields

(**) M[I|J] = q^{|C∩J|−|R∩I|}[I|J]M + lin. comb. of terms [I_{2}|J_{2}]X_{ij}N and [I_{3}|J_{3}]X_{st}N_{1}.

Observe that the terms [I_{2}|J_{2}]X_{ij}N and [I_{3}|J_{3}]X_{st}N_{1} have the same bidegree as M[I|J], and that the terms X_{ij}N and X_{st}N_{1} are monomials of length |R|. We already have |I_{2}| = |I_{3}| = |I| while (I_{2}, J_{2}) < (I, J) and (I_{3}, J_{3}) < (I, J). Therefore (**) gives us the desired relation. □

VI. Appendix: Laplace and exchange relations

We adapt some of the relations derived in [12]. (Although the base field is taken to be ℂ in that paper, the arguments are valid over any field.) For subsets I, J ⊆ {1, …, n}, set

ℓ(I; J) = |{(i, j) ∈ I × J | i > j}|.

In the following formulas, we use ⊔ to denote disjoint unions. Notation and conventions from (1.1)–(1.4) are again in force.

6.1. Lemma. (Laplace expansions) Let I, J ⊆ {1, …, n} with |I| = |J|.

(a) If J = J_{1} ⊔ J_{2}, then

[I|J] = (−q)^{−ℓ(J_{1};J_{2})} ∑_{\begin{matrix}I_{1}⊔I_{2}=I \\ |I_{1}|=|J_{1}| \end{matrix}}(−q)^{ℓ(I_{1};I_{2})}[I_{1}|J_{1}][I_{2}|J_{2}].

(b) If I = I_{1} ⊔ I_{2}, then

[I|J] = (−q)^{−ℓ(I_{1};I_{2})} ∑_{\begin{matrix}J_{1}⊔J_{2}=J \\ |J_{1}|=|I_{1}| \end{matrix}}(−q)^{ℓ(J_{1};J_{2})}[I_{1}|J_{1}][I_{2}|J_{2}].

Proof. The nontrivial cases (J_{1}, J_{2} ≠ ∅ in (a), and I_{1}, I_{2} ≠ ∅ in (b)) are given in [12, Proposition 1.1]. The trivial cases are clear. □

6.2. The sums in the next formulas run over certain partitions of index sets; we take these sums to run over only those partitions for which the terms in the formulas are defined. For instance, in part (a) the only allowable partitions K′ ⊔ K′′ = K are those for which J_{1} ∩ K′ = J_{2} ∩ K′′ = ∅ while |J_{1}| + |K′| = |I_{1}| and |K′′| + |J_{2}| = |I_{2}|.

Observe that in each formula, all terms on both sides of the equation have the same bidegree.

Proposition. (Exchange formulas) Let I_{1}, I_{2}, J_{1}, J_{2}, K ⊆ {1, …, n}.

(a) If |J_{ν}| ≤ |I_{ν}| and |K| = |I_{1}| + |I_{2}| − |J_{1}| − |J_{2}|, then

(*) ∑_{K′⊔K′′=K}(−q)^{ℓ(J_{1};K′)+ℓ(K′;K′′)+ℓ(K′′;J_{2})}[I_{1}|J_{1} ⊔ K′][I_{2}|K′′ ⊔ J_{2}]

= ∑_{I′_{ν}⊔I′′_{ν}=I_{ν}}(−q)^{ℓ(I′_{1};I′′_{1})+ℓ(I′′_{1};I′′_{2})+ℓ(I′′_{2};I′_{2})}[I′_{1}|J_{1}][I′′_{1} ⊔ I′′_{2}|K][I′_{2}|J_{2}].

(b) If |I_{ν}| ≤ |J_{ν}| and |K| = |J_{1}| + |J_{2}| − |I_{1}| − |I_{2}|, then

∑_{K′⊔K′′=K}(−q)^{ℓ(I_{1};K′)+ℓ(K′;K′′)+ℓ(K′′;I_{2})}[I_{1} ⊔ K′|J_{1}][K′′ ⊔ I_{2}|J_{2}]

= ∑_{J′_{ν}⊔J′′_{ν}=J_{ν}}(−q)^{ℓ(J′_{1};J′′_{1})+ℓ(J′′_{1};J′′_{2})+ℓ(J′′_{2};J′_{2})}[I_{1}|J′_{1}][K|J′′_{1} ⊔ J′′_{2}][I_{2}|J′_{2}].

Proof. (a) The case in which 1 ≤ |J_{ν}| < |I_{ν}| is given in the proof of [12, Proposition 1.2]; our version of this case includes only the terms with nonzero coefficients. The same proof yields the general case, as follows. First expand the left hand side of (*) by applying Lemma 6.1(a) to both [I_{1}|J_{1} ⊔ K′] and [I_{2}|K′′ ⊔ J_{2}]. This yields

(†)  ∑_{\begin{matrix}K′⊔K′′=K \\ I′_{1}⊔I′′_{1}=I_{1} \\ I′′_{2}⊔I′_{2}=I_{2} \end{matrix}}(−q)^{ℓ(I′_{1};I′′_{1})+ℓ(I′′_{2};I′_{2})+ℓ(K′;K′′)}[I′_{1}|J_{1}][I′′_{1}|K′][I′′_{2}|K′′][I′_{2}|J_{2}].

We can also expand the right hand side of (*) by applying Lemma 6.1(b) to [I′′_{1} ⊔ I′′_{2}|K]. Since this also yields (†), part (a) is proved.

(b) This is proved in the same manner. □

6.3. Note that if |I_{1} ∪ I_{2}| < |K| in Proposition 6.2, there do not exist disjoint subsets I′′_{ν} ⊆ I_{ν} such that |I′′_{1} ⊔ I′′_{2}| = |K|, and so the right hand side of formula (a) is zero. Similarly, if |J_{1} ∪ J_{2}| < |K|, the right hand side of (b) is zero. These cases are called generalized Plücker relations [12, Proposition 1.2].

Acknowledgment

We thank James Zhang for several very helpful conversations, and Jacques Alev and Laurent Rigal for useful comments.

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Department of Mathematics, University of California, Santa Barbara, CA 93106, USA E-mail address: goodearl@math.ucsb.edu

Department of Mathematics, J.C.M.B., Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland

E-mail address: tom@maths.ed.ac.uk