Ring theoretic properties of quantum grassmannians

A C Kelly^{∗}, T H Lenagan and L Rigal^{†}

Abstract The m × n quantum grassmannian, G_{q}(m, n), with m ≤ n, is the subalgebra of the algebra O_{q}(M_{mn}) of quantum m × n matrices that is generated by the maximal m×m quantum minors. Several properties of G_{q}(m, n) are established. In particular, a k-basis of G_{q}(m, n) is obtained, and it is shown that G_{q}(m, n) is a noetherian domain of Gelfand-Kirillov dimension m(n−m)+1. The algebra G_{q}(m, n) is identified as the subalgebra of coinvariants of a natural left coaction of O_{q}(SL_{m}) on O_{q}(M_{mn}) and it is shown that G_{q}(m, n) is a maximal order.

2000 Mathematics subject classification: 16W35, 16P40,16P90, 16S38, 17B37, 20G42

Introduction

Fix a base field k, a nonzero scalar q ∈ k and positive integers m, n with m ≤ n. The coordinate ring of quantum m × n matrices, O_{q}(M_{mn}), is the k-algebra generated by mn indeterminates X_{ij}, 1 ≤ i ≤ m and 1 ≤ j ≤ n, subject to the following relations:

\begin{aligned}X_{ij}X_{il} & = & qX_{il}X_{ij}, \\ X_{ij}X_{kj} & = & qX_{kj}X_{ij}, \\ X_{il}X_{kj} & = & X_{kj}X_{il}, \\ X_{ij}X_{kl} − X_{kl}X_{ij} & = & (q − q^{−1})X_{il}X_{kj}, \end{aligned}  (1)

for 1 ≤ i < k ≤ m and 1 ≤ j < l ≤ n. It is well-known that O_{q}(M_{mn}) can be presented as an iterated skew polynomial algebra over k with the generators added in lexicographic order. As a consequence of this presentation, it is easy to establish that O_{q}(M_{mn}) is a noetherian domain of Gelfand-Kirillov dimension mn.

We will usually write O_{q}(M_{n}) for the algebra O_{q}(M_{nn}). In this algebra the quantum determinant, D_{q} = det_{q} is defined by

D_{q} := ∑_{σ∈S_{n}}(−q)^{l(σ)}X_{1,σ(1)} … X_{n,σ(n)};

∗Some of the results in this paper appear in the first author’s PhD thesis (Edinburgh, 2001). She thanks EPSRC for financial support.

†Part of this work was done while the third author was visiting the University of Edinburgh. He thanks the Edinburgh Mathematical Society for the financial support of this visit.

from [13, Theorem 4.6.1], we know that D_{q} is in the centre of O_{q}(M_{n}).

Following [6], we use the notation [I | J] to denote the quantum determinant of the quantum matrix subalgebra O_{q}(M_{I,J}) of O_{q}(M_{mn}) generated by the elements X_{ij} with i ∈ I and j ∈ J, where I and J are index sets with |I| = |J|. The element [I|J] is the quantum minor determined by the index sets I and J. If I = {i_{1}, …, i_{s}} and J = {j_{1}, …, j_{s}} where the indices are written in ascending order, then we will often denote [I | J] by [i_{1} … i_{s} | j_{1} … j_{s}].

In this paper we are interested in studying the ring theoretic properties of a certain subalgebra of O_{q}(M_{mn}), the quantum deformation of the homogeneous coordinate ring of the m × n grassmannian, G_{q}(m, n). This is a deformation of the classical homogeneous coordinate ring of the grassmannian of m-dimensional k-subspaces of n-dimensional k-space and is generated by the maximal quantum minors of O_{q}(M_{mn}); to be more specific, G_{q}(m, n) is the subalgebra of O_{q}(M_{mn}) generated by the m × m quantum minors of O_{q}(M_{mn}). In the quantum grassmannian G_{q}(m, n), any m×m quantum minor will involve rows 1, …, m of the quantum matrix (X_{ij}) associated to O_{q}(M_{mn}). Thus, to simplify notation, we may denote a quantum minor by its columns only; that is, the quantum minor given by the row set {1, …, m} and column set J will be denoted by [J].

Example G_{q}(2, 4) is the k-algebra generated by the 2 × 2 minors of the 2 × 4 quantum matrix of O_{q}(M_{2,4}): [12] , [13] , [14] , [23] , [24] and [34] . Using the relations for O_{q}(M_{mn}) and [6, Lemma A.1] we can calculate the following commutation relations:

[12] [13] = q [13] [12] , [12] [14] = q [14] [12] , [12] [23] = q [23] [12] ,
[12] [24] = q [24] [12] , [12] [34] = q^{2} [34] [12] , [13] [14] = q [14] [13] ,
[13] [23] = q [23] [13] , [13] [24] = [24] [13] + (q − q^{−1}) [14] [23] ,
[13] [34] = q [34] [13] , [14] [23] = [23] [14] , [14] [24] = q [24] [14] ,
[14] [34] = q [34] [14] , [23] [24] = q [24] [23] , [23] [34] = q [34] [23] ,
[24] [34] = q [34] [24] ,

and the Quantum Plücker relation

[12] [34] − q [13] [24] + q^{2} [14] [23] = 0.

Remark Quantum matrices and quantum grassmannians can be defined in an exactly similar manner over any commutative ring R with an invertible element q ∈ R. In the next section, we shall need to consider quantum grassmannians defined over a Laurent polynomial extension either of a field or of the integers.

1 Fioresi’s commutation relations

In [3], Fioresi has developed useful commutation relations for the m × m quantum minors which generate G_{q}(m, n). However, Fioresi works in the following setting. The field k that she considers is required to be algebraically closed of characteristic zero, and the quantum matrix algebra that she considers is generated as an algebra over the ring k[q, q^{−1}], where q is transcendental over k. The first thing that we need to do is to observe that these commutation relations hold over any field k and for any 0 ≠ q ∈ k. A couple of warnings about notation for readers comparing [3] with this paper. First, because of the choice of relations for O_{q}(M_{mn}), it is necessary to replace q by q^{−1} in any relation taken from [3]. Secondly, Fioresi works with the quantum grassmannian defined by the maximal m × m minors of O_{q}(M_{nm}); thus, in any maximal minor, she uses all of the m columns, and a generating quantum minor of the grassmannian is specified by choosing m rows. To deal with this second difference, we can think of both versions of the quantum grassmannian as being subalgebras in the quantum matrix algebra O_{q}(M_{n}) and observe that the transpose automorphism, τ, see [13, 3.7.1], transforms Fioresi’s quantum grassmannian to our quantum grassmannian.

Recall the following total lexicographic ordering on quantum minors: [j_{1}j_{2} … j_{m}] <_{lex} [i_{1}i_{2} … i_{m}] if and only if there exists an index α such that j_{l} = i_{l} for l < α, but j_{α} < i_{α}.

Let [I] = [i_{1} … i_{m}] denote an m×m quantum minor. If [I] ≠ [1 … m], consider the least integer s such that i_{s} > s. Let σ([I]) be the quantum minor obtained from [I] by replacing i_{s} by i_{s} − 1 and leaving the other indices unchanged. Obviously, σ([I]) <_{lex} [I]. The standard tower of [I] is the sequence of quantum minors [I_{N}] >_{lex} [I_{N−1}] >_{lex} … >_{lex} [I_{1}] >_{lex} [I_{0}] where [I_{N}] = [I], [I_{l−1}] = σ([I_{l}]), and [I_{0}] = [1, …, r]. If [I] = [1 … r] then the standard tower is defined to be the single quantum minor [I].

We will denote the version of the m × n quantum grassmannian constructed by Fioresi by G_{h}(m, n). Note also that the relations in [3] use h where we would use h^{−1}; thus we should interchange h and h^{−1}.

Proposition 1.1 Let K be an algebraically closed field of characteristic zero, and let h be an indeterminate over K. Set G_{h}(m, n) to be the quantum grassmannian subalgebra of O_{h}(M(K[h, h^{−1}])_{mn}). Let I, J ⊆ {1, …, n} with |I| = |J| = m, and [I] <_{lex} [J]. Set s = m − |I ∩ J|. Then in G_{h}(m, n),

[I] [J] = h^{s} [J] [I] + ∑_{[L]<_{lex}[I]} λ_{[L]} (h − h^{−1})^{i_{[L]}} (−h)^{j_{[L]}} [L] [L′] ,

where i_{[L]}, j_{[L]} ∈ ℕ and λ_{[L]} is either 0 or 1, while L′ is the set (I ∩ J) ∪ ((I ∪ J) \ L).

Proof In [3, Proposition 2.21 and Theorem 3.6], Fioresi obtains commutation relations of the above form, but with the products [L] [L′] on the right hand side of the equation above more carefully stated. In Proposition 2.21 she first obtains the result for the case that I ∩ J = ∅. In this case, the quantum minors [L] involved are members of the standard tower of [I], and so [L] <_{lex} [I], as we require. The general case where I ∩ J ≠ ∅ is dealt with in Theorem 3.6. Set [Ĩ] to be the quantum minor obtained from columns I\(I ∩ J), and similarly, define [J̃]. Proposition 2.21 provides a commutation rule for [Ĩ][J̃] with terms on the right hand side [L̃][L̃′] where [L̃] <_{lex} [Ĩ]. In Theorem 3.6, a commutation rule with the same coefficients is then obtained for [I] [J] by replacing each [L̃][L̃′] by [L̃ ∪ (I ∩ J)][L̃′ ∪ (I ∩ J)]. Thus, all that needs to be done is to make the easy observation that if [L̃] <_{lex} [Ĩ] then [L̃ ∪ (I ∩ J)] <_{lex} [Ĩ∪ (I ∩ J)] = [I].

Corollary 1.2 Let k be any field and q any nonzero element of k. Set G_{q}(m, n) to be the quantum grassmannian subalgebra of O_{q}(M_{mn}). Let I, J ⊆ {1, …, n} with |I| = |J| = m, and [I] <_{lex} [J]. Set s = m − |I ∩ J|. Then in G_{q}(m, n),

[I] [J] = q^{s} [J] [I] + ∑_{[L]<_{lex}[I]} λ_{[L]} (q − q^{−1})^{i_{[L]}} (−q)^{j_{[L]}} [L] [L′] ,

where λ_{[L]} ∈ k, i_{[L]}, j_{[L]} ∈ ℕ and λ_{[L]} is either 0 or 1, while L′ is the set (I ∩ J) ∪ ((I ∪ J) \ L).

Proof Proposition 1.1 applies in the case that K = ℂ. In this case, observe that the coefficients of the monomials in the maximal minors are all in ℤ[h, h^{−1}]; so that these relations hold in the quantum grassmannian over ℤ[h, h^{−1}]. There is then a natural homomorphism from this quantum grassmannian to G_{q}(m, n), such that z ↦ z1_{k} for z ∈ ℤ and h ↦ q, which produces the required relations.

Recall that an element a of an algebra A is a normal element if aA = Aa. The next result follows immediately from the previous Corollary.

Corollary 1.3 An m × m quantum minor [I] ∈ G_{q}(m, n) is normal modulo the ideal generated by the set {[J] | [J] <_{lex} [I]}.

The algebra O_{q}(M_{mn}) is a connected ℕ-graded algebra, graded by the total degree in the canonical generators. Since G_{q}(m, n) is a subalgebra generated by homogeneous elements of degree m with respect to this grading, G_{q}(m, n) inherits a connected ℕ-graded structure in which its canonical generators have degree one.

Theorem 1.4 The quantum grassmannian G_{q}(m, n) is a noetherian domain.

Proof The quantum grassmannian G_{q}(m, n) is generated by the \binom{n}{m} quantum minors of size m in O_{q}(M_{mn}). Denote these quantum minors by u_{1} <_{lex} u_{2} <_{lex} … <_{lex} u\binom{n}{m}. Then by Corollary 1.3, \{u_{1}, …, u\binom{n}{m}\} is a normalising sequence of G_{q}(m, n); that is, u_{1} is normal and u_{l} is normal modulo the ideal generated by {u_{1}, …, u_{l−1}}, for l > 1. The factor by the ideal generated by this normalising sequence is the base field; so the fact that G_{q}(m, n) is noetherian follows by repeated use of [1, Lemma 8.2].

Finally, G_{q}(m, n) is a domain since it is a subalgebra of O_{q}(M_{mn}) which is a domain.

Remark If A is a noetherian, connected ℕ-graded k-algebra such that every nonsimple graded prime factor ring A/P contains a nonzero homogeneous normal element in ⊕_{i≥1} (A/P)_{i} then we say that A has enough normal elements ([14]). Thus, the two previous results show that the quantum grassmannian has enough normal elements.

There is a useful isomorphism between G_{q}(m, n) and G_{q^{−1}}(m, n) which we now describe. Notice that, if 1 ≤ i_{1} < ⋯ < i_{m} ≤ n, G_{q}(m, n) is isomorphic to the subalgebra of O_{q}(M_{n}) generated by the m × m minors that use rows i_{1}, …, i_{m}, that is, the minors [I|J] with I = {i_{1}, …, i_{m}} and J ⊆ {1, …, n}, |J| = m. Let A := O_{q}(M_{n}) with generators X_{ij} and A′ := O_{q^{−1}}(M_{n}) with generators X′_{ij}. Take a copy R of G_{q}(m, n) inside A generated by the m×m quantum minors that use the first m rows of A, and take a copy R′ of G_{q^{−1}}(m, n) that uses the last m rows of A′. Following the proof of [7, Corollary 5.9], we see that there is an isomorphism δ : A ⟶ A′ which takes [I|J] to [ω_{0}I|ω_{0}J]′, where [−|−]′ denotes a quantum minor in A′ := O_{q^{−1}}(M_{n}) and ω_{0} is the longest element of the symmetric group S_{n}; that is, ω_{0}(i) = n − i + 1. Note that the isomorphism δ restricted to R produces an isomorphism from R to R′ that takes a generating minor [I] to the minor [ω_{0}I]′. In particular, note that under this isomorphism, [12 … m], the leftmost minor of R = G_{q}(m, n), is translated into the rightmost minor [n − m + 1 … n]′ of the quantum grassmannian R′ = G_{q^{−1}}(m, n). We denote this induced isomorphism from G_{q}(m, n) to G_{q^{−1}}(m, n) by δ also.

As an example of the use of the isomorphism δ, we record the following lemma which we need later.

Lemma 1.5 Let I ⊆ {1, …, n} with |I| = m. Then

[I] [n − m + 1 … n] = q^{s}[n − m + 1 … n] [I]

where s = m − |I ∩ {n − m + 1, …, n}|, and thus [n − m + 1 … n] is normal in G_{q}(m, n).

Proof Note that ω_{0}{n − m + 1, …, n} = {1, …, m}. Note also that |I ∩ {n − m + 1, …, n}| = |ω_{0}I ∩ ω_{0}{n − m + 1, …, n}| = |ω_{0}I ∩ {1, …, m}|.

By Corollary 1.2, [1 … m][ω_{0}I] = q^{s}[ω_{0}I][1 … m]. Applying δ to this equation gives [n − m + 1 … n]′[I]′ = q^{s}[I]′[n − m + 1 … n]′ in G_{q^{−1}}(m, n). This can be rewritten as [I]′[n − m + 1 … n]′ = q^{−s}[n − m + 1 … n]′[I]′ in G_{q^{−1}}(m, n). Finally, replacing q^{−1} by q, we obtain

[I][n − m + 1 … n] = q^{s}[n − m + 1 … n][I]

in G_{q}(m, n).

2 A basis for G_{q}(m, n)

In this section, we obtain a basis for G_{q}(m, n). This basis is a subset of the basis of preferred products of O_{q}(M_{mn}) obtained in [6, Section 1]. First, we adapt the language used in that paper to the grassmannian subalgebra G_{q}(m, n). Recall from Section 1 that if J is an m-element subset of {1, …, n} then [J] denotes the quantum minor [1, …, m | J] of O_{q}(M_{mn}). Thus, let m, n ∈ ℕ^{∗} with n ≥ m. We define a partial ordering on m-element subsets of {1, …, n}.

Definition 2.1 Let A, B ⊆ {1, …, n} with |A| = m = |B|. We define a partial ordering, denoted by ≤_{∗}. Write A and B in ascending order:

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

Define A ≤_{∗} B to mean that a_{i} ≤ b_{i} for i = 1, …, m.

This naturally defines a partial ordering on the generators of G_{q}(m, n).

Definition 2.2 Let [I] and [J] belong to the generating set of G_{q}(m, n). Then we write that [I] ≤_{c} [J] if and only if I ≤_{∗} J.

For example, Figure 1 shows the ordering on generators of G_{q}(3, 6).

[figure: [456]; [356]; [346]; [256]; [345]; [246]; [156]; [245]; [236]; [146]; [235]; [145]; [136]; [234]; [135]; [126]; [134]; [125]; [124]; [123]]

Figure 1: The partial ordering ≤_{c} on G_{q}(3, 6)

Recall that a tableau is a Young diagram with entries in each box. If each row of a tableau T has length m then we will say that T is an m-tableau. Here, we consider tableaux with entries from {1, …, n} and no repetitions in each row. An allowable m-tableau T is an m-tableau with strictly increasing rows. If an allowable m-tableau T has rows J_{1}, …, J_{s}, then T is preferred if and only if J_{1} ≤_{∗} J_{2} ≤_{∗} … ≤_{∗} J_{s}.

Let I = {m, m − 1, …, 1} and let S be an m-tableau which has the same number of rows as T and such that each row of S is I. Then T is an allowable (preferred) m-tableau if and only if the bitableau (S | T) is allowable (preferred) in the sense of [6]. With this in mind, we define the following ordering on allowable m-tableau. Let

T = (J_{1} ; J_{2} ; ⋮ ; J_{t}) , S = (L_{1} ; L_{2} ; ⋮ ; L_{s}) .

Then T ≺ S if t > s, or if s = t and

{J_{1}, …, J_{t}} <_{lex} {L_{1}, …, L_{s}};

that is, there exists an index i such that J_{α} = L_{α} for α < i, but J_{i} <_{∗} L_{i}.

Any allowable m-tableau determines a product of quantum minors in the quantum grassmannian as follows.

Definition 2.3 For any (allowable) m-tableau

T = (J_{1} ; J_{2} ; ⋮ ; J_{s}) ,

define [T] = [J_{1}][J_{2}] … [J_{s}].

Definition 2.4 The content of an m-tableau T is the multiset {1^{t_{1}}, 2^{t_{2}}, ⋯, n^{t_{n}}}, where t_{i} is the number of times i appears in T.

We will use the content of a tableau to define a natural ℤ^{n}-grading on the m × n quantum grassmannian. There is a ℤ^{n}-grading on O_{q}(M_{mn}) defined by assigning degree ε_{j} to X_{ij}, where ε_{j} for j = 1, …, n form the natural basis of ℤ^{n}. Since the maximal minors of O_{q}(M_{mn}) are homogeneous with respect to this basis, there is an induced ℤ^{n}-grading on G_{q}(m, n): consider a product of minors [T] in G_{q}(m, n), if the tableau T has content {1^{t_{1}}, 2^{t_{2}}, …, n^{t_{n}}}, then [T] is homogeneous of degree (t_{1}, t_{2}, …, t_{n}). Thus, the degree of a product is dependent on the number of times each column of the m × n quantum matrix appears in it.

Theorem 2.5 (Generalised Quantum Plücker Relations for Quantum grassmannians) Let J_{1}, J_{2}, K ⊆ {1, 2, …, n} be such that |J_{1}|, |J_{2}| ≤ m and |K| = 2m − |J_{1}| − |J_{2}| > m. Then

∑_{K′⊔K′′=K} (−q)^{ℓ(J_{1};K′)+ℓ(K′;K′′)+ℓ(K′′;J_{2})} [J_{1} ⊔ K′][K′′ ⊔ J_{2}] = 0,

where ℓ (I; J) = |{(i, j) ∈ I × J : i > j}|.

Proof We work in the algebra O_{q}(M_{n}) and apply [6, Proposition B2(a)] with I_{1} = I_{2} = {1, …, m} =: I. Thus,

∑_{K′⊔K′′=K} (−q)^{ℓ(J_{1};K′)+ℓ(K′;K′′)+ℓ(K′′;J_{2})} [I|J_{1} ⊔ K′][I|K′′ ⊔ J_{2}] = 0,

since |K| > m = |I_{1} ∪ I_{2}|, see [6, B3]. This is the desired relation.

Lemma 2.6 Let T be an m-tableau with content γ and suppose that T is not preferred. Then

(a) T is not minimal with respect to ≺ among m-tableaux with content γ;

(b) [T] can be expressed as a linear combination of products [S], where each S is an m-tableau with content γ such that S ≺ T.

Proof Follow the proof of [6, Lemma 1.7]. Note that in the proof the only place where the shape of a bitableau might change is near the end of the proof where the right-hand side of the Exchange Formula is considered. In our situation, the right-hand side is zero, as noted in Theorem 2.5.

Note that fixing the content of an m-tableau fixes its shape and thus fixes the number of rows in the m-tableau.

Let ∂ = (c_{1}, …, c_{n}) ∈ ℕ^{n}. Let V be the homogeneous component of degree ∂ in G_{q}(m, n). Note that V might be zero, and that this is the case if and only if there is no product [T] where T is an m-tableau of content (1^{c_{1}} … n^{c_{n}}). Also, an element of G_{q}(m, n) belongs to V if and only if it is a linear combination of products [T], where T runs over all m-tableau with content (1^{c_{1}} … n^{c_{n}}); that is, the products [T], where T runs over all m-tableau with content (1^{c_{1}} … n^{c_{n}}) span V.

Theorem 2.7 Let ∂ = (c_{1}, …, c_{n}) ∈ ℕ^{n}, let V be the homogeneous component of G_{q}(m, n) with degree ∂, and set γ = (1^{c_{1}}2^{c_{2}} ⋯ n^{c_{n}}). The products [T], as T runs over all preferred m-tableau with content γ, form a basis for V.

Proof It is enough to prove that for any m-tableau T with content γ the product [T] is a linear combination of products [S] where S is a preferred m-tableau with content γ. Let E be the set of m-tableau with content γ; clearly, E is a finite set and we order it by ≺. We use induction on ≺ to show the result. Let U ∈ E. If U is minimal, then it is preferred, by part (a) of the previous result. Otherwise, by part (b) of the previous result, [U] is a linear combination of products [S], where S ∈ E and S ≺ U. Thus, by an induction argument applied to S, we may conclude that [U] is a linear combination of products [S] where S is a preferred m-tableau with content γ.

Recall that G_{q}(m, n) is a subalgebra of O_{q}(M_{mn}) and notice that the products [T], as T runs over all preferred m-tableaux of content γ, form a subset of the basis of O_{q}(M_{mn}) constructed in [6]. Therefore, they are linearly independent and we have the result.

Corollary 2.8 The products [T], as T runs over all preferred m-tableaux, form a basis for G_{q}(m, n).

This basis can be used to calculate the Gelfand-Kirillov dimension of the m×n quantum grassmannian.

Consider the partial ordering ≤_{c} on the generating minors of G_{q}(m, n). A saturated path between two minors a <_{c} b will be an ‘upwards path’ a = a_{1} <_{c} a_{2} <_{c} … <_{c} a_{l} = b of minors such that no additional terms can be added; that is, for any index i there is no minor d such that a_{i} <_{c} d <_{c} a_{i+1}. The length of such a saturated path is defined to be l. For example, a saturated path between the minors [134] and [256] in G_{q}(3, 6) is

[134] <_{c} [234] <_{c} [235] <_{c} [236] <_{c} [246] <_{c} [256] .

The length of this saturated path is 6.

A maximal path is a saturated path between the two minors [1 … m] and [n − m + 1 … n]. It is easy to check that any maximal path has length m(n − m) + 1.

Proposition 2.9 Let G = G_{q}(m, n) and let α be the length of a maximal path in G. Then

GKdim(G_{q}(m, n)) = α = m(n − m) + 1.

Proof Let V be the k-subspace of G spanned by the m × m minors which generate G. Then GKdim(G) = \overline{lim} log_{n} d_{V} (n) where d_{V} (n) = dim_{k} (∑_{i=0}^{n} V^{i}). Let a_{1}, a_{2}, …, a_{α} be a maximal path in G. Then a_{1}^{s_{1}}a_{2}^{s_{2}} … a_{α}^{s_{α}} ∈ V^{n+1} whenever ∑_{i=1}^{α} s_{i} = n + 1. The set {a_{1}^{s_{1}}a_{2}^{s_{2}} … a_{α}^{s_{α}} | Σs_{i} = n + 1} is linearly independent. Therefore

dim_{k} (V^{n+1}) ≥ |\{a_{1}^{s_{1}}a_{2}^{s_{2}} … a_{α}^{s_{α}} | Σs_{i} = n + 1\}| = \binom{n + α}{α − 1}

which is a polynomial in n of degree α − 1. It follows that GKdim(G) ≥ α.

Let a_{i_{1}} … a_{i_{n}} ∈ V^{n}. By Theorem 2.7, a_{i_{1}} … a_{i_{n}} may be rewritten as a linear combination of preferred products from V^{n}.

There are finitely many maximal paths in G_{q}(m, n). Suppose there are c such paths and index them 1, …, c. Let a_{1} <_{c} a_{2} <_{c} … <_{c} a_{α} be the ith maximal path and let W_{i}^{(n)} denote the subspace generated by monomials a_{1}^{s_{1}} … a_{α}^{s_{α}} such that Σs_{j} = n. The above observation shows that V^{n} ⊆ ∑_{i=1}^{c} W_{i}^{(n)}. Consider dim(W_{i}^{(n)}). The products a_{1}^{s_{1}} … a_{α}^{s_{α}} such that Σs_{j} = n are linearly independent. Therefore

dim(W_{i}^{(n)}) = |{a_{1}^{s_{1}} … a_{α}^{s_{α}} | ∑ s_{i} = n}| = |{(s_{1}, …, s_{α}) ∈ ℕ^{α} | ∑ s_{i} = n}|.

Therefore dim(W_{i}^{(n)}) = dim(W_{j}^{(n)}) for all i, j ∈ {1, …, c}. Thus

dim (V^{n}) ≤ dim (∑_{i=1}^{c} W_{i}^{(n)}) ≤ c dim (W_{1}^{(n)}) = c \binom{n + α − 1}{α − 1}

and d_{V} (n) ≤ c ∑_{i=0}^{n} \binom{i+α−1}{α−1}, a polynomial of degree α. It follows that GKdim(G) ≤ α. Hence, GKdim(G) = α = m(n − m) + 1.

For example, GKdim(G_{q}(2, 4)) = 2(4 − 2) + 1 = 5.

3 Noncommutative Dehomogenisation

If R is a commutative ℕ-graded algebra, and x is a homogeneous nonzerodivisor in degree one, then the dehomogenisation of R at x is usually defined to be the factor algebra R/(x−1)R, [2, Appendix 16.D]. This definition is unsuitable in a noncommutative algebra if the element x is merely normal rather than central: in this case, the factor algebra is often too small to be useful. For example, let R be the quantum plane k_{q}[x, y] with xy = qyx and q ≠ 1. Setting x = 1 forces y = 0; so that the factor algebra R/⟨x − 1⟩ is isomorphic to k rather than being an algebra of Gelfand-Kirillov dimension 1, as one might hope. However, in the commutative case, an alternative approach is to observe that the localised algebra S := R[x^{−1}] is ℤ-graded, S = ⊕_{i∈ℤ} S_{i}, and that S_{0} ≅ R/(x − 1)R. If x is a normal nonzerodivisor of degree one in a noncommutative ℕ-graded algebra R = ⊕_{i∈ℕ} R_{i}, then one can form the Ore localisation R[x^{−1}] =: S, and then this second approach does yield a useful algebra in the noncommutative case. Indeed, for i, j ∈ ℕ denote by R_{i}x^{−j} the k-subspace of elements of S that can be written as rx^{−j} with r ∈ R_{i}; clearly, R_{i}x^{−j} ⊆ R_{i+1}x^{−(j+1)}. For l ∈ ℤ, set S_{l} = ∑_{t≥0} R_{l+t}x^{−t} = ∪_{t≥0} R_{l+t}x^{−t}. Then S is a ℤ-graded algebra with S = ⊕_{l∈ℤ} S_{l}.

Definition 3.1 Let R = ⊕R_{i} be an ℕ-graded k-algebra and let x be a regular homogeneous normal element of R of degree one. Then the dehomogenisation of R at x, written Dhom(R, x), is defined to be the zero degree subalgebra S_{0} of the ℤ-graded algebra S := R[x^{−1}].

It is easy to check that Dhom(R, x) = ∑_{i=0}^{∞} R_{i}x^{−i} = ∪_{i=0}^{∞} R_{i}x^{−i}. In particular, if R = k[R_{1}] then Dhom(R, x) = ∑_{i=0}^{∞} (R_{1}x^{−1})^{i} = ∪_{i=0}^{∞} (R_{1}x^{−1})^{i}, and further, if R_{1} = ka_{1} + ⋯ + ka_{s} then Dhom(R, x) = k[a_{1}x^{−1}, …, a_{s}x^{−1}].

Denote by σ the automorphism of S given by σ(s) = xsx^{−1} for s ∈ S. Note that σ induces an automorphism of S_{0}, also denoted by σ.

Lemma 3.2 Let R be an ℕ-graded algebra and let x be a regular normal homogeneous element of degree 1. Then there is an isomorphism

θ : Dhom(R, x)[y, y^{−1}; σ] ⟶ R[x^{−1}]

which is the identity on Dhom(R, x) and sends y to x.

Proof The existence of θ is clear from the universal property of skew-Laurent extensions. It is easy to check that θ is an isomorphism.

Some properties of dehomogenisation follow in an elementary way from this result.

Corollary 3.3 Let R = ⊕_{i≥0} R_{i} be an ℕ-graded algebra and let x be a regular homogeneous normal element of degree one.

(i) R is a domain if and only if Dhom(R, x) is a domain. (ii) If R is noetherian then Dhom(R, x) is noetherian. (iii) If R is locally finite (that is, dim(R_{i}) < ∞ for all i ∈ ℕ) then GKdim(R) = GKdim(Dhom(R, x)) + 1.

Proof Point (i) follows at once from the isomorphism in Lemma 3.2. (ii) If R is noetherian then so is R[x^{−1}] and thus Dhom(R, x)[y, y^{−1}; σ] is noetherian, by Lemma 3.2. As is well-known, since σ is an automorphism of Dhom(R, x), this implies that Dhom(R, x) is noetherian. (iii) Let σ be the automorphism of R induced by conjugation by x. It is clear that σ is a graded automorphism; and so from the local finiteness of R, we see that the elements x^{i}, for i ≥ 1, are local normal elements in the sense of [9, p168]. By using [9, 12.4.4], it follows that GKdim(R[x^{−1}]) = GKdim(R). On the other hand, the automorphism σ induced on S_{0} by conjugation by x in S is locally algebraic in the sense of [9, p164]. Indeed, S_{0} = ∪_{t≥0} R_{t}x^{−t} and for all t ∈ ℕ the k-subspace R_{t}x^{−t} is a finite dimensional σ-stable subspace of S_{0}. It follows from [9, p164] that GKdim(S_{0}[y, y^{−1}; σ]) = GKdim(S_{0}) + 1. The conclusion follows from Lemma 3.2.

4 Dehomogenisation of G_{q}(m, n)

In the classical commutative theory it is a well-known and basic result that the dehomogenisation of the homogeneous coordinate ring of the m × n grassmannian at the minor [n − m + 1, …, n] is isomorphic to the coordinate ring of m × (n − m) matrices; that is,

\frac{O(G(m, n))}{⟨[n − m + 1, …, n] − 1⟩} ≅ O(M_{m,n−m}(k)).

In this section, we show that the corresponding result holds for G_{q}(m, n) when we use the noncommutative dehomogenisation defined in the previous section. Recall from Lemma 1.5 that [n − m + 1, …, n] is a normal element of G_{q}(m, n): in fact, it q-commutes with the other maximal minors, and this will be important in calculations.

Recall that we may consider G_{q}(m, n) to be a ℕ-graded algebra with each m × m quantum minor given degree 1. Set x = [n − m + 1, …, n] and S := G_{q}(m, n)[x^{−1}], and note that Dhom(G_{q}(m, n), [n − m + 1, …, n]) = S_{0} is generated by elements of the form {I} := [I] [n − m + 1, …, n]^{−1} with I ⊆ {1, …, n} and |I| = m, see Section 3.

Now let u be a positive integer and consider O_{q}(M_{u}). If I ⊆ {1, …, u} then Ĩ := {1, …, u}\I. In an exponent I denotes the sum of the indices occuring in the index set I.

Let D_{q} be the quantum determinant of O_{q}(M_{u}). Since D_{q} is a central element, we can invert it to form the u × u quantum general linear group O_{q}(GL_{u}) := O_{q}(M_{u})[D_{q}^{−1}]. The algebra O_{q}(GL_{u}) is a Hopf algebra, with antipode S, and counit ε.

There is a useful antiendomorphism Γ : O_{q}(M_{u}) ⟶ O_{q}(M_{u}) defined on generators by Γ(X_{ij}) = (−q)^{i−j}[˜{j}| ˜{i}], see [13, Corollary 5.2.2]. We need to know the effect of Γ on quantum minors. This is given in the following lemma, which is presumably well-known but we give a proof since we have been unable to find a clear exposition. Recall that Δ([I|J]) = ∑_{|K|=|I|} [I|K]⊗[K|J], where Δ is the comultiplication map on O_{q}(M_{u}), by [12, (1.9)]. Recall also that ε([I|J]) equals 1 if I = J and 0 otherwise.

Lemma 4.1 Let [I|J] be an r × r quantum minor in O_{q}(M_{u}). Then,

(i) S([I|J]) = (−q)^{I−J}[J̃|Ĩ]D_{q}^{−1} (ii) Γ([I|J]) = (−q)^{I−J}[J̃|Ĩ]D_{q}^{r−1}

Proof We establish the first claim by calculating the expression

∑_{K,L} (−q)^{L−J}S([I|K])[K|L][J̃|L̃]D_{q}^{−1}

in two different ways.

First,

∑_{K,L} (−q)^{L−J}S([I|K])[K|L][J̃|L̃]D_{q}^{−1} = ∑_{K} S([I|K]) {∑_{L} (−q)^{L−J}[K|L][J̃|L̃]D_{q}^{−1}}
= ∑_{K} S([I|K])ε([K|J])1 = S([I|J]),

by using the first equality of [13, 4.4.3].

Secondly,

∑_{K,L} (−q)^{L−J}S([I|K])[K|L][J̃|L̃]D_{q}^{−1} = ∑_{L} {∑_{K} S([I|K])[K|L]} (−q)^{L−J}[J̃|L̃]D_{q}^{−1}
= ∑_{L} ε([I|L])(−q)^{L−J}[J̃|L̃]D_{q}^{−1}
= (−q)^{I−J}[J̃|Ĩ]D_{q}^{−1},

by using the defining property of the antipode.

The second claim follows easily from the first, since S([I|J]) = Γ([I|J])D_{q}^{−r} for r × r quantum minors [I|J]. This is easily established from the fact that it holds on the generators X_{ij} and that S and Γ are anti-endomorphisms.

We will need the anti-endomorphism Γ∘τ : O_{q}(M_{u}) ⟶ O_{q}(M_{u}) defined by Γ∘τ(X_{ij}) = (−q)^{j−i}[ ˜{i}|˜{j}] for 1 ≤ i, j ≤ u. Here, τ is the transposition automorphism given in [13, Proposition 3.7.1(1)]. Note that, by Lemma 4.1, the effect of Γ ∘ τ on the r × r quantum minor [I | J] is given by Γ ∘ τ([I | J]) = (−q)^{J−I}[Ĩ | J̃]D_{q}^{r−1}.

Given I = {i_{1}, …, i_{m}} ⊆ {1, …, n} the set I\{i_{k}} is denoted by {i_{1}, …, î_{k}, …, i_{m}}. Given two sets I, J ⊆ {1, …, n} recall that

ℓ (I; J) := |{(i, j) ∈ I × J : i > j}|.

In the next proof, and throughout the paper, (−q)^{•} denotes a power of −q that is not necessary to keep track of explicitly.

Lemma 4.2 The k-algebra Dhom(G_{q}(m, n), [n − m + 1, …, n]) = S_{0} is generated as an algebra by the elements {j n − m + 1 … î… n} for 1 ≤ j ≤ n − m < i ≤ n.

Proof We know that S_{0} is generated by the elements {I} := [I] [n − m + 1, …, n]^{−1}, where I ⊆ {1, …, n} and |I| = m. We show that each such element can be expressed as a k-linear combination of products of elements of the form {j n − m + 1 … î… n}, where 1 ≤ j ≤ n − m < i ≤ n. Denote by A the subalgebra of S_{0} generated by the elements {j n − m + 1 … î… n}.

Let I = {i_{1} ≤ … ≤ i_{m}} ≠ {n − m + 1, …, n} be an ordered subset of {1, …, n} and let 2 ≤ t ≤ m + 1 be such that i_{t} ≥ n − m + 1 but i_{t−1} < n − m + 1; that is, I ∩ {1, …, n − m} = {i_{1}, … i_{t−1}}. We will use induction on t to show that {I} ∈ A.

If t = 2, then I is of the form {j n − m + 1 … î… n} and so {I} ∈ A. Consider a fixed t ∈ {3, …, m + 1} and suppose that the result is true for t − 1. Now consider [I] = [i_{1} … i_{m}] with I ∩ {1, …, n − m} = {i_{1}, … i_{t−1}}. We use the generalised Quantum Plücker relations (Theorem 2.5) to rewrite the product [n − m + 1, …, n] [i_{1} … i_{m}].

Let K = {i_{1}, n − m + 1, …, n}, J_{1} = ∅ and J_{2} = {i_{2}, …, i_{m}}. Then

∑_{K′⊔K′′=K} (−q)^{ℓ(K′;K′′)+ℓ(K′′;J_{2})} [K′] [K′′ ⊔ J_{2}] = 0

where either

K′ = {n − m + 1, …, n} and K′′ = {i_{1}},

or

K′ = {i_{1}} ∪ {n − m + 1, …, l̂, …, n} and K′′ = {l}

where n − m + 1 ≤ l ≤ n and l ∉ {i_{2}, …, i_{m}}. Let S = {n − m + 1, …, n} \ {i_{2}, …, i_{m}}. By re-arranging the above equation, we obtain

[n − m + 1, …, n] [i_{1} … i_{m}] = − ∑_{l∈S} (−q)^{•} [i_{1} n − m + 1 … l̂… n] [l i_{2} … i_{m}] .

Multiplying through by [n − m + 1, …, n]^{−2} from the right, and using Lemma 1.5 gives

{i_{1} … i_{m}} = ∑_{l∈S} ± (−q)^{•} {i_{1} n − m + 1 … l̂… n}{l i_{2} … i_{m}}.

Now {l, i_{2}, …, i_{m}} ∩ {1, …, n − m} = {i_{2}, …, i_{t−1}} and so, by the inductive hypothesis, {l i_{2} … i_{m}} ∈ A. Clearly {i_{1} n − m + 1 … l̂… n} ∈ A, therefore {i_{1} … i_{m}} ∈ A. This completes the inductive step and the result follows.

Theorem 4.3 There is an isomorphism

ρ : O_{q}(M_{m,n−m}) ⟶ Dhom(G_{q}(m, n), [n − m + 1, …, n])

which is defined on generators by ρ(X_{ij}) = \{j n − m + 1 … { \atop n − i + 1} … n\}, for 1 ≤ i ≤ m and 1 ≤ j ≤ n − m.

Proof In order to show that ρ is a homomorphism we have to show that the images of the X_{ij} under ρ still obey the relevant commutation relations. We will make repeated use of the anti-endomorphism Γ ∘ τ defined before Lemma 4.2. There are four types of products to consider.

(1) Let 1 ≤ i < l ≤ m and 1 ≤ j ≤ n − m. Then X_{ij}X_{lj} = qX_{lj}X_{ij}, and so we must show that ρ (X_{ij}) ρ (X_{lj}) = qρ (X_{lj}) ρ (X_{ij}). Let t = n + 1 − i and s = n + 1 − l. Note that s < t, and consider the product

[j n − m + 1 … t̂… n] [j n − m + 1 … ŝ… n]

in G_{q}(m, n). We can think of this as a product in O_{q}(M_{m+1}) where the rows are indexed by 1, …, m+1 and the columns by j, n−m+1, …, n. Apply the anti-endomorphism Γ∘τ to the commutation relation X_{m+1,s}X_{m+1,t} = qX_{m+1,t}X_{m+1,s} we obtain:

[j n − m + 1 … t̂… n] [j n − m + 1 … ŝ… n]
= q [j n − m + 1 … ŝ… n] [j n − m + 1 … t̂… n] .

Multiplying through this equation on the right by [n − m + 1, …, n]^{−2} on each side and using Lemma 1.5 gives

{j n − m + 1 … t̂… n}{j n − m + 1 … ŝ… n}
= q{j n − m + 1 … ŝ… n}{j n − m + 1 … t̂… n};

that is, ρ (X_{ij}) ρ (X_{lj}) = qρ (X_{lj}) ρ (X_{ij}).

(2) Let 1 ≤ j < r ≤ n − m and 1 ≤ i ≤ m. Then X_{ij}X_{ir} = qX_{ir}X_{ij}. Let t = n + 1 − i and, as in (1), think of the product

[j n − m + 1 … t̂… n] [r n − m + 1 … t̂… n]

as sitting inside O_{q}(M_{m+1}) where the rows are indexed by 1, …, m + 1 and the columns by j, r, n−m+1, … t̂, …, n. Then Γ∘τ applied to the relation X_{m+1,j}X_{m+1,r} = qX_{m+1,r}X_{m+1,j} in O_{q}(M_{m+1}) gives us

[j n − m + 1 … t̂… n] [r n − m + 1 … t̂… n]
= q [r n − m + 1 … t̂… n] [j n − m + 1 … t̂… n] .

Therefore, multiplying through this equation on the right by [n−m+1, …, n]^{−2} and using Lemma 1.5, we get

{j n − m + 1 … t̂… n}{r n − m + 1 … t̂… n}
= q{r n − m + 1 … t̂… n}{j n − m + 1 … t̂… n};

that is, ρ(X_{ij})ρ(X_{ir}) = qρ(X_{ir})ρ(X_{ij})

(3) Let 1 ≤ i < l ≤ m, and 1 ≤ j < r ≤ n − m. Then

X_{ij}X_{lr} = X_{lr}X_{ij} + (q − q^{−1}) X_{lj}X_{ir}.

Let t = n+1−i and s = n+1−l. Note that n−m+1 ≤ s < t ≤ n, and that j < r < s < t. Consider the product

[j n − m + 1 … t̂… n] [r n − m + 1 … ŝ… n]

as a product in O_{q}(M_{m+2}), where the m + 2 rows are indexed by 1, …, m + 2 and the columns by j, r, n − m + 1, …, n.

The relation

[13] [24] = [24] [13] + (q − q^{−1}) [14] [23]

that we calculated earlier for G_{q}(2, 4) shows that, in O_{q}(M_{m+2}),

[I | js][I | rt] = [I | rt][I | js] + (q − q^{−1})[I | jt][I | rs]

where I = {m + 1, m + 2}, since j < r < s < t. By applying the anti-endomorphism Γ ∘ τ to this relation, we obtain

[j n − m + 1 … t̂… n] [r n − m + 1 … ŝ… n]
= [r n − m + 1 … ŝ… n] [j n − m + 1 … t̂… n]
+ (q − q^{−1}) [j n − m + 1 … ŝ… n] [r n − m + 1 … t̂… n]

in G_{q}(m, n). Multiplying through by [n − m + 1, …, n]^{−2} and using Lemma 1.5 we get

{j n − m + 1 … t̂… n}{r n − m + 1 … ŝ… n}
= {r n − m + 1 … ŝ… n}{j n − m + 1 … t̂… n}
+ (q − q^{−1}) {j n − m + 1 … ŝ… n}{r n − m + 1 … t̂… n};

that is, ρ(X_{ij})ρ(X_{lr}) = ρ(X_{lr})ρ(X_{ij}) + (q − q^{−1})ρ(X_{lj})ρ(X_{ir}), as required.

(4) Let 1 ≤ i < l ≤ m and 1 ≤ j < r ≤ n − m. Then

X_{ir}X_{lj} = X_{lj}X_{ir}.

Let t = n + 1 − i and s = n + 1 − l so that n − m + 1 ≤ s < t ≤ n and j < r < s < t. Arguing as in (3), the relation [23][14] = [14][23] in G_{q}(2, 4) produces, in O_{q}(M_{m+2}), the relation

[I | rs][I | jt] = [I | jt][I | rs].

Applying Γ ∘ τ to this relation gives

[r n − m + 1 … t̂… n] [j n − m + 1 … ŝ… n]
= [j n − m + 1 … ŝ… n] [r n − m + 1 … t̂… n] .

Multiplying through by [n − m + 1, …, n]^{−2} we get

{r n − m + 1 … t̂… n}{j n − m + 1 … ŝ… n}
= {j n − m + 1 … ŝ… n}{r n − m + 1 … t̂… n};

that is, ρ(X_{ir})ρ(X_{lj}) = ρ(X_{lj})ρ(X_{ir}), as required.

Thus, ρ extends to a homomorphism. The images of the generators under ρ generate Dhom(G_{q}(m, n), [n − m + 1, …, n]), by Lemma 4.2; so ρ is surjective. We show that ρ is injective by comparing Gelfand-Kirillov dimensions. If ρ was not injective, then GKdim(Dhom(G_{q}(m, n), [n − m + 1, …, n]) < GKdim(O_{q}(M_{m,n−m})) = m(n − m), since O_{q}(M_{m,n−m}) is a domain. However, by Corollary 3.3 and Proposition 2.9, we know that GKdim(Dhom(G_{q}(m, n), [n − m + 1, …, n]) = GKdim(G_{q}(m, n)) − 1 = m(n − m) + 1 − 1 = m(n − m). Thus, ρ is injective and hence ρ is an isomorphism.

Corollary 4.4 Let ϕ be the automorphism of O_{q}(M_{m,n−m}) defined by ϕ(X_{ij}) = q^{−1}X_{ij}, for 1 ≤ i ≤ m and 1 ≤ j ≤ n − m. Then

O_{q}(M_{m,n−m})[y, y^{−1}; ϕ] ⟶ G_{q}(m, n) [[n − m + 1, …, n]^{−1}]

defined by X_{ij} ↦ \{j n − m + 1 … { \atop n + 1 − i} … n\} and y ↦ [n − m + 1, …, n] is an isomor-

phism of algebras.

Proof Recall from Lemma 3.2 that there is an isomorphism

θ : Dhom(G_{q}(m, n), [n − m + 1, …, n])[y, y^{−1}; σ] ⟶ G_{q}(m, n) [[n − m + 1, …, n]^{−1}]

given by y ↦ [n − m + 1, …, n] and {j n − m + 1 … t̂… n} ↦ {j n − m + 1 … t̂… n}, where σ is the automorphism of Dhom(G_{q}(m, n), [n − m + 1, …, n]) given by conjugation by the quantum minor [n − m + 1, …, n]. On the other hand, by Theorem 4.3, there is an isomorphism ρ : O_{q}(M_{m,n−m}) ⟶ Dhom(G_{q}(m, n), [n − m + 1, …, n]), and it is easy to see, by using Lemma 1.5, that the automorphism induced in O_{q}(M_{m,n−m}) by σ via ρ is ϕ. Thus, ρ extends to an isomorphism

\overline{ρ} : O_{q}(M_{m,n−m})[y, y^{−1}; ϕ] ⟶ Dhom(G_{q}(m, n), [n − m + 1, …, n])[y, y^{−1}; σ]

such that \overline{ρ}(y) = y. Clearly, θ ∘ \overline{ρ} is the desired isomorphism.

Note that in [4] Fioresi proves a restricted version of Theorem 4.3. More specifically, operating over the ring K[q, q^{−1}], where K is algebraically closed of characteristic zero and q is transcendental over K, she shows that O_{q}(M_{n}) is isomorphic to the subalgebra of G_{q}(n, 2n)[[n + 1 … 2n]^{−1}] generated by the elements {j n + 1 … î… 2n}, but does not show that this subalgebra is the dehomogenisation of G_{q}(n, 2n) at [n + 1 … 2n].

Example Let S = G_{q}(2, 4)[[34]^{−1}]. Then Dhom(G_{q}(2, 4), [34]) = S_{0} and S_{0} is generated by the elements

[12] [34]^{−1} , [13] [34]^{−1} , [14] [34]^{−1} , [23] [34]^{−1} , [24] [34]^{−1} .

Recall that {ij} = [ij] [34]^{−1}. From the commutation relations for G_{q}(2, 4) given in the introduction, we can calculate the following commutation relations:

{13}{23} = q{23}{13}; {13}{14} = q{14}{13};
{13}{24} = {24}{13} + (q − q^{−1}) {23}{14};
{14}{23} = {23}{14}; {14}{24} = q{24}{14}; {23}{24} = q{24}{23}

and from the Quantum Plücker relation;

{12} = {13}{24} − q{23}{14}.

We can immediately see the correspondence (or we can use ρ to find the correspondence):

\begin{aligned}O_{q}(M(2)) & ⟷ & S_{0} \\ X_{11} & ⟷ & \{13\} \\ X_{12} & ⟷ & \{23\} \\ X_{21} & ⟷ & \{14\} \\ X_{22} & ⟷ & \{24\} \\ D_{q} & ⟷ & \{12\}, \end{aligned}

and from Theorem 4.3

Dhom(G_{q}(2, 4), [34]) ≅ O_{q}(M(2)).

5 G_{q}(m, n) as coinvariants of O_{q}(SL_{m})

Recall that the m × m quantum special linear group, O_{q}(SL_{m}), is defined by O_{q}(SL_{m}) := O_{q}(M_{m})/⟨D_{q} − 1⟩.

In this section we show that G_{q}(m, n) is the algebra of coinvariants of a natural left coaction of O_{q}(SL_{m}) on O_{q}(M_{mn}). There is a natural epimorphism π : O_{q}(GL_{m}) ⟶ O_{q}(SL_{m}) which sends D_{q} to 1. In order to distinguish generators in the various algebras, we will often denote the canonical generators in O_{q}(M_{n}) by X_{ij}, in O_{q}(M_{nm}) by Y_{ij}, in O_{q}(M_{mn}) by Z_{ij} and in O_{q}(GL_{m}) by T_{ij}. Further, set U_{ij} := π(T_{ij}) ∈ O_{q}(SL_{m}). Note that both O_{q}(GL_{m}) and O_{q}(SL_{m}) are Hopf algebras.

It is easy to check that one can define a morphism of algebras satisfying the following rule:

λ : O_{q}(M_{mn}) ⟶ O_{q}(GL_{m}) ⊗ O_{q}(M_{mn}), Z_{ij} ↦ ∑_{k=1}^{m} T_{ik} ⊗ Z_{kj}

and that this induces a morphism of algebras

Λ : O_{q}(M_{mn}) ⟶ O_{q}(SL_{m}) ⊗ O_{q}(M_{mn}), Z_{ij} ↦ ∑_{k=1}^{m} U_{ik} ⊗ Z_{kj}

where Λ := (π ⊗ id) ∘ λ.

The morphisms λ and Λ endow O_{q}(M_{mn}) with left comodule algebra structures over O_{q}(GL_{m}) and O_{q}(SL_{m}), respectively. Recall that if H is a Hopf algebra and M is a left Hcomodule via the coaction γ : M ⟶ H ⊗M then m ∈ M is a coinvariant if γ(m) = 1⊗m. In this section we show that G_{q}(m, n) is the set of coinvariants of the O_{q}(SL_{m})-comodule O_{q}(M_{mn}) under the comodule map Λ. In fact, this result is an easy consequence of [8, Theorem 6.6], once we have described the set-up of that paper.

The map Y_{ij} ↦ ∑_{k=1}^{m} Y_{ik} ⊗ T_{kj} induces a morphism of algebras ρ : O_{q}(M_{nm}) ⟶ O_{q}(M_{nm}) ⊗ O_{q}(GL_{m}) which endows O_{q}(M_{nm}) with a right comodule algebra stucture over O_{q}(GL_{m}). Let O_{q}(V) denote the algebra O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}). The coactions λ and ρ defined above can be combined to give a left comodule structure on O_{q}(V) which we denote by γ. To be precise,

γ : O_{q}(V) ⟶ O_{q}(GL_{m}) ⊗ O_{q}(V)

is given by the rule

γ(a ⊗ b) := ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}

for a ∈ O_{q}(M_{nm}) and b ∈ O_{q}(M_{mn}), where λ(b) = ∑_{(b)} b_{−1} ⊗ b_{0} and ρ(a) = ∑_{(a)} a_{0} ⊗ a_{1}. Here, we are using the Sweedler notation and S is the antipode of O_{q}(GL_{m}). In turn, this coaction induces a coaction Γ : O_{q}(V) ⟶ O_{q}(SL_{m}) ⊗ O_{q}(V) given by Γ := (π ⊗ id) ∘ γ; so that

Γ(a ⊗ b) := ∑_{(a),(b)} π(S(a_{1})b_{−1}) ⊗ a_{0} ⊗ b_{0}.

The main results of [8] identify the coinvariants of the coactions γ and Γ. In particular, Theorem 6.6 of [8] identifies the coinvariants of the coaction Γ in the following way. There is a morphism of algebras μ : O_{q}(M_{n}) ⟶ O_{q}(V) = O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}) given by X_{ij} ↦ ∑_{k=1}^{m} Y_{ik} ⊗ Z_{kj}. Let R denote μ(O_{q}(M_{n})). It is proved in [6] that R ≅ O_{q}(M_{n})/I, where I is the ideal generated by the (m + 1) × (m + 1) quantum minors of O_{q}(M_{n}). We have the following theorem.

Theorem 5.1 [8, Theorem 6.6] Let G_{1} and G_{2} denote the respective grassmannian subalgebras of O_{q}(M_{nm}) and O_{q}(M_{mn}) generated by all the m × m quantum minors. The set of Γ-coinvariants in O_{q}(V) = O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}) is the subalgebra generated by G_{1} ⊗ G_{2} and R. More precisely,

(O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}))^{co O_{q}(SL_{m})} = (G_{1} ⊗ G_{2}) · R.

The result we are aiming for follows easily from this.

Theorem 5.2

(O_{q}(M_{mn}))^{co O_{q}(SL_{m})} = G_{q}(m, n).

Proof It is easily seen that there is a commutative diagram

O_{q}(M_{mn})  ⟶^{i}  O_{q}(M_{nm}) ⊗ O_{q}(M_{mn})
Λ ↓  Γ ↓
O_{q}(SL_{m}) ⊗ O_{q}(M_{mn}) ⟶^{id⊗i} O_{q}(SL_{m}) ⊗ O_{q}(M_{nm}) ⊗ O_{q}(M_{mn})

where i is the canonical injection. Moreover, let j : O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}) ⟶ O_{q}(M_{mn}) be the canonical projection; that is,

j : O_{q}(M_{nm}) ⊗ O_{q}(M_{mn}) ⟶^{p⊗id} k ⊗ O_{q}(M_{mn}) ≅ O_{q}(M_{mn})

where p is the projection modulo the irrelevant ideal of O_{q}(M_{nm}). Clearly, we have that j ∘ i = id. We see from the above commutative diagram that, if b ∈ O_{q}(M_{mn}) is a Λcoinvariant, then i(b) = 1 ⊗ b is a Γ-coinvariant. Thus, it follows from Theorem 5.1 that 1 ⊗ b ∈ (G_{1} ⊗ G_{2}).R. Hence, b = j(1 ⊗ b) ∈ j(G_{1} ⊗ G_{2})j(R). Clearly, j(R) ⊆ k and j((G_{1} ⊗ G_{2})) ⊆ G_{2}; and so b ∈ G_{2} = G_{q}(m, n). This shows that O_{q}(M_{mn})^{coO_{q}(SL_{m})} ⊆ G_{q}(m, n). Since it is clear that an m × m quantum minor of O_{q}(M_{mn}) is a Λ-coinvariant, the converse inclusion follows from the fact that Λ is a morphism of algebras.

Note that Fioresi and Hacon, [5], have a version of this result, with the usual restrictions as described earlier in this paper.

6 G_{q}(m, n) is a maximal order

Let R be a noetherian domain with division ring of fractions Q. Then R is said to be a maximal order in Q if the following condition is satisfied: if T is a ring such that R ⊆ T ⊆ Q and such that there exist nonzero elements a, b ∈ R with aT b ⊆ R, then T = R. This condition is the natural noncommutative analogue of normality for commutative domains, see, for example, [11, Section 5.1].

Recall that an element d in a ring R is said to be left regular if rd = 0 implies that r = 0 for r ∈ R. The following is a general result that we will be able to apply to show that the quantum grassmannian G_{q}(m, n) is a maximal order.

Proposition 6.1 Suppose that R is a noetherian domain with division ring of fractions Q. Suppose that a, b ∈ R are nonzero normal elements such that R[a^{−1}] and R[b^{−1}] are both maximal orders, that b is left regular modulo aR and that ab = λba for some central unit λ ∈ R. Then R is a maximal order.

Proof First, we show that R[a^{−1}] ∩ R[b^{−1}] = R. Suppose that this is not the case, and choose q ∈ R[a^{−1}] ∩ R[b^{−1}] \R. Write q = ra^{−d} = sb^{−e} with d, e ≥ 1 and r ∈ R\Ra, s ∈ R\Rb. Cross multiply to get rb^{e} = λ^{•}sa^{d} (remember that ab = λba). Since b is left regular modulo aR, this gives r ∈ Ra, a contradiction. Thus, R[a^{−1}] ∩ R[b^{−1}] = R.

Now, to show that R is a maximal order, it is enough to show that if J is a nonzero ideal of R and q ∈ Q with either qJ ⊆ J or Jq ⊆ J then q ∈ R, [11, Proposition 5.1.4]. Suppose, without loss of generality, that qJ ⊆ J. By assumption, S := R[a^{−1}] and T := R[b^{−1}] are maximal orders. Also, SJ = JS is an ideal of S and T J = JT is an ideal of T. We have qJS ⊆ JS and so q ∈ S. Similarly, q ∈ T. Thus, q ∈ S ∩ T = R; and so R is a maximal order.

Theorem 6.2 G_{q}(m, n) is a maximal order.

Proof We will apply the previous result to R := G_{q}(m, n) with a := [1, …, m] and b := [n − m + 1, …, n]. Observe that b is normal by Lemma 1.5 and that a is normal by Corollary 1.2. Note that ab = (−q)^{•}ba, by Lemma 1.5. First we observe that b is left regular modulo aR. The reason is that since a is the minimal minor in the preferred ordering, a basis for aR is given by preferred products that start with a. If r ∈ R is such that rb ∈ aR, then when we write r as a linear combination of preferred products then multiplying each preferred product that occurs by b on the right still gives a preferred product, since b is the maximal element with respect to the preferred order. Thus, since rb ∈ aR each of these preferred products must begin with a, and so the original ones also begin with a, hence r ∈ aR.

In Corollary 4.4, we have shown that R[b^{−1}] ≅ O_{q}(M_{m,n−m})[y, y^{−1}; ϕ] and so R[b^{−1}] is a maximal order ([10, V. Proposition 2.5, IV. Proposition 2.1]). Also R[a^{−1}] is a maximal order by using the isomorphism δ introduced in Section 1 and the fact that R[b^{−1}] is a maximal order.

Thus, the hypotheses of Proposition 6.1 are satisfied, and we deduce that G_{q}(m, n) is a maximal order.

References

[1] M Artin, J Tate and M van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, 33-85, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990.

[2] W Bruns and U Vetter, Determinantal rings, Springer Lecture Notes in Mathematics, 1327, Springer-Verlag, Berlin, 1988.

[3] R Fioresi, Quantum deformation of the Grassmannian manifold, J. Algebra 214 (1999), 418-447.

[4] R Fioresi, A deformation of the big cell inside the Grassmannian manifold G(r, n), Rev. Math. Phys. 11 (1999), 25-40.

[5] R Fioresi and C Hacon, Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties, J. Algebra 242 (2001), 433-446.

[6] K R Goodearl and T H Lenagan. Quantum determinantal ideals, Duke Mathematical Journal 103 165-190, 2000.

[7] K R Goodearl and T H Lenagan. Winding-invariant prime ideals in quantum 3 × 3 matrices, to appear in Journal of Algebra, preprint available at math.QA/0112051.

[8] K R Goodearl, T H Lenagan and L Rigal, The first fundamental theorem of coinvariant theory for the quantum general linear group, Publ. RIMS (Kyoto) 36 (2000), 269-296.

[9] G R Krause and T H Lenagan. Growth of algebras and Gelfand-Kirillov dimension, Revised edition. Graduate Studies in Mathematics, 22. American Mathematical Society, Providence, RI, 2000.

[10] G Maury and J Raynaud, Ordres maximaux au sens de K Asano, Springer Lecture Notes in Mathematics Vol 808, Springer-Verlag, Berlin, 1980.

[11] J C McConnell and J C Robson. Noncommutative Noetherian Rings. Wiley, Chichester, 1987.

[12] M Noumi, H Yamada and K Mimachi, Finite-dimensional representations of the quantum group GL_{q}(n; ℂ) and the zonal spherical functions on U_{q}(n − 1)\U_{q}(n), Japanese J. Math 19 (1993), 31-80.

[13] B Parshall and J Wang. Quantum linear groups. Mem. Amer. Math. Soc 89 (1991), no. 439.

[14] J J Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra 189 (1997), 390-405.

A. C. Kelly, T. H. Lenagan: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland E-mail: ACK@bosinternet.com, tom@maths.ed.ac.uk

L. Rigal: Université Jean Monnet (Saint-Étienne), Faculté des Sciences et Techniques, Département de Mathématiques 23 rue du Docteur Paul Michelon 42023 Saint-Étienne Cédex 2, France E-mail: Laurent.Rigal@univ-st-etienne.fr