THE FIRST FUNDAMENTAL THEOREM OF COINVARIANT

THEORY FOR THE QUANTUM GENERAL LINEAR GROUP

K. R. Goodearl, T. H. Lenagan, and L. Rigal

Abstract. We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum groups O_{q}(GL_{t}(K)) and O_{q}(SL_{t}(K)) on the quantized algebra O_{q}(M_{m,t}(K)) ⊗ O_{q}(M_{t,n}(K)). (Here K is an arbitrary field and q an arbitrary nonzero scalar.) In both cases, the set of coinvariants is a subalgebra of O_{q}(M_{m,t}(K))⊗O_{q}(M_{t,n}(K)), which we identify.

Introduction

One of the highlights of classical invariant theory is the determination of the algebra of invariant functions for the standard action of the general linear group on the variety of pairs of matrices over a field K. More precisely, the standard action of GL_{t} = GL_{t}(K) on the variety V := M_{m,t}(K) × M_{t,n}(K) induces an action of GL_{t} on O(V), and the classical theorem determines the algebra of invariants, O(V)^{GL_{t}}. We recall the details below, since, if we assume that K is algebraically closed, the method of proof we follow has an easy geometric translation.

The main theorem of this paper, Theorem 4.5, gives a quantum analog of the above theorem. Since the quantum group O_{q}(GL_{t}) is not a group but a Hopf algebra, we first place the classical situation into a Hopf algebra context. This is standard: the action of GL_{t} on V induces a coaction of O(GL_{t}) on O(V), under which O(V) becomes an O(GL_{t})- comodule algebra, and O(V)^{GL_{t}} equals the algebra of O(GL_{t})-coinvariants in O(V). It is this situation which has a natural quantization: the coordinate ring O(V) becomes the algebra O_{q}(V) := O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}), and the coaction of O(GL_{t}) on O(V) becomes a coaction of O_{q}(GL_{t}) on O_{q}(V). We prove the First Fundamental Theorem of Coinvariant Theory for this coaction, that is, we identify the set of coinvariants, O_{q}(V)^{co O_{q}(GL_{t})}.

There is a natural comultiplication map θ_{q}^{∗} : O_{q}(M_{m,n}) → O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}), which is the quantum analog of matrix multiplication M_{m,t} × M_{t,n} → M_{m,n}. We prove that the

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16W30, 17B37.

The research of the first author was partially supported by National Science Foundation research grant DMS-9622876, and that of the first two authors by NATO Collaborative Research Grant 960250. The first author also thanks Mme. M.-P. Malliavin for inviting him to the Université de Paris VI during January 1999, where a portion of this work was done. Another portion was done while the third author was attending the L.M.S. Symposium on Quantum Groups (Durham, July 1999) and during a subsequent visit to the University of Edinburgh. He wishes to thank the London Mathematical Society for partial financial support of both trips.

set of coinvariants is equal to the image of θ_{q}^{∗}. In an earlier paper, [2], the first two authors have shown that the kernel of θ_{q}^{∗} is the ideal generated by the (t + 1) × (t + 1) quantum minors of O_{q}(M_{m,n}): this is the Second Fundamental Theorem of Coinvariant Theory for this comodule action. Taken together, these two results give a complete description of O_{q}(V)^{co O_{q}(GL_{t})}. Further, we investigate the coaction of O_{q}(SL_{t}) on O_{q}(V) induced by that of O_{q}(GL_{t}) and identify the coinvariants of this coaction.

The basic structure of our proof follows the outline of one of the possible proofs in the classical invariant theoretical setting. However, there are significant problems that arise due to the noncommutative setting. The most striking one is that, unlike in the commutative case, the O_{q}(GL_{t})-comodule O_{q}(V) is not a comodule algebra. For this reason, it is not even obvious at the outset that the set of coinvariants forms a subalgebra. More generally, the quantum analogs of several maps that we need are not algebra morphisms, and so their properties cannot be analyzed simply by checking how they behave on sets of algebra generators.

Nevertheless, it is useful to start by reviewing the classical situation, to provide a skeleton for our approach.

The classical situation. We fix an algebraically closed field K and positive integers m, n, t. For integers u, v > 0, we write M_{u,v} = M_{u,v}(K) for the set of u × v matrices with entries in K. We will be mainly interested in the general linear group GL_{t} = GL_{t}(K) and its standard action on the algebraic variety V = V_{m,t,n} := M_{m,t} × M_{t,n}. This action is given by:

\xymatrix{GL_{t} × V \ar[r]^{γ} & V \\ (g, (A, B)) \ar@{|->}[r] & (Ag^{−1}, gB).}

Thus GL_{t} acts on O(V) ≅ O(M_{m,t}) ⊗ O(M_{t,n}). Classical invariant theory is interested in computing the subalgebra O(V)^{GL_{t}} of invariants for this action. The description of this algebra goes as follows. Consider the morphism of varieties

\xymatrix{M_{m,t} × M_{t,n} \ar[r]^{θ} & M_{m,n} \\ (A, B) \ar@{|->}[r] & AB}

and its associated comorphism θ^{∗} : O(M_{m,n}) → O(M_{m,t}) ⊗ O(M_{t,n}). Let X_{ij} (for 1 ≤ i ≤ m and 1 ≤ j ≤ n) stand for the usual coordinate functions on the variety M_{m,n}, and let I_{t+1} denote the ideal of O(M_{m,n}) generated by all the (t + 1) × (t + 1) minors of the generic matrix (X_{ij}) over O(M_{m,n}). (This ideal is zero if t ≥ min{m, n}.)

Theorem 0.1. The ring of invariants O(V)^{GL_{t}} equals Im θ^{∗}. □

Theorem 0.2. The kernel of θ^{∗} is I_{t+1}. □

Theorems 0.1 and 0.2 are respectively known as the First Fundamental Theorem of Invariant Theory and the Second Fundamental Theorem of Invariant Theory (for GL_{t}). They give a complete description of O(V)^{GL_{t}}.

We denote by M_{m,n}^{≤t} the subvariety of M_{m,n} of m × n matrices with rank at most t. This variety is just the image of the morphism θ, and so we can factor θ in the form

M_{m,t} × M_{t,n} ⟶^{μ} M_{m,n}^{≤t} ⟶^{⊆} M_{m,n}.

COINVARIANT THEORY FOR QUANTUM GL_{t} 3

Since the restriction map r : O(M_{m,n}) → O(M_{m,n}^{≤t}) is surjective, the comorphism μ^{∗} is injective and has the same image as θ^{∗}. Thus, Theorem 0.1 can be rephrased in the form O(V)^{GL_{t}} = Im μ^{∗}. Further, Theorem 0.2 shows that ker(r) = ker(θ^{∗}) = I_{t+1}, and therefore O(M_{m,n}^{≤t}) = O(M_{m,n})/I_{t+1}.

The proof of De Concini and Procesi for Theorem 0.1. We briefly describe the proof of Theorem 0.1 given by De Concini and Procesi in [1]; more precisely, we follow the exposition of that proof given in [8].

The general case can be easily reduced to that where t < min{m, n}. So, we restrict attention to that particular case. Let us fix some notation. We will denote by M_{m,n}^{≤t,∘} the open subset of M_{m,n}^{≤t} consisting of matrices with rank at most t whose upper leftmost t × t minor is invertible. In a similar way, M_{m,t}^{∘} denotes the set of m × t matrices whose uppermost t × t minor is invertible, and M_{t,n}^{∘} denotes the set of t × n matrices whose leftmost t × t minor is invertible. Finally, we set V^{∘} = M_{m,t}^{∘} × M_{t,n}^{∘}. Clearly the action γ of GL_{t} on V restricts to an action γ^{∘} on V^{∘} and we can ask for the invariants of that restricted action. It turns out that they are easy to compute.

Let us adopt the convention that if (A, B) is an element of V = M_{m,t} × M_{t,n} we write A and B in the forms

A = [ \begin{matrix}A_{0} \\ A_{1} \end{matrix} ]  and  B = [ B_{0} B_{1} ]

with A_{0}, B_{0} ∈ M_{t} while A_{1} ∈ M_{m−t,t} and B_{1} ∈ M_{t,n−t}.

With these notations, we define a morphism of varieties:

\xymatrix{V^{∘} \ar[r]^{i} & M_{m,n}^{≤t,∘} × GL_{t} \\ (A, B) \ar@{|->}[r] & (AB, B_{0}).}

It turns out that i is actually an isomorphism of varieties (this is an elementary fact, and the inverse morphism can be explicitly written down) and that the action of GL_{t} on M_{m,n}^{≤t,∘} × GL_{t} induced by γ^{∘} is just the natural one:

\xymatrix{GL_{t} × (M_{m,n}^{≤t,∘} × GL_{t}) \ar[r]^{ξ} & M_{m,n}^{≤t,∘} × GL_{t} \\ (g, C, h) \ar@{|->}[r] & (C, gh).}

The invariants for ξ are easy to compute, using for instance the coinvariants of the associated O(GL_{t})-comodule structure on O(M_{m,n}^{≤t,∘} × GL_{t}) (see the quantum case below). One then proves that O(M_{m,n}^{≤t,∘} × GL_{t})^{GL_{t}} = O(M_{m,n}^{≤t,∘}) ⊆ O(M_{m,n}^{≤t,∘} × GL_{t}).

To recover from this the invariants for γ^{∘} it is enough to use the comorphism i^{∗} of i. Let us consider the morphism

μ^{∘} : M_{m,t}^{∘} × M_{t,n}^{∘} = V^{∘} ⟶ M_{m,n}^{≤t,∘}

given by multiplication of matrices, and denote by (μ^{∘})^{∗} its comorphism:

(μ^{∘})^{∗} : O(M_{m,n}^{≤t,∘}) ⟶ O(V^{∘}) = O(M_{m,t}^{∘}) ⊗ O(M_{t,n}^{∘}).

Then we can describe i^{∗} as the composition

i^{∗} : O(M_{m,n}^{≤t,∘}) ⊗ O(GL_{t}) ⟶^{(μ^{∘})^{∗}⊗id} O(V^{∘}) ⊗ O(GL_{t}) ⟶^{id ⊗⊆} O(V^{∘}) ⊗ O(M_{t,n}^{∘})

⟶^{=} O(M_{m,t}^{∘}) ⊗ O(M_{t,n}^{∘}) ⊗ O(M_{t,n}^{∘}) ⟶^{id ⊗\underline{m}} O(V^{∘}),

where \underline{m} denotes multiplication in the algebra O(M_{t,n}^{∘}). From the above we get that the ring of invariants for the action γ^{∘} is just i^{∗}(O(M_{m,n}^{≤t,∘}) ⊗ 1) = (μ^{∘})^{∗}(O(M_{m,n}^{≤t,∘})).

The ring of invariants for γ^{∘} can thus be described as the localisation of Im μ^{∗} with respect to the multiplicative set generated by d_{Y} ⊗ d_{Z} where d_{Y} is the uppermost t × t minor of the generic matrix (Y_{ij}) of generators of O(M_{m,t}), and d_{Z} is the leftmost t × t minor of the generic matrix (Z_{ij}) of generators of O(M_{t,n}).

The second step of the proof is to show that one can “remove denominators” to deduce invariants for the action γ from invariants for the localised action γ^{∘}. Indeed, if ϕ is an invariant function in O(V) (i.e., for the action γ) then of course it is an invariant function in the localised ring O(V^{∘}) = O(V)_{d_{Y} ⊗d_{Z}} (for the action γ^{∘}). Hence, there is a non-negative integer s such that ϕ(d_{Y} ⊗ d_{Z})^{s} ∈ Im μ^{∗}. Thus, in order to establish that O(V)^{GL_{t}} = Im μ^{∗}, it is enough to prove that for ψ ∈ O(V), if there is a non-negative integer s such that ψ(d_{Y} ⊗ d_{Z})^{s} ∈ Im μ^{∗}, then ψ ∈ Im μ^{∗}. This last result is proved using the theory of standard bases.

In the quantum situation, the lower left quantum minors play a special rôle, since they are normal elements. It is for this reason that, unlike in the commutative case, we will invert leftmost lower t × t minors instead of leftmost upper ones.

Throughout the paper, we work over an arbitrary base field K and make an arbitrary choice of a nonzero element q ∈ K. We will have to deal with the following four quantized coordinate rings: O_{q}(M_{m,n}), O_{q}(M_{m,t}), O_{q}(M_{t,n}) and O_{q}(GL_{t}). In order to avoid confusion, we will denote their respective canonical generators by X_{ij}, Y_{ij}, Z_{ij} and T_{ij}. The definitions of these algebras will be as in [6, 2]; for instance, X_{ij}X_{ik} = qX_{ik}X_{ij} when j < k. Thus, each time we use results from [7] we must replace q by q^{−1}. Finally, a convention concerning notation: each time that we have to deal with the multiplication map in an algebra, we will denote it by \underline{m}. The context will make clear which algebra is concerned.

1. The general setup

Below, we will have to deal with the following situation combining a right and a left comodule algebra. Let (H, \underline{m}, η, Δ, ε, S) be a Hopf algebra, (A, ρ^{∗}) a right comodule algebra over H, and (B, λ^{∗}) a left comodule algebra over H. Here,

ρ^{∗} : A ⟶ A ⊗ H  and  λ^{∗} : B ⟶ H ⊗ B

are the comodule structure maps; that A and B are comodule algebras means that ρ^{∗} and λ^{∗} are also algebra morphisms. It is well known that A can be turned into a left H-comod-ule using the structure map A ⟶^{ρ^{∗}} A ⊗ H ⟶^{id ⊗S} A ⊗ H ⟶^{τ_{12}} H ⊗ A, where τ_{12} is the flip.

Thus, A ⊗ B can be equipped with the structure of a left H-comodule via the following structure map:

γ^{∗} : A ⊗ B ⟶^{ρ^{∗}⊗λ^{∗}} A ⊗ H ⊗ H ⊗ B ⟶^{id ⊗S⊗id ⊗ id} A ⊗ H ⊗ H ⊗ B

⟶^{τ_{(132)}} H ⊗ H ⊗ A ⊗ B ⟶^{\underline{m}⊗id ⊗ id} H ⊗ A ⊗ B,

where τ_{(132)} is the isomorphism which permutes the factors according to the cycle (132), that is, τ_{(132)}(a ⊗ h ⊗ h′ ⊗ b) = h ⊗ h′ ⊗ a ⊗ b. Using the standard comodule notations ρ^{∗}(a) = ∑_{(a)} a_{0} ⊗ a_{1} and λ^{∗}(b) = ∑_{(b)} b_{−1} ⊗ b_{0} for a ∈ A and b ∈ B (cf. [5, p. 11]), one thus has

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

Of course, (A ⊗ B, γ^{∗}) is not a comodule algebra any longer. Nevertheless, this comodule continues to have nice properties that are now described. Let us recall that if (M, ν^{∗}) is a left H-comodule, then the set of H-coinvariants of (M, ν^{∗}) (or ν^{∗}-coinvariants for short) is the sub-comodule of M defined by M^{co H} := {x ∈ M | ν^{∗}(x) = 1 ⊗ x}. It is immediate that if (M, ν^{∗}) is a comodule algebra then M^{co H} is a subalgebra of M. In the more general situation described above this property is not automatic but still true.

Proposition 1.1. In the above notation:

(a) If v ∈ A ⊗ B is such that γ^{∗}(v) = z ⊗ v for some central element z ∈ H, and if w ∈ A ⊗ B is any element, then γ^{∗}(vw) = γ^{∗}(v)γ^{∗}(w). In particular, this holds when v is a γ^{∗}-coinvariant.

(b) If v, w ∈ A ⊗ B are γ^{∗}-coinvariants, then vw is again a γ^{∗}-coinvariant.

(c) The set (A ⊗ B)^{co H} is a subalgebra of A ⊗ B.

Proof. Without loss of generality, we may assume that w = a′ ⊗ b′ is a pure tensor. Moreover, let us write v = ∑_{i=1}^{r} a_{i} ⊗ b_{i}. Since both ρ^{∗} and λ^{∗} are algebra morphisms, we have

(ρ^{∗} ⊗ λ^{∗})(vw) = ∑_{i=1}^{r} ∑_{(a_{i}),(b_{i})} ∑_{(a′),(b′)} a_{i,0}a′_{0} ⊗ a_{i,1}a′_{1} ⊗ b_{i,−1}b′_{−1} ⊗ b_{i,0}b′_{0},

where a_{i,0} = (a_{i})_{0} etc. Hence,

(γ^{∗})(vw) = ∑_{i=1}^{r} ∑_{(a_{i}),(b_{i})} ∑_{(a′),(b′)} S(a_{i,1}a′_{1})b_{i,−1}b′_{−1} ⊗ a_{i,0}a′_{0} ⊗ b_{i,0}b′_{0}

= ∑_{i=1}^{r} ∑_{(a_{i}),(b_{i})} ∑_{(a′),(b′)} S(a′_{1})S(a_{i,1})b_{i,−1}b′_{−1} ⊗ a_{i,0}a′_{0} ⊗ b_{i,0}b′_{0}

= ∑_{(a′),(b′)}(S(a′_{1}) ⊗ 1 ⊗ 1)(∑_{i=1}^{r} ∑_{(a_{i}),(b_{i})} S(a_{i,1})b_{i,−1} ⊗ a_{i,0} ⊗ b_{i,0})(b′_{−1} ⊗ a′_{0} ⊗ b′_{0})

= ∑_{(a′),(b′)}(S(a′_{1}) ⊗ 1 ⊗ 1)γ^{∗}(v)(b′_{−1} ⊗ a′_{0} ⊗ b′_{0}).

But, since γ^{∗}(v) = z ⊗ v and z is central, we have

(γ^{∗})(vw) = ∑_{(a′),(b′)}(S(a′_{1}) ⊗ 1 ⊗ 1)(z ⊗ v)(b′_{−1} ⊗ a′_{0} ⊗ b′_{0})

= (z ⊗ v) ∑_{(a′),(b′)} S(a′_{1})b′_{−1} ⊗ a′_{0} ⊗ b′_{0} = γ^{∗}(v)γ^{∗}(w).

This proves statement (a). Clearly, (b) as well as (c) follow at once from (a). □

We now record a special case of a coaction for which computing the coinvariants is easy. Again, (H, \underline{m}, η, Δ, ε, S) is a Hopf algebra, and M denotes a vector space. Clearly, the map

\xymatrix{M \ar[r] & H ⊗ M \\ a \ar@{|->}[r] & 1 ⊗ a}

defines a (trivial) left coaction of H on M. Moreover, the comultiplication Δ of H makes H itself into a left H-comodule. The tensor product of these two coactions gives a left coaction of H on M ⊗ H:

\xymatrix{M ⊗ H \ar[r]^{ξ^{∗}} & H ⊗ M ⊗ H \\ a ⊗ h \ar@{|->}[r] & ∑_{(h)} h_{1} ⊗ a ⊗ h_{2}.}

Lemma 1.2. With the above notation, (M ⊗ H)^{co H} = M ⊗ 1.

Proof. Consider x ∈ (M ⊗ H)^{co H}, and write x = ∑_{i=1}^{s} a_{i} ⊗ h_{i} where the a_{i} are linearly independent. Now

∑_{i=1}^{s} ∑_{(h_{i})} h_{i,1} ⊗ a_{i} ⊗ h_{i,2} = ξ^{∗}(x) = 1 ⊗ x = ∑_{i=1}^{s} 1 ⊗ a_{i} ⊗ h_{i}.

Since the a_{i} are linearly independent, we obtain

Δ(h_{i}) = ∑_{(h_{i})} h_{i,1} ⊗ h_{i,2} = 1 ⊗ h_{i}

for each i, whence h_{i} = (id ⊗ε)Δ(h_{i}) = 1·ε(h_{i}). This shows that the h_{i} are scalars, and so x ∈ M ⊗ 1. Thus, (M ⊗ H)^{co H} ⊆ M ⊗ 1. The reverse inclusion is obvious. □

2. Quantization of the standard action of GL_{t} on V

As in the introduction, we fix our base field K and positive integers m, n, t. By analogy with the commutative situation, we put

O_{q}(V) = O_{q}(V_{m,t,n}) := O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}).

It is easily checked that O_{q}(V) is an iterated skew polynomial extension of the base field K, and so O_{q}(V) is a noetherian domain. A quantum analogue of the action γ is obtained as a coaction of O_{q}(GL_{t}) on O_{q}(V) that we now describe.

It is easy to check that one can define morphisms of algebras satisfying the following rules:

\xymatrix{O_{q}(M_{m,t}) \ar[r]^{ρ_{q}^{∗}} & O_{q}(M_{m,t}) ⊗ O_{q}(GL_{t}) & O_{q}(M_{t,n}) \ar[r]^{λ_{q}^{∗}} & O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}) \\ Y_{ij} \ar@{|->}[r] & ∑_{k=1}^{t} Y_{ik} ⊗ T_{kj} & Z_{ij} \ar@{|->}[r] & ∑_{k=1}^{t} T_{ik} ⊗ Z_{kj}.}

Moreover, ρ_{q}^{∗} endows O_{q}(M_{m,t}) with a right O_{q}(GL_{t})-comodule algebra structure, while λ_{q}^{∗} endows O_{q}(M_{t,n}) with a left O_{q}(GL_{t})-comodule algebra structure. It follows that τ_{12}∘(id ⊗S)∘ρ_{q}^{∗} gives O_{q}(M_{m,t}) the structure of a left O_{q}(GL_{t})-comodule, where S denotes the antipode in O_{q}(GL_{t}). The tensor product of these two left coactions allows us to define a left comodule structure on O_{q}(V) that we denote by γ_{q}^{∗}. So, as in Section 1,

γ_{q}^{∗} : O_{q}(V) ⟶ O_{q}(GL_{t}) ⊗ O_{q}(V)

is given by the rule

γ_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}

for a ∈ O_{q}(M_{m,t}) and b ∈ O_{q}(M_{t,n}).

The main objective of this paper is to calculate O_{q}(V)^{co O_{q}(GL_{t})}, the set of coinvariants of γ_{q}^{∗}. As we see by Proposition 1.1, this set is a subalgebra of O_{q}(V).

Quantum minors. Recall that quantum minors in a quantum matrix algebra O_{q}(M_{u,v}) correspond to quantum determinants in subalgebras of O_{q}(M_{u,v}). More precisely, if 1 ≤ r_{1} < r_{2} < ⋯ < r_{l} ≤ u and 1 ≤ c_{1} < c_{2} < ⋯ < c_{l} ≤ v, then the subalgebra of O_{q}(M_{u,v}) generated by {X_{r_{i}c_{j}} | 1 ≤ i, j ≤ l} is naturally isomorphic to O_{q}(M_{l}), and the element of this subalgebra corresponding to the quantum determinant in O_{q}(M_{l}) is called the quantum minor of O_{q}(M_{u,v}) corresponding to the rows r_{1}, …, r_{l} and columns c_{1}, …, c_{l} (see [6, §1.2; 7, §4.3]). Since we require a number of the formulas and results developed in [2], we shall use the notation of that paper for quantum minors. Thus, the l × l quantum minor described above will be denoted by the symbol

[r_{l}r_{l−1} ⋯ r_{1} | c_{1}c_{2} ⋯ c_{l}],

or by [R|C] where R and C denote the sets {r_{1}, …, r_{l}} and {c_{1}, …, c_{l}}, respectively.

Remark 2.1. (i) The ideal of O_{q}(M_{m,n}) generated by all the (t + 1) × (t + 1) quantum minors will be denoted I_{t+1}, or I_{t+1}^{m,n} when m and n require emphasis. (In case t ≥ min{m, n}, there are no (t + 1) × (t + 1) quantum minors in O_{q}(M_{m,n}), and I_{t+1} = 0.) Moreover, we put

O_{q}(M_{m,n}^{≤t}) := O_{q}(M_{m,n})/I_{t+1}^{m,n}

and x_{ij} := X_{ij} + I_{t+1} ∈ O_{q}(M_{m,n}^{≤t}).

(ii) For any m′ ≥ m and n′ ≥ n, we will identify O_{q}(M_{m,n}) with the subalgebra of O_{q}(M_{m′,n′}) generated by those X_{ij} with i ≤ m and j ≤ n. Now there is a K-algebra retraction

π_{m,n} = π_{m,n}^{m′,n′} : O_{q}(M_{m′,n′}) ⟶ O_{q}(M_{m,n})

such that π_{m,n}(X_{ij}) = X_{ij} for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and π_{m,n}(X_{ij}) = 0 for i > m or j > n. It is worth mentioning here that the inclusion map O_{q}(M_{m,n}) → O_{q}(M_{m′,n′}) (corresponding to the identification above) provides a section for π_{m,n}; this fact will be used without further comment. Note that π_{m,n}[I|J] = [I|J] when I ⊆ {1, …, m} and J ⊆ {1, …, n}, while π_{m,n}[I|J] = 0 otherwise. It follows that π_{m,n}(I_{t+1}^{m′,n′}) = I_{t+1}^{m,n}, and consequently I_{t+1}^{m,n} = I_{t+1}^{m′,n′} ∩ O_{q}(M_{m,n}).

(iii) As is easily checked, there is a morphism of algebras

\xymatrix{O_{q}(M_{m,n}) \ar[r]^{θ_{q}^{∗}} & O_{q}(V) \\ X_{ij} \ar@{|->}[r] & ∑_{k=1}^{t} Y_{ik} ⊗ Z_{kj}.}

For instance, this can be checked by using the commutative diagram

\xymatrix{O_{q}(M_{l}) \ar[r]^{Δ} & O_{q}(M_{l}) ⊗ O_{q}(M_{l}) \ar[d]^{π_{m,t}⊗π_{t,n}} \\ O_{q}(M_{m,n}) \ar[r]^{θ_{q}^{∗}} \ar[u]^{⊆} & O_{q}(V)}

where l ≥ max{m, n} and Δ denotes the comultiplication of O_{q}(M_{l}).

(iv) The comultiplication rule for quantum minors in a square quantum matrix algebra, say O_{q}(M_{l}), states that

Δ[I|J] = ∑_{\begin{matrix}L⊆\{1,...,l\} \\ |L|=|I| \end{matrix}}[I|L] ⊗ [L|J]

for all quantum minors [I|J] ∈ O_{q}(M_{l}) (e.g., [6, (1.9)]).

(v) From points (ii), (iii) and (iv), it is clear that I_{t+1} is contained in the kernel of θ_{q}^{∗}. Hence, there is an induced morphism of algebras

μ_{q}^{∗} : O_{q}(M_{m,n}^{≤t}) ⟶ O_{q}(V)

such that μ_{q}^{∗}(x_{ij}) = ∑_{k=1}^{t} Y_{ik} ⊗ Z_{kj}. The work of the first and second authors in [2] leads to the following result:

Theorem 2.2. In the above notation, the morphism μ_{q}^{∗} is an injection. In particular, O_{q}(M_{m,n}^{≤t}) is a domain.

Proof. We must show that ker(θ_{q}^{∗}) = I_{t+1}. In view of Remark 2.1(ii), there is no loss of generality in assuming that m = n > t. That case is proved in [2, Proposition 2.4]. □

Clearly, Theorem 2.2 gives a quantum analogue of the Second Fundamental Theorem of Invariant Theory; see Theorem 0.2. The rest of this work is devoted to proving a quantum analogue of the First Fundamental Theorem of Invariant Theory; see Theorem 0.1. A first easy proposition is in order.

Proposition 2.3. The set of coinvariants O_{q}(V)^{co O_{q}(GL_{t})} is a subalgebra of O_{q}(V) containing Im μ_{q}^{∗}.

Proof. We are in the setup defined at the begining of Section 1, so we may apply Proposition 1.1(c) to see that O_{q}(V)^{co O_{q}(GL_{t})} is a subalgebra of O_{q}(V). For the rest of the statement, since the map μ_{q}^{∗} is a morphism of algebras, it is enough to prove that μ_{q}^{∗}(x_{ij}) is a γ_{q}^{∗}-coinvariant, for 1 ≤ i ≤ m and 1 ≤ j ≤ n. However,

γ_{q}^{∗}(μ_{q}^{∗}(x_{ij})) = ∑_{k=1}^{t} γ_{q}^{∗}(Y_{ik} ⊗ Z_{kj}) = ∑_{r,s,k=1}^{t} S(T_{rk})T_{ks} ⊗ Y_{ir} ⊗ Z_{sj}

= ∑_{r,s=1}^{t} ε(T_{rs}) ⊗ Y_{ir} ⊗ Z_{sj} = ∑_{r=1}^{t} 1 ⊗ Y_{ir} ⊗ Z_{rj} = 1 ⊗ μ_{q}^{∗}(x_{ij}). □

Remark 2.4. By using the retraction maps π_{•,•} discussed in Remark 2.1(ii), most results proved for quantized coordinate rings of square matrices immediately carry over to the rectangular case, as in the proof of Theorem 2.2. For the purposes of the present paper, rather than carry over a large number of results in this way, it is more efficient to check that it suffices to prove our main theorem in the case m = n. This also allows us to assume that t < n, as follows.

Choose an integer l ≥ max{m, n} (later, it will be helpful to take l > max{m, n}), and observe that we have a commutative diagram as follows:

[figure: O_{q}(M_{m,n}^{≤t}); μ_{q}^{∗}; ⊆; O_{q}(M_{m,n}^{≤t}); μ_{q}^{∗}; O_{q}(M_{l}^{≤t}); μ_{q}^{∗}; _{\overline{π}_{m,n}}; O_{q}(V_{m,t,n}); γ_{q}^{∗}; ⊆; O_{q}(V_{m,t,n}); γ_{q}^{∗}; O_{q}(V_{l,t,l}); γ_{q}^{∗}; π_{m,t}⊗π_{t,n}; O_{q}(GL_{t}) ⊗ O_{q}(V_{m,t,n}); id ⊗⊆; O_{q}(GL_{t}) ⊗ O_{q}(V_{m,t,n}); O_{q}(GL_{t}) ⊗ O_{q}(V_{l,t,l}); id ⊗π_{m,t}⊗π_{t,n}]

Here ‘⊆’ is shorthand for various embeddings induced by identifications (cf. Remark 2.1(ii)), and \overline{π}_{m,n} denotes the map induced by the retraction π_{m,n} : O_{q}(M_{l}) → O_{q}(M_{m,n}) (recall from Remark 2.1(ii) that π_{m,n}(I_{t+1}^{l,l}) = I_{t+1}^{m,n}). It is clear from the above diagram that

O_{q}(V_{l,t,l})^{co O_{q}(GL_{t})} = μ_{q}^{∗}(O_{q}(M_{l}^{≤t})) ⟹ O_{q}(V_{m,t,n})^{co O_{q}(GL_{t})} = μ_{q}^{∗}(O_{q}(M_{m,n}^{≤t})).

Hence, to prove our main theorem it suffices to consider the case (m = n > t).

3. A localisation of γ_{q}^{∗}

In view of Remark 2.4, we assume, until further notice, that m = n > t. Certain notations contract slightly in this case; e.g., O_{q}(M_{m,n}^{≤t}) becomes O_{q}(M_{n}^{≤t}).

Remark 3.1. (i) We label the following t × t quantum minors:

d_{X} := [n, …, n − t + 1 | 1, …, t] ∈ O_{q}(M_{n})
d_{Y} := [n, …, n − t + 1 | 1, …, t] ∈ O_{q}(M_{n,t})
d_{Z} := [t, …, 1 | 1, …, t] ∈ O_{q}(M_{t,n})
d_{T} := [t, …, 1 | 1, …, t] ∈ O_{q}(GL_{t}).

Further, we set d_{x} = d_{X} + I_{t+1} ∈ O_{q}(M_{n}^{≤t}). It is known that d_{X} ∉ I_{t+1} (for instance, it is clear from Remark 3.2(i) below that θ_{q}^{∗}(d_{X}) ≠ 0), and so d_{x} ≠ 0.

(ii) It is well known that in any O_{q}(M_{u,v}), the “lower left” quantum minors – i.e., those corresponding to the lowest l rows and the leftmost l columns, for any l – are normal elements (e.g., see [2, Corollary 5.2]). In particular, the quantum minors d_{X}, d_{Y}, d_{Z} are normal in O_{q}(M_{n}), O_{q}(M_{n,t}), O_{q}(M_{t,n}), respectively. Since d_{T} is the quantum determinant in O_{q}(GL_{t}), it is central. It follows from the normality of d_{X} that d_{x} is normal in O_{q}(M_{n}^{≤t}).

(iii) Since d_{Y} is a normal element in O_{q}(M_{n,t}), we can consider the localisation of O_{q}(M_{n,t}) with respect to the multiplicative set generated by d_{Y}. By analogy with the commutative case, we denote this localisation, O_{q}(M_{n,t})[d_{Y}^{−1}], by O_{q}(M_{n,t}^{∘}). Similarly, the localisations O_{q}(M_{t,n})[d_{Z}^{−1}] and O_{q}(M_{n}^{≤t})[d_{x}^{−1}] will be denoted O_{q}(M_{t,n}^{∘}) and O_{q}(M_{n}^{≤t,∘}), respectively. At this stage it is worth mentioning that

O_{q}(M_{n}^{≤t,∘}) ≅ (O_{q}(M_{n})[d_{X}^{−1}])/(I_{t+1}[d_{X}^{−1}]);

this is an easy consequence of the universal properties of quotients and localisations. We will use this isomorphism without further comment.

Remark 3.2. (i) It follows from Remark 2.1(ii,iv) that in the notation of Section 2, one has

ρ_{q}^{∗}(d_{Y}) = d_{Y} ⊗ d_{T}  and  λ_{q}^{∗}(d_{Z}) = d_{T} ⊗ d_{Z}.

Further, θ_{q}^{∗}(d_{X}) = d_{Y} ⊗ d_{Z}, and so

μ_{q}^{∗}(d_{x}) = d_{Y} ⊗ d_{Z}.

(ii) In view of Remark 3.1(iii) and point (i) above, ρ_{q}^{∗} and λ_{q}^{∗} extend uniquely to morphisms of algebras

ρ_{q}^{∗∘} : O_{q}(M_{n,t}^{∘}) ⟶ O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t})
λ_{q}^{∗∘} : O_{q}(M_{t,n}^{∘}) ⟶ O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘}).

(iii) Clearly, λ_{q}^{∗∘} defines a left O_{q}(GL_{t})-comodule algebra structure on O_{q}(M_{t,n}^{∘}), while ρ_{q}^{∗∘} defines a right O_{q}(GL_{t})-comodule algebra structure on O_{q}(M_{n,t}^{∘}). Again, the composition τ_{12} ∘ (id ⊗S) ∘ ρ_{q}^{∗∘} defines a left O_{q}(GL_{t})-comodule structure on O_{q}(M_{n,t}^{∘}). Finally, if we set

O_{q}(V^{∘}) := O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘})

and tensor the above two left coactions, we obtain a map

γ_{q}^{∗∘} : O_{q}(V^{∘}) ⟶ O_{q}(GL_{t}) ⊗ O_{q}(V^{∘}),

endowing O_{q}(V^{∘}) with the structure of a left O_{q}(GL_{t})-comodule. The map γ_{q}^{∗∘} follows the same formula as γ_{q}^{∗}; namely,

γ_{q}^{∗∘}(a ⊗ b) = ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}

for a ∈ O_{q}(M_{n,t}^{∘}) and b ∈ O_{q}(M_{t,n}^{∘}). Obviously, the restriction of γ_{q}^{∗∘} to O_{q}(V) is just γ_{q}^{∗}.

(iv) Recall from Theorem 2.2 that the morphism μ_{q}^{∗} : O_{q}(M_{n}^{≤t}) ⟶ O_{q}(V) is injective. Because of Remark 3.1(iii) and point (i) above, we can extend μ_{q}^{∗} to the corresponding localised algebras and obtain another injective morphism of algebras:

μ_{q}^{∗∘} : O_{q}(M_{n}^{≤t,∘}) ⟶ O_{q}(V^{∘}).

We shall need a technical lemma. We have already mentioned that γ_{q}^{∗} is not an algebra morphism, and so neither is γ_{q}^{∗∘}. Nevertheless, we have the following statement, the first part of which is a special case of Proposition 1.1.

Lemma 3.3. The set of coinvariants O_{q}(V^{∘})^{co O_{q}(GL_{t})} is a subalgebra of O_{q}(V^{∘}), containing Im μ_{q}^{∗∘}.

Proof. As noted, Proposition 1.1(c) implies that O_{q}(V^{∘})^{co O_{q}(GL_{t})} is a subalgebra of O_{q}(V^{∘}).

Let us observe that for r ∈ ℕ, the element (d_{Y} ⊗ d_{Z})^{−r} is a coinvariant for γ_{q}^{∗∘}. Indeed, since both ρ_{q}^{∗∘} and λ_{q}^{∗∘} are morphism of algebras, we have ρ_{q}^{∗∘}(d_{Y}^{−r}) = d_{Y}^{−r} ⊗ d_{T}^{−r} and λ_{q}^{∗∘}(d_{Z}^{−r}) = d_{T}^{−r} ⊗ d_{Z}^{−r}. Recall that S(d_{T}) = d_{T}^{−1} [6, p. 40; 7, proof of Theorem 5.3.2]. Hence,

γ_{q}^{∗∘}((d_{Y} ⊗ d_{Z})^{−r}) = γ_{q}^{∗∘}(d_{Y}^{−r} ⊗ d_{Z}^{−r}) = S(d_{T}^{−r})d_{T}^{−r} ⊗ d_{Y}^{−r} ⊗ d_{Z}^{−r}
= d_{T}^{r} d_{T}^{−r} ⊗ d_{Y}^{−r} ⊗ d_{Z}^{−r} = 1 ⊗ d_{Y}^{−r} ⊗ d_{Z}^{−r}.

This proves the claim. Now since μ_{q}^{∗∘} is a morphism of algebras, any element c ∈ Im μ_{q}^{∗∘} has the form c = a(d_{Y} ⊗ d_{Z})^{−r} for some a ∈ Im μ_{q}^{∗} and r ∈ ℕ. So, from the observation above, the fact that Im μ_{q}^{∗} ⊆ O_{q}(V)^{co O_{q}(GL_{t})} (see Proposition 2.3), and the fact that O_{q}(V^{∘})^{co O_{q}(GL_{t})} is a subalgebra of O_{q}(V^{∘}), we conclude that c ∈ O_{q}(V^{∘})^{co O_{q}(GL_{t})}. □

Coaction of O_{q}(GL_{t}) on O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}). One key point in the commutative case is that the action of GL_{t} on V^{∘} turns out to identify with the natural action of GL_{t} on M_{n}^{≤t,∘} ×GL_{t} by left translation on the second factor. Here, we define and study a quantum analogue for the latter action.

We can apply the situation described at the end of Section 1 with M = O_{q}(M_{n}^{≤t,∘}) and H = O_{q}(GL_{t}). We thus obtain a left O_{q}(GL_{t})-comodule structure on O_{q}(M_{n}^{≤t,∘})⊗O_{q}(GL_{t}) via:

\xymatrix{O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) \ar[r]^{ξ_{q}^{∗}} & O_{q}(GL_{t}) ⊗ O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) \\ x ⊗ h \ar@{|->}[r] & ∑_{(h)} h_{1} ⊗ x ⊗ h_{2}.}

According to what we proved in Lemma 1.2, we have the following statement:

Proposition 3.4. The set of ξ_{q}^{∗}-coinvariants is a subalgebra of O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}). More precisely,

(O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}))^{co O_{q}(GL_{t})} = O_{q}(M_{n}^{≤t,∘}) ⊗ 1. □

Link between the coactions γ_{q}^{∗∘} and ξ_{q}^{∗}. In light of the commutative case, one expects that O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) and O_{q}(V^{∘}) are isomorphic as O_{q}(GL_{t})-comodules. This is, in fact, true, although we do not need the full statement here, only that O_{q}(V^{∘}) is a comodule retraction of O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}). The morphism that appears as the most natural candidate for an isomorphism between these two comodules is a quantum analog of i^{∗} that we describe below.

Remark 3.5. (i) The following injective homomorphisms of algebras will be of constant use in the sequel. One should pay attention to the fact that, due to the choice of the minors we have inverted, these are not the usual maps one might expect to use. These embeddings are all into the lower left corner of the target algebra, in order to make use of the normality of the elements d_{Y} etc. We denote all these morphisms by “inj” in order to avoid introducing too much notation; the context will make clear which one we are using.

\xymatrix{ & T_{ij} \ar@{|->}[r] & Y_{n−t+i,j} \\ T_{ij} \ar@{|->}[d] & O_{q}(GL_{t}) \ar[r]^{inj} \ar[d]_{inj} & O_{q}(M_{n,t}^{∘}) \ar[d]^{inj} & Y_{ij} \ar@{|->}[d] \\ Z_{ij} & O_{q}(M_{t,n}^{∘}) \ar[r]^{inj} & O_{q}(M_{n}^{≤t,∘}) & x_{ij} \\  & Z_{ij} \ar@{|->}[r] & x_{n−t+i,j}}

The injectivity of these maps is easy to prove, since in each case we can exhibit a left inverse. We show how to deal with the case of the left hand one; similar arguments hold for the others. The retraction π_{t,t} (see Remark 2.1(ii)) induces a morphism of algebras O_{q}(M_{t,n}^{∘}) ⟶ O_{q}(GL_{t}) that maps Z_{ij} to T_{ij} if j ≤ t and to 0 if j > t. Clearly, this map provides a left inverse for inj.

(ii) Identify O_{q}(M_{n,t}^{∘}) (respectively, O_{q}(M_{t,n}^{∘})) with its image in O_{q}(M_{n}^{≤t,∘}). Then, the restriction of μ_{q}^{∗∘} to O_{q}(M_{n,t}^{∘}) (respectively, O_{q}(M_{t,n}^{∘})) coincides with ρ_{q}^{∗∘} (respectively, λ_{q}^{∗∘}), as one can easily check on generators (this is enough since the maps involved are morphism of algebras). More precisely, we are claiming that the following diagram is commutative:

\xymatrix{O_{q}(M_{n,t}^{∘}) \ar[r]^{inj} \ar[d]_{ρ_{q}^{∗∘}} & O_{q}(M_{n}^{≤t,∘}) \ar[d]^{μ_{q}^{∗∘}} & O_{q}(M_{t,n}^{∘}) \ar[l]_{inj} \ar[d]^{λ_{q}^{∗∘}} \\ O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t}) \ar[r]^{id ⊗ inj} & O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘}) & O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘}) \ar[l]_{inj ⊗ id}}

For example, to see that the left hand square commutes, it suffices to check that μ_{q}^{∗∘} ∘ inj and (id ⊗ inj) ∘ ρ_{q}^{∗∘} agree on the Y_{ij} and on d_{Y}. This is clear – both maps send Y_{ij} to ∑_{k=1}^{t} Y_{ik} ⊗ Z_{kj} and d_{Y} to d_{Y} ⊗ d_{Z}. The commutativity of the right hand square is proved analogously.

(iii) Finally, the restriction of λ_{q}^{∗∘} (respectively, μ_{q}^{∗∘}) to O_{q}(GL_{t}) coincides with Δ, i.e., the following diagram is commutative:

\xymatrix{O_{q}(GL_{t}) \ar[r]^{inj} \ar[d]_{Δ} & O_{q}(M_{t,n}^{∘}) \ar[r]^{inj} \ar[d]^{λ_{q}^{∗∘}} & O_{q}(M_{n}^{≤t,∘}) \ar[d]^{μ_{q}^{∗∘}} \\ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) \ar[r]^{id ⊗ inj} & O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘}) \ar[r]^{inj ⊗ id} & O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘})}

In the following, we will identify the algebras O_{q}(GL_{t}), O_{q}(M_{n,t}^{∘}), O_{q}(M_{t,n}^{∘}) with their images under inj and we will use the commutativity of the above diagrams without explicit mention.

Quantization of i^{∗}. Due to our identification of O_{q}(GL_{t}) with a subalgebra of O_{q}(M_{t,n}^{∘}), we can use the multiplication map \underline{m} on O_{q}(M_{t,n}^{∘}) to endow O_{q}(M_{t,n}^{∘}) with the structure of a right O_{q}(GL_{t})-module. We now consider the following quantum analogue of i^{∗}:

i_{q}^{∗} : O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) ⟶^{μ_{q}^{∗∘}⊗id} O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘}) ⊗ O_{q}(GL_{t})

⟶^{id ⊗\underline{m}} O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘}) = O_{q}(V^{∘}).

It can be proved that i_{q}^{∗} is an isomorphism of left O_{q}(GL_{t})-comodules, but we shall not need the full result – for our purposes, it suffices to construct a right inverse for i_{q}^{∗}.

Recall that the antipode S in O_{q}(GL_{t}) is bijective (e.g., [7, Theorem 5.4.2]).

Consider the map j_{q}^{∗} : O_{q}(V^{∘}) → O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) defined as the composition of the following maps:

O_{q}(V^{∘}) = O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(M_{t,n}^{∘}) ⟶^{ρ_{q}^{∗∘}⊗λ_{q}^{∗∘}} O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘})

⟶^{id ⊗ id ⊗Δ⊗id} O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘})

⟶^{id ⊗S⊗id ⊗S^{−1}⊗id} O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n}^{∘})

⟶^{τ_{(135)(24)}} O_{q}(M_{t,n}^{∘}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(M_{n,t}^{∘}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t})

⟶^{inj ⊗ inj ⊗ inj ⊗\underline{m}} O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t})

⟶^{\underline{m}(\underline{m}⊗id)⊗id} O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}).

Here τ_{(135)(24)} denotes the isomorphism which shuffles the factors via the permutation (135)(24), that is,

τ_{(135)(24)}(x_{1} ⊗ x_{2} ⊗ x_{3} ⊗ x_{4} ⊗ x_{5}) = x_{5} ⊗ x_{4} ⊗ x_{1} ⊗ x_{2} ⊗ x_{3}.

For a pure tensor a ⊗ b ∈ O_{q}(V^{∘}), we can give the following formula for j_{q}^{∗}(a ⊗ b) using the notation ρ_{q}^{∗}(a) = ∑_{(a)} a_{0} ⊗ a_{1} and (Δ ⊗ id)λ_{q}^{∗}(b) = ∑_{(b)} b_{−2} ⊗ b_{−1} ⊗ b_{0}. Namely,

j_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} b_{0}S^{−1}(b_{−1})a_{0} ⊗ S(a_{1})b_{−2}.

Lemma 3.6. The map j_{q}^{∗} is a morphism of left O_{q}(GL_{t})-comodules.

Proof. We must prove that the following diagram is commutative:

\xymatrix{O_{q}(V^{∘}) \ar[r]^{j_{q}^{∗}} \ar[d]_{γ_{q}^{∗∘}} & O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}) \ar[d]^{ξ_{q}^{∗}} \\ O_{q}(GL_{t}) ⊗ O_{q}(V^{∘}) \ar[r]^{id ⊗j_{q}^{∗}} & O_{q}(GL_{t}) ⊗ O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t})}

If a ∈ O_{q}(M_{n,t}^{∘}) and b ∈ O_{q}(M_{t,n}^{∘}), then (with the usual conventions for expanding components)

(id ⊗j_{q}^{∗})γ_{q}^{∗∘}(a ⊗ b) = ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ j_{q}^{∗}(a_{0} ⊗ b_{0})

= ∑_{(a),(b)} S(a_{2})b_{−3} ⊗ b_{0}S^{−1}(b_{−1})a_{0} ⊗ S(a_{1})b_{−2}.

Since Δ(S(c)d) = ∑_{(c),(d)} S(c_{2})d_{1} ⊗ S(c_{1})d_{2} for c, d ∈ O_{q}(GL_{t}), we also have

ξ_{q}^{∗}j_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} ξ_{q}^{∗}(b_{0}S^{−1}(b_{−1})a_{0} ⊗ S(a_{1})b_{−2})

= ∑_{(a),(b)} S(a_{2})b_{−3} ⊗ b_{0}S^{−1}(b_{−1})a_{0} ⊗ S(a_{1})b_{−2}.

Therefore (id ⊗j_{q}^{∗})γ_{q}^{∗∘}(a ⊗ b) = ξ_{q}^{∗}j_{q}^{∗}(a ⊗ b), as desired. □

In the sequel, we shall need the identity

∑_{(c)} c_{2}S^{−1}(c_{1}) = ∑_{(c)} S^{−1}(c_{2})c_{1} = ε(c) · 1

for c ∈ O_{q}(GL_{t}). This follows from the defining property of the antipode S by applying the antiautomorphism S^{−1}. Moreover, in view of the identity Δ ∘ S = τ_{12} ∘ (S ⊗ S) ∘ Δ (e.g., [5, Proposition 1.5.10]), we also have

(S^{−1} ⊗ S^{−1}) ∘ τ_{12} ∘ Δ = Δ ∘ S^{−1}.

Lemma 3.7. The composition i_{q}^{∗} ∘ j_{q}^{∗} is the identity map on O_{q}(V^{∘}).

Proof. Let a ⊗ b ∈ O_{q}(V^{∘}). Using Remark 3.5(ii,iii) and the fact that μ_{q}^{∗∘} is a morphism of algebras, we have

(μ_{q}^{∗∘} ⊗ id)j_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)}(μ_{q}^{∗∘}(b_{0})(μ_{q}^{∗∘}S^{−1}(b_{−1}))μ_{q}^{∗∘}(a_{0})) ⊗ S(a_{1})b_{−2}

= ∑_{(a),(b)}(λ_{q}^{∗∘}(b_{0})(ΔS^{−1}(b_{−1}))ρ_{q}^{∗∘}(a_{0})) ⊗ S(a_{1})b_{−2}

= ∑_{(a),(b)}((b_{−1} ⊗ b_{0})(ΔS^{−1}(b_{−2}))(a_{0} ⊗ a_{1})) ⊗ S(a_{2})b_{−3}.

Consequently, with the help of the identities above, we see that

(μ_{q}^{∗∘} ⊗ id)j_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} b_{−1}S^{−1}(b_{−2})a_{0} ⊗ b_{0}S^{−1}(b_{−3})a_{1} ⊗ S(a_{2})b_{−4}

= ∑_{(a),(b)} ε(b_{−1})a_{0} ⊗ b_{0}S^{−1}(b_{−2})a_{1} ⊗ S(a_{2})b_{−3}

= ∑_{(a),(b)} a_{0} ⊗ b_{0}S^{−1}(b_{−2}ε(b_{−1}))a_{1} ⊗ S(a_{2})b_{−3}

= ∑_{(a),(b)} a_{0} ⊗ b_{0}S^{−1}(b_{−1})a_{1} ⊗ S(a_{2})b_{−2}.

Thus,

i_{q}^{∗}j_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} a_{0} ⊗ b_{0}S^{−1}(b_{−1})a_{1}S(a_{2})b_{−2} = ∑_{(a),(b)} a_{0} ⊗ b_{0}S^{−1}(b_{−1})ε(a_{1})b_{−2}

= ∑_{(a),(b)} a_{0}ε(a_{1}) ⊗ b_{0}S^{−1}(b_{−1})b_{−2} = ∑_{(b)} a ⊗ b_{0}ε(b_{−1}) = a ⊗ b. □

Proposition 3.8. O_{q}(V^{∘})^{co O_{q}(GL_{t})} = Im μ_{q}^{∗∘}.

Proof. We have the inclusion “⊇” by Lemma 3.3. To prove the reverse inclusion, consider an arbitrary γ_{q}^{∗∘}-coinvariant c ∈ O_{q}(V^{∘}). Since j_{q}^{∗} is a comodule morphism (Lemma 3.6), j_{q}^{∗}(c) is a ξ_{q}^{∗}-coinvariant in O_{q}(M_{n}^{≤t,∘}) ⊗ O_{q}(GL_{t}). Hence, j_{q}^{∗}(c) ∈ O_{q}(M_{n}^{≤t,∘}) ⊗ 1 by Proposition 3.4. Since i_{q}^{∗}j_{q}^{∗}(c) = c by Lemma 3.7, and since clearly i_{q}^{∗}(O_{q}(M_{n}^{≤t,∘}) ⊗ 1) = Im μ_{q}^{∗∘}, we conclude that c ∈ Im μ_{q}^{∗∘}. This verifies the inclusion “⊆”. □

4. The First Fundamental Theorem for O_{q}(GL_{t})-coinvariants

In this section, we complete the proof of our main theorem – that the set of γ_{q}^{∗}-coin-variants in O_{q}(V) equals the image of μ_{q}^{∗}. We first discuss how to obtain γ_{q}^{∗}-coinvariants from γ_{q}^{∗∘}-coinvariants by “removing denominators”.

Remark 4.1. (i) As we already mentioned in Remark 3.2(iii), the restriction of γ_{q}^{∗∘} to O_{q}(V) is just γ_{q}^{∗}. So, if w ∈ O_{q}(V) is a γ_{q}^{∗}-coinvariant, it is also a γ_{q}^{∗∘}-coinvariant. Thus, according to Proposition 3.8, w ∈ Im μ_{q}^{∗∘}. This means that there exists s ∈ ℕ such that w(d_{Y} ⊗d_{Z})^{s} ∈ Im μ_{q}^{∗}. So, to prove that the set of γ_{q}^{∗}-coinvariants equals Im μ_{q}^{∗}, it is enough to show the following property (cf. Proposition 2.3):

(†)  ∀w ∈ O_{q}(V), if w(d_{Y} ⊗ d_{Z}) ∈ Im μ_{q}^{∗}, then w ∈ Im μ_{q}^{∗}.

(ii) Recall from Remark 3.2(i) that the image under μ_{q}^{∗} of the (normal) element d_{x} ∈ O_{q}(M_{n}^{≤t}) is the (normal) element d_{Y} ⊗ d_{Z} ∈ O_{q}(V). We claim that to prove (†), it is enough (in fact, equivalent) to verify the following property:

(‡)  (μ_{q}^{∗})^{−1}(⟨d_{Y} ⊗ d_{Z}⟩) = ⟨d_{x}⟩.

Thus, assume that (‡) holds, and let w ∈ O_{q}(V) such that w(d_{Y} ⊗ d_{Z}) ∈ Im μ_{q}^{∗}. Then w(d_{Y} ⊗ d_{Z}) = μ_{q}^{∗}(y) for some y ∈ O_{q}(M_{n}^{≤t}), and (‡) implies that y = zd_{x} for some z ∈ O_{q}(M_{n}^{≤t}). Since μ_{q}^{∗} is a morphism of algebras, w(d_{Y} ⊗ d_{Z}) = μ_{q}^{∗}(y) = μ_{q}^{∗}(z)(d_{Y} ⊗ d_{Z}), and so w = μ_{q}^{∗}(z) because O_{q}(V) is a domain. This shows that (‡) implies (†). The converse follows easily from the injectivity of μ_{q}^{∗} (see Theorem 2.2).

Remark 4.2. (i) The computations in this section and in Section 6 below rely on the preferred bases for the algebras O_{q}(M_{n}), O_{q}(M_{n,t}), and O_{q}(M_{t,n}) developed in [2, Section 1], and we follow the notation of that paper. See, in particular, [2, Corollary 1.11] for the rectangular case. We recall the notation [T |T′] for the product of quantum minors corresponding to an allowable bitableaux (T, T′), and we recall also that it is sometimes convenient to label rows of (T, T′) in the form (I, J) where I and J are sets of row and column indices, respectively.

(ii) On tensoring the preferred basis elements of O_{q}(M_{n,t}) with those of O_{q}(M_{t,n}), we obtain a preferred basis for O_{q}(V) consisting of all pure tensors [S|S′]⊗[T |T′] where (S, S′) and (T, T′) are preferred bitableaux with appropriate entries. In particular, S and T each have at most t columns.

(iii) Because of the quantum Laplace expansions [6, Proposition 1.1 and Corollary; 7, Corollary 4.4.4], the ideal I_{t+1} in O_{q}(M_{n}) contains all the r × r quantum minors for t + 1 ≤ r ≤ n. Hence, [T |T′] ∈ I_{t+1} whenever (T, T′) is a preferred bitableau and T has at least t + 1 columns.

The statement and proof of [2, Proposition 2.4] show that I_{t+1} does not contain any nonzero linear combinations of distinct products [S|S′] where the (S, S′) are preferred bitableaux and S has at most t columns. Consequently, I_{t+1} has a K-basis consisting of the products [T |T′] such that (T, T′) is a preferred bitableau and T has at least t + 1 columns.

(iv) In view of point (iii), the cosets [T |T′]+I_{t+1} such that (T, T′) is a preferred bitableau and T has at most t columns form a K-basis for O_{q}(M_{n}^{≤t}). To simplify the notation, we shall write cosets a + I_{t+1} in the form \overline{a}. Thus, \overline{[T |T′]} = 0 whenever T has more than t columns.

It is convenient to label two special index sets:

Ĩ = {n − t + 1, …, n}  and  J̃ = {1, …, t}.

With this notation, observe that d_{X} and d_{Y} can both be labeled [Ĩ|J̃], while d_{Z} and d_{T} can both be labeled [J̃|J̃].

Lemma 4.3. (a) A K-basis for ⟨d_{Y} ⟩ consists of all [T |T′] in O_{q}(M_{n,t}) where (T, T′) is a preferred bitableau with first row (Ĩ, J̃).

(b) A K-basis for ⟨d_{Z}⟩ consists of all [T |T′] in O_{q}(M_{t,n}) where (T, T′) is a preferred bitableau with first row (J̃, J̃).

(c) A K-basis for ⟨d_{Y} ⊗ d_{Z}⟩ consists of all pure tensors [S|S′] ⊗ [T |T′] in O_{q}(V) where (S, S′) and (T, T′) are preferred bitableaux with first rows (Ĩ, J̃) and (J̃, J̃), respectively.

(d) A K-basis for ⟨d_{x}⟩ consists of all \overline{[T |T′]} in O_{q}(M_{n}^{≤t}) where (T, T′) is a preferred bitableau with first row (Ĩ, J̃).

Proof. (a) Obviously any such [T |T′] lies in ⟨d_{Y} ⟩, because d_{Y} = [Ĩ|J̃]. Recall that d_{Y} is a normal element of O_{q}(M_{n,t}), so that ⟨d_{Y} ⟩ = d_{Y} O_{q}(M_{n,t}). Since (Ĩ, J̃) is the minimum index pair labeling quantum minors in O_{q}(M_{n,t}), the preferred basis of O_{q}(M_{n,t}) is closed under left multiplication by d_{Y}. Hence, ⟨d_{Y} ⟩ is spanned by products d_{Y} [S|S′] as (S, S′) runs through all preferred bitableaux, and each such d_{Y} [S|S′] = [T |T′] for some preferred bitableau (T, T′) with first row (Ĩ, J̃).

(b) This is proved in the same manner as part (a).

(c) Note that because d_{Y} and d_{Z} are normal elements of O_{q}(M_{n,t}) and O_{q}(M_{t,n}), respectively,

⟨d_{Y} ⊗ d_{Z}⟩ = (d_{Y} ⊗ d_{Z})O_{q}(V) = d_{Y} O_{q}(M_{n,t}) ⊗ d_{Z}O_{q}(M_{t,n}) = ⟨d_{Y} ⟩ ⊗ ⟨d_{Z}⟩.

Hence, part (c) follows directly from parts (a) and (b).

(d) By Remark 4.2(iv), a K-basis for O_{q}(M_{n}^{≤t}) consists of all \overline{[T |T′]} such that (T, T′) is a preferred bitableau and T has at most t columns. Among index pairs (I, J) with |I| ≤ t, the minimum element is (Ĩ, J̃). Since d_{x} = \overline{[Ĩ|J̃]}, we therefore obtain part (d) in exactly the same manner as part (a). □

In the following proof, we shall need the natural multigradings on quantum matrix algebras (cf. [2, §1.5]). For instance, O_{q}(M_{n,t}) is graded by ℤ^{n} × ℤ^{t} with each generator Y_{ij} having degree (ϵ_{i}, ϵ_{j}), where ϵ_{1}, ϵ_{2}, … denote the standard basis elements in ℤ^{n} and ℤ^{t}. In the proof, we use the label ‘homogeneous’ to refer to homogeneous elements with respect to the above gradings.

Proposition 4.4. (μ_{q}^{∗})^{−1}(⟨d_{Y} ⊗ d_{Z}⟩) = ⟨d_{x}⟩.

Proof. The inclusion ‘⊇’ is clear since μ_{q}^{∗}(d_{x}) = d_{Y} ⊗d_{Z}. If this inclusion is proper, choose an element

x = ∑_{i=1}^{r} α_{i}\overline{[T_{i}|T′_{i}]} ∈ (μ_{q}^{∗})^{−1}(⟨d_{Y} ⊗ d_{Z}⟩) \ ⟨d_{x}⟩

where the α_{i} are nonzero scalars, the (T_{i}, T′_{i}) are distinct preferred bitableaux, and the T_{i} have at most t columns. We may assume that none of the \overline{[T_{i}|T′_{i}]} lie in ⟨d_{x}⟩. Thus, by Lemma 4.3(d), none of the (T_{i}, T′_{i}) has first row (Ĩ, J̃). Define n-tuples \overline{ρ}(T_{i}) as in [2, §2.2], and let \overline{ρ} be the minimum of the \overline{ρ}(T_{i}) under reverse lexicographic order.

After re-indexing, we may assume that there is some r′ such that \overline{ρ}(T_{i}) = \overline{ρ} for i ≤ r′ and \overline{ρ}(T_{i}) >_{rlex} \overline{ρ} for i > r′. Applying [2, Lemma 2.3] to each μ_{q}^{∗}\overline{[T_{i}|T′_{i}]} (= θ_{q}^{∗}[T_{i}|T′_{i}]) and collecting terms, we see (using the notation of [2, §2.2]) that

μ_{q}^{∗}(x) = ∑_{i=1}^{r′} 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}. We then observe (as in the proof of [2, Proposition 2.4]) that all of the X_{j} belong to different homogeneous components than the [T_{i}|μ(T_{i})] for i ≤ r′. Since μ_{q}^{∗}(x) ∈ ⟨d_{Y} ⊗ d_{Z}⟩ = ⟨d_{Y} ⟩ ⊗ ⟨d_{Z}⟩ and the ideal ⟨d_{Y} ⟩ is homogeneous, it follows that

∑_{i=1}^{r′} α_{i}[T_{i}|μ(T_{i})] ⊗ [μ′(T_{i})|T′_{i}] ∈ ⟨d_{Y} ⊗ d_{Z}⟩.

For 1 ≤ i < j ≤ r′, either T_{i} ≠ T_{j} or T′_{i} ≠ T′_{j}, whence (T_{i}, μ(T_{i})) ≠ (T_{j}, μ(T_{j})) or (μ′(T_{i}), T′_{i}) ≠ (μ′(T_{j}), T′_{j}). Recall from [2, §2.2] that the (T_{i}, μ(T_{i})) and the (μ′(T_{i}), T′_{i}) are preferred bitableaux. In view of Lemma 4.3(c), it follows that T_{i} has first row Ĩ and T′_{i} has first row J̃ for 1 ≤ i ≤ r′. However, this contradicts our choices above, and therefore the proposition is proved. □

Theorem 4.5. Let m, n, t be arbitrary positive integers. The set of γ_{q}^{∗}-coinvariants in O_{q}(V) = O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}) is Im μ_{q}^{∗}, that is,

(O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}))^{co O_{q}(GL_{t})} = μ_{q}^{∗}(O_{q}(M_{m,n}^{≤t})).

Proof. In view of Remark 2.4, there is no loss of generality in assuming that m = n > t. The theorem is then immediate using Remark 4.1 and Proposition 4.4. □

5. O_{q}(SL_{t})-coinvariants

For arbitrary positive integers m, n, t, there is a natural coaction Γ_{q}^{∗} of O_{q}(SL_{t}) on O_{q}(V) induced from the O_{q}(GL_{t})-coaction γ_{q}^{∗} that we have been studying. Denote by π the natural quotient map O_{q}(GL_{t}) → O_{q}(SL_{t}). The coaction Γ_{q}^{∗} : O_{q}(V) → O_{q}(SL_{t}) ⊗ O_{q}(V) is given by Γ_{q}^{∗} := (π⊗id)∘γ_{q}^{∗}. In this section and the next, we compute the Γ_{q}^{∗}-coinvariants.

In fact, Γ_{q}^{∗} can be constructed from the left coaction of O_{q}(SL_{t}) on O_{q}(M_{t,n}) given by

(π ⊗ id) ∘ λ_{q}^{∗} : O_{q}(M_{t,n}) → O_{q}(SL_{t}) ⊗ O_{q}(M_{t,n})

and the right coaction of O_{q}(SL_{t}) on O_{q}(M_{m,t}) given by

(id ⊗π) ∘ ρ_{q}^{∗} : O_{q}(M_{m,t}) → O_{q}(M_{m,t}) ⊗ O_{q}(SL_{t})

by using the construction described in Section 1.

Proposition 5.1. The set of coinvariants O_{q}(V)^{co O_{q}(SL_{t})} is a subalgebra of O_{q}(V) containing Im μ_{q}^{∗}.

Proof. That O_{q}(V)^{co O_{q}(SL_{t})} is a subalgebra of O_{q}(V) follows from Proposition 1.1(c). It contains Im μ_{q}^{∗} because the elements of Im μ_{q}^{∗} are γ_{q}^{∗}-coinvariants and hence also Γ_{q}^{∗}- coinvariants. □

In order to describe the Γ_{q}^{∗}-coinvariants, we may assume that m = n and that t < n, as in Remark 2.4. We start by observing that there is a natural ℤ-grading on quantum matrices which will simplify the problem. The algebra O_{q}(M_{n,t}) can be graded by total degree in the variables Y_{ij} and we put O_{q}(M_{n,t}) = ⨁_{i≥0} O_{q}(M_{n,t})_{i}, where O_{q}(M_{n,t})_{i} is the subspace spanned by monomials of total degree i in O_{q}(M_{n,t}). In the same way, and with obvious notation, O_{q}(M_{t,n}) = ⨁_{j≥0} O_{q}(M_{t,n})_{j}. It follows that

O_{q}(V) = ⨁_{i,j=0}^{∞} O_{q}(V)_{i,j},

where O_{q}(V)_{i,j} = O_{q}(M_{n,t})_{i} ⊗ O_{q}(M_{t,n})_{j}.

Recall that ρ_{q}^{∗} : O_{q}(M_{n,t}) → O_{q}(M_{n,t}) ⊗ O_{q}(GL_{t}) is the algebra morphism defined by ρ_{q}^{∗}(Y_{ij}) = ∑_{k=1}^{t} Y_{ik} ⊗ T_{kj}. Thus, it is clear that ρ_{q}^{∗}(O_{q}(M_{n,t})_{i}) ⊆ O_{q}(M_{n,t})_{i} ⊗ O_{q}(GL_{t}). Similarly, one has λ_{q}^{∗}(O_{q}(M_{t,n})_{j}) ⊆ O_{q}(GL_{t}) ⊗ O_{q}(M_{t,n})_{j}. It follows that

γ_{q}^{∗}(O_{q}(V)_{i,j}) ⊆ O_{q}(GL_{t}) ⊗ O_{q}(V)_{i,j}
Γ_{q}^{∗}(O_{q}(V)_{i,j}) ⊆ O_{q}(SL_{t}) ⊗ O_{q}(V)_{i,j}

for all i, j.

Given w ∈ O_{q}(V), we will write w = ∑_{i,j} w_{ij} with w_{ij} ∈ O_{q}(V)_{i,j}.

Lemma 5.2. Let w = ∑_{i,j} w_{ij} ∈ O_{q}(V). Then (a) w is a γ_{q}^{∗}-coinvariant if and only if each w_{ij} is a γ_{q}^{∗}-coinvariant.

(b) w is a Γ_{q}^{∗}-coinvariant if and only if each w_{ij} is a Γ_{q}^{∗}-coinvariant.

Proof. If each w_{ij} is a γ_{q}^{∗}-coinvariant then, obviously, w is a γ_{q}^{∗}-coinvariant. Conversely, suppose that w is a γ_{q}^{∗}-coinvariant. One has ∑_{i,j} 1⊗w_{ij} = 1⊗w = γ_{q}^{∗}(w) = ∑_{i,j} γ_{q}^{∗}(w_{ij}), and from the discussion above, we see that γ_{q}^{∗}(w_{ij}) ∈ O_{q}(GL_{t}) ⊗ O_{q}(V)_{i,j}. Thus, since O_{q}(GL_{t}) ⊗ O_{q}(V) = ⨁_{i,j} O_{q}(GL_{t}) ⊗ O_{q}(V)_{i,j}, one has γ_{q}^{∗}(w_{ij}) = 1 ⊗ w_{ij} for all i, j. This finishes the proof of (a); the proof of (b) is similar. □

The previous result shows that, in order to describe the Γ_{q}^{∗}-coinvariants, it is enough to describe the coinvariants which are in each O_{q}(V)_{i,j}. It is obvious from the definition of Γ_{q}^{∗} that any γ_{q}^{∗}-coinvariant will be a Γ_{q}^{∗}-coinvariant. Our first aim is to show that any homogeneous Γ_{q}^{∗}-coinvariant w is a γ_{q}^{∗}-semi-coinvariant, meaning that γ_{q}^{∗}(w) = d_{T}^{s} ⊗ w for some integer s; this is achieved in Theorem 5.5. We begin by constructing some maps which will help us to prove this fact.

The above grading on O_{q}(V) relates to the following comodule structure. Consider the Hopf algebra O_{q}(K^{×}) = K[T^{±1}] and the algebra morphism π′ : O_{q}(GL_{t}) ↠ O_{q}(K^{×}) such that T_{ii} ↦ T for all i, while T_{ij} ↦ 0 for i ≠ j. It is easily checked that π′ is, in fact, a morphism of Hopf algebras. The composite map

O_{q}(V) ⟶^{γ_{q}^{∗}} O_{q}(GL_{t}) ⊗ O_{q}(V) ⟶^{π′⊗id} O_{q}(K^{×}) ⊗ O_{q}(V)

thus gives O_{q}(V) the structure of a left comodule over O_{q}(K^{×}). The following proposition shows that, in fact, O_{q}(V) is an O_{q}(K^{×})-comodule algebra.

Proposition 5.3. (a) (π′ ⊗ id) ∘ γ_{q}^{∗} is an algebra morphism.

(b) (π′ ⊗ id)γ_{q}^{∗}(Y_{ij} ⊗ 1) = T^{−1} ⊗ Y_{ij} ⊗ 1 and (π′ ⊗ id)γ_{q}^{∗}(1 ⊗ Z_{ij}) = T ⊗ 1 ⊗ Z_{ij} for all i, j.

(c) (π′ ⊗ id)γ_{q}^{∗}(w) = T^{j−i} ⊗ w for all w ∈ O_{q}(V)_{i,j}.

Proof. (a) Recall that, for a ∈ O_{q}(M_{n,t}) and b ∈ O_{q}(M_{t,n}), if we put ρ_{q}^{∗}(a) = ∑_{(a)} a_{0} ⊗ a_{1} and λ_{q}^{∗}(b) = ∑_{(b)} b_{−1} ⊗ b_{0}, then γ_{q}^{∗}(a ⊗ b) = ∑_{(a),(b)} S(a_{1})b_{−1} ⊗ a_{0} ⊗ b_{0}. If, also, a′ ⊗ b′ ∈ O_{q}(V) then, since ρ_{q}^{∗} and λ_{q}^{∗} are algebra morphisms, one has

γ_{q}^{∗}((a ⊗ b)(a′ ⊗ b′)) = ∑_{(a′),(b′)} S(a′_{1})S(a_{1})b_{−1}b′_{−1} ⊗ a_{0}a′_{0} ⊗ b_{0}b′_{0}.

Using the fact that O_{q}(K^{×}) is a commutative algebra we then get

(π′ ⊗ id)γ_{q}^{∗}((a ⊗ b)(a′ ⊗ b′)) = ((π′ ⊗ id)γ_{q}^{∗})(a ⊗ b)((π′ ⊗ id)γ_{q}^{∗})(a′ ⊗ b′).

(b) If A_{jk} denotes the (t − 1) × (t − 1) quantum minor of O_{q}(GL_{t}) obtained by deleting row j and column k, then S(T_{kj}) = (−q)^{k−j}A_{jk}d_{T}^{−1}, by [7, Theorem 5.3.2], and so

γ_{q}^{∗}(Y_{ij} ⊗ 1) = ∑_{k=1}^{t} S(T_{kj}) ⊗ Y_{ik} ⊗ 1 = ∑_{k=1}^{t} (−q)^{k−j}A_{jk}d_{T}^{−1} ⊗ Y_{ik} ⊗ 1

and γ_{q}^{∗}(1 ⊗ Z_{ij}) = ∑_{k=1}^{t} T_{ik} ⊗ 1 ⊗ Z_{kj}. It follows that (π′ ⊗ id)γ_{q}^{∗}(Y_{ij} ⊗ 1) = T^{−1} ⊗ Y_{ij} ⊗ 1 and that (π′ ⊗ id)γ_{q}^{∗}(1 ⊗ Z_{ij}) = T ⊗ 1 ⊗ Z_{ij}.

(c) This is clear from parts (a) and (b). □

We shall need the algebra morphism

ϕ_{q}^{∗} : O_{q}(GL_{t}) ⊗ O_{q}(K^{×}) → O_{q}(GL_{t})[\overline{χ}] := O_{q}(GL_{t})[χ]/⟨χ^{t} − d_{T}^{−1}⟩,

where χ is an indeterminate and \overline{χ} = χ + ⟨χ^{t} − d_{T}^{−1}⟩, such that each T_{ij} ⊗ 1 ↦ T_{ij}\overline{χ} and 1 ⊗ T ↦ \overline{χ}^{−1}. It is easy to check that ϕ_{q}^{∗}(d_{T} ⊗ 1) = d_{T} \overline{χ}^{t} = 1, and so there is an induced algebra morphism

φ_{q}^{∗} : O_{q}(SL_{t}) ⊗ O_{q}(K^{×}) → O_{q}(GL_{t})[\overline{χ}]

such that φ_{q}^{∗} ∘ (π ⊗ id) = ϕ_{q}^{∗}.

Finally, let α_{q} : O_{q}(GL_{t}) → O_{q}(GL_{t})[\overline{χ}] denote the canonical injection.

Lemma 5.4. (a) ϕ_{q}^{∗} ∘ (id ⊗π′) ∘ Δ = α_{q}.

(b) (φ_{q}^{∗} ⊗ id) ∘ (id ⊗π′ ⊗ id) ∘ (id ⊗γ_{q}^{∗}) ∘ Γ_{q}^{∗} = (α_{q} ⊗ id) ∘ γ_{q}^{∗}.

Proof. (a) It is enough to check this on the generators T_{ij}, since each of the maps involved is an algebra morphism. One has

ϕ_{q}^{∗}(id ⊗π′)Δ(T_{ij}) = ϕ_{q}^{∗}(id ⊗π′)(∑_{k=1}^{t} T_{ik} ⊗ T_{kj}) = ϕ_{q}^{∗}(T_{ij} ⊗ T) = T_{ij}.

(b) Since γ_{q}^{∗} is the structure map of a left O_{q}(GL_{t})-comodule, we have (Δ ⊗ id) ∘ γ_{q}^{∗} = (id ⊗γ_{q}^{∗}) ∘ γ_{q}^{∗}. In view of part (a), it follows that

(α_{q} ⊗ id) ∘ γ_{q}^{∗} = (ϕ_{q}^{∗} ⊗ id) ∘ (id ⊗π′ ⊗ id) ∘ (Δ ⊗ id) ∘ γ_{q}^{∗}
= (ϕ_{q}^{∗} ⊗ id) ∘ (id ⊗π′ ⊗ id) ∘ (id ⊗γ_{q}^{∗}) ∘ γ_{q}^{∗}.

We also have the following commutative diagram:

\xymatrix{ & O_{q}(V) \ar[dl]_{γ_{q}^{∗}} \ar[dr]^{Γ_{q}^{∗}} \\ O_{q}(GL_{t}) ⊗ O_{q}(V) \ar[d]_{id ⊗γ_{q}^{∗}} \ar[rr]^{π⊗id} &  & O_{q}(SL_{t}) ⊗ O_{q}(V) \ar[d]^{id ⊗γ_{q}^{∗}} \\ O_{q}(GL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(V) \ar[d]_{id ⊗π′⊗id} \ar[rr]^{π⊗id ⊗ id} &  & O_{q}(SL_{t}) ⊗ O_{q}(GL_{t}) ⊗ O_{q}(V) \ar[d]^{id ⊗π′⊗id} \\ O_{q}(GL_{t}) ⊗ O_{q}(K^{×}) ⊗ O_{q}(V) \ar[rr]^{π⊗id ⊗ id} \ar@{-->}[dr]_{ϕ_{q}^{∗}⊗id} &  & O_{q}(SL_{t}) ⊗ O_{q}(K^{×}) ⊗ O_{q}(V) \ar@{-->}[dl]^{φ_{q}^{∗}⊗id} \\  & O_{q}(GL_{t})[\overline{χ}] ⊗ O_{q}(V)}

Thus (ϕ_{q}^{∗} ⊗ id) ∘ (id ⊗π′ ⊗ id) ∘ (id ⊗γ_{q}^{∗}) ∘ γ_{q}^{∗} = (φ_{q}^{∗} ⊗ id) ∘ (id ⊗π′ ⊗ id) ∘ (id ⊗γ_{q}^{∗}) ∘ Γ_{q}^{∗}, which completes the proof of part (b). □

Theorem 5.5. Let w ∈ O_{q}(V)_{i,j} be a Γ_{q}^{∗}-coinvariant.

(a) The difference i − j is divisible by t, and γ_{q}^{∗}(w) = d_{T}^{−s} ⊗ w where s = (i − j)/t.

(b) If i ≥ j, then (1 ⊗ d_{Z}^{s})w is a γ_{q}^{∗}-coinvariant.

(c) If i ≤ j, then (d_{Y}^{−s} ⊗ 1)w is a γ_{q}^{∗}-coinvariant.

Proof. (a) By Proposition 5.3(c), we have (π′ ⊗ id)γ_{q}^{∗}(w) = T^{j−i} ⊗ w. From this and the assumption that Γ_{q}^{∗}(w) = 1 ⊗ w, it follows that

(φ_{q}^{∗} ⊗ id)(id ⊗π′ ⊗ id)(id ⊗γ_{q}^{∗})Γ_{q}^{∗}(w) = (φ_{q}^{∗} ⊗ id)(id ⊗π′ ⊗ id)(id ⊗γ_{q}^{∗})(1 ⊗ w)
= (φ_{q}^{∗} ⊗ id)(1 ⊗ T^{j−i} ⊗ w) = \overline{χ}^{i−j} ⊗ w.

Thus, by Lemma 5.4(b), (α_{q} ⊗ id)γ_{q}^{∗}(w) = \overline{χ}^{i−j} ⊗ w. In particular,

\overline{χ}^{i−j} ⊗ w ∈ Im(α_{q} ⊗ id) = O_{q}(GL_{t}) · 1 ⊗ O_{q}(V).

Since O_{q}(GL_{t})[\overline{χ}] is a free O(GL_{t})-module with basis \{1, \overline{χ}, …, \overline{χ}^{t−1}\}, we have t | i − j. Set s = (i − j)/t. Now

(α_{q} ⊗ id)γ_{q}^{∗}(w) = \overline{χ}^{ts} ⊗ w = (α_{q} ⊗ id)(d_{T}^{−s} ⊗ w),

and therefore γ_{q}^{∗}(w) = d_{T}^{−s} ⊗ w, as required.

(b) Since λ_{q}^{∗} is an algebra morphism, we see using Remark 3.2(i) that λ_{q}^{∗}(d_{Z}^{s}) = d_{T}^{s} ⊗d_{Z}^{s}, and hence γ_{q}^{∗}(1⊗d_{Z}^{s}) = d_{T}^{s} ⊗1⊗d_{Z}^{s}. As d_{T}^{s} is central in O_{q}(GL_{t}), we may apply Proposition 1.1(a) to conclude that

γ_{q}^{∗}((1 ⊗ d_{Z}^{s})w) = γ_{q}^{∗}(1 ⊗ d_{Z}^{s})γ_{q}^{∗}(w) = (d_{T}^{s} ⊗ 1 ⊗ d_{Z}^{s})(d_{T}^{−s} ⊗ w) = 1 ⊗ ((1 ⊗ d_{Z}^{s})w).

(c) By Remark 3.2(i), ρ_{q}^{∗}(d_{Y}^{−s}) = d_{Y}^{−s} ⊗ d_{T}^{−s}. Since S(d_{T}) = d_{T}^{−1}, we thus obtain

γ_{q}^{∗}(d_{Y}^{−s} ⊗ 1) = d_{T}^{s} ⊗ d_{Y}^{−s} ⊗ 1,

and therefore part (c) follows from a second application of Proposition 1.1(a). □

6. The First Fundamental Theorem for O_{q}(SL_{t})-coinvariants

In this final section, we determine the Γ_{q}^{∗}-coinvariants in O_{q}(V). Let A_{1} and A_{2} denote the respective subalgebras of O_{q}(M_{m,t}) and O_{q}(M_{t,n}) generated by all the t × t quantum minors. We shall prove that the set of Γ_{q}^{∗}-coinvariants in O_{q}(V) is the subalgebra generated by A_{1}⊗A_{2} together with Im μ_{q}^{∗}. In fact, this subalgebra turns out to be equal to the product (A_{1} ⊗ A_{2}) · Im μ_{q}^{∗}. On the road to this goal, Theorem 5.5 puts us in a position roughly similar to that of Remark 4.1(ii), and to finish we need some computations analogous to Proposition 4.4.

Until further notice, we continue to assume that m = n > t.

Remark 6.1. (i) As in Section 4, we set Ĩ = {n − t + 1, …, n} and J̃ = {1, …, t}. Let P_{1} and P_{2} denote the following ideals:

P_{1} = ⟨ \overline{[Ĩ|J]} | J ⊆ \{1, …, n\} and |J| = t ⟩ ◁ O_{q}(M_{n}^{≤t})
P_{2} = ⟨ \overline{[I|J̃]} | I ⊆ \{1, …, n\} and |I| = t ⟩ ◁ O_{q}(M_{n}^{≤t}).

These turn out to be completely prime ideals of O_{q}(M_{n}^{≤t}) – see Remark 6.4. In view of [2, Corollary 5.2], the quantum minors [Ĩ|J] can be arranged in a polynormal sequence, as can the [I|J̃]. Hence,

P_{1} = ∑_{\begin{matrix}J⊆\{1,...,n\} \\ |J|=t \end{matrix}} \overline{[Ĩ|J]}O_{q}(M_{n}^{≤t})  and  P_{2} = ∑_{\begin{matrix}I⊆\{1,...,n\} \\ |I|=t \end{matrix}} \overline{[I|J̃]}O_{q}(M_{n}^{≤t}).

(ii) It can be proved that a K-basis for P_{1} consists of all \overline{[T |T′]} where (T, T′) is a preferred bitableaux with first row of the form (Ĩ,?), and that a K-basis for P_{2} consists of all \overline{[T |T′]} where (T, T′) is a preferred bitableaux with first row of the form (?, J̃). However, we shall not need these bases.

(iii) In view of point (i), powers of the P_{i} can be expressed as follows:

P_{1}^{s} = ∑_{\begin{matrix}J_{1},...,J_{s}⊆\{1,...,n\} \\ |J_{1}|=···=|J_{s}|=t \end{matrix}} \overline{[Ĩ|J_{1}][Ĩ|J_{2}] ⋯ [Ĩ|J_{s}]}O_{q}(M_{n}^{≤t})

P_{2}^{s} = ∑_{\begin{matrix}I_{1},...,I_{s}⊆\{1,...,n\} \\ |I_{1}|=···=|I_{s}|=t \end{matrix}} \overline{[I_{1}|J̃][I_{2}|J̃] ⋯ [I_{s}|J̃]}O_{q}(M_{n}^{≤t})

for s = 1, 2, . . ..

Lemma 6.2. Let s be a nonnegative integer.

(a) A K-basis for ⟨d_{Y}^{s} ⊗ 1⟩ consists of all pure tensors [S|S′] ⊗ [T |T′] in O_{q}(V) where (S, S′) and (T, T′) are preferred bitableaux and the first s rows of (S, S′) equal (Ĩ, J̃).

(b) A K-basis for ⟨1 ⊗ d_{Z}^{s}⟩ consists of all pure tensors [S|S′] ⊗ [T |T′] in O_{q}(V) where (S, S′) and (T, T′) are preferred bitableaux and the first s rows of (T, T′) equal (J̃, J̃).

Proof. Since ⟨d_{Y}^{s} ⊗ 1⟩ = ⟨d_{Y}^{s} ⟩ ⊗ O_{q}(M_{t,n}) and ⟨1 ⊗ d_{Z}^{s}⟩ = O_{q}(M_{n,t}) ⊗ ⟨d_{Z}^{s}⟩, it suffices to check that the ideals ⟨d_{Y}^{s} ⟩ and ⟨d_{Z}^{s}⟩ have bases of the appropriate forms. This can be done just as in Lemma 4.3(a)(b). □

Proposition 6.3. If s is any nonnegative integer, then

(μ_{q}^{∗})^{−1}(⟨d_{Y}^{s} ⊗ 1⟩) = P_{1}^{s}  and  (μ_{q}^{∗})^{−1}(⟨1 ⊗ d_{Z}^{s}⟩) = P_{2}^{s}.

Proof. We prove the first equality; the proof of the second is analogous. The proof closely mimics that of Proposition 4.4.

In view of Remark 2.1(iii-v), we see that

μ_{q}^{∗}\overline{[Ĩ|J]} = [Ĩ|J̃] ⊗ [J̃|J] = d_{Y} ⊗ [J̃|J]

for all t-element subsets J ⊆ {1, …, n}. Using Remark 6.1(iii) and the fact that μ_{q}^{∗} is an algebra morphism, it follows that P_{1}^{s} ⊆ (μ_{q}^{∗})^{−1}(⟨d_{Y}^{s} ⊗ 1⟩).

If this inclusion is proper, choose an element

x = ∑_{i=1}^{r} α_{i}\overline{[T_{i}|T′_{i}]} ∈ (μ_{q}^{∗})^{−1}(⟨d_{Y}^{s} ⊗ 1⟩) \ P_{1}^{s}

where the α_{i} are nonzero scalars, the (T_{i}, T′_{i}) are distinct preferred bitableaux, and the T_{i} have at most t columns. We may assume that none of the \overline{[T_{i}|T′_{i}]} lie in P_{1}^{s}. Let \overline{ρ} be the minimum of the \overline{ρ}(T_{i}) under reverse lexicographic order.

After re-indexing, we may assume that there is some r′ such that \overline{ρ}(T_{i}) = \overline{ρ} for i ≤ r′ and \overline{ρ}(T_{i}) >_{rlex} \overline{ρ} for i > r′. Applying [2, Lemma 2.3] to each μ_{q}^{∗}\overline{[T_{i}|T′_{i}]} and collecting terms, we see that

μ_{q}^{∗}(x) = ∑_{i=1}^{r′} 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}. We then observe that all of the X_{j} belong to different homogeneous components than the [T_{i}|μ(T_{i})] for i ≤ r′. Since

μ_{q}^{∗}(x) ∈ ⟨d_{Y}^{s} ⊗ 1⟩ = ⟨d_{Y}^{s} ⟩ ⊗ O_{q}(M_{t,n})

and the ideal ⟨d_{Y}^{s} ⟩ is homogeneous, it follows that

∑_{i=1}^{r′} α_{i}[T_{i}|μ(T_{i})] ⊗ [μ′(T_{i})|T′_{i}] ∈ ⟨d_{Y}^{s} ⊗ 1⟩.

For 1 ≤ i < j ≤ r′, either (T_{i}, μ(T_{i})) ≠ (T_{j}, μ(T_{j})) or (μ′(T_{i}), T′_{i}) ≠ (μ′(T_{j}), T′_{j}). In view of Lemma 6.2(a), it follows that for 1 ≤ i ≤ r′, the first s rows of T_{i} equal Ĩ. But then \overline{[T_{i}|T′_{i}]} ∈ P_{1}^{s} for 1 ≤ i ≤ r′, which contradicts our choices above and therefore establishes the desired equality. □

Remark 6.4. Although we shall not need the fact here, we note that the case s = 1 of Proposition 6.3 implies that P_{1} and P_{2} are completely prime ideals of O_{q}(M_{n}^{≤t}). To see this, it obviously suffices to show that ⟨d_{Y} ⊗ 1⟩ and ⟨1 ⊗ d_{Z}⟩ are completely prime ideals of O_{q}(V). Observe that O_{q}(M_{n,t})/⟨d_{Y} ⟩ is an iterated skew polynomial ring over an isomorphic copy of O_{q}(M_{t}) modulo the ideal generated by its quantum determinant. Since the latter algebra is a domain ([3], [4, p. 182], or see [2, Theorem 2.5]), it follows that O_{q}(M_{n,t})/⟨d_{Y} ⟩ is a domain. Now the algebra

O_{q}(V)/⟨d_{Y} ⊗ 1⟩ ≅ (O_{q}(M_{n,t})/⟨d_{Y} ⟩) ⊗ O_{q}(M_{t,n})

is an iterated skew polynomial ring over O_{q}(M_{n,t})/⟨d_{Y} ⟩, and therefore O_{q}(V)/⟨d_{Y} ⊗ 1⟩ is a domain. Thus ⟨d_{Y} ⊗ 1⟩ is completely prime, and a similar argument shows that ⟨1 ⊗ d_{Z}⟩ is completely prime.

Lemma 6.5. Let w ∈ O_{q}(V)_{i,j} be a Γ_{q}^{∗}-coinvariant.

(a) If i ≥ j, then w ∈ (A_{1} ⊗ 1) · Im μ_{q}^{∗}.

(b) If i ≤ j, then w ∈ (1 ⊗ A_{2}) · Im μ_{q}^{∗}.

Proof. By Theorem 5.5(a), the ratio s = (i − j)/t is an integer.

(a) In this case, Theorem 5.5(b) shows that (1 ⊗ d_{Z}^{s})w is a γ_{q}^{∗}-coinvariant. Thus, by Theorem 4.5, (1 ⊗ d_{Z}^{s})w = μ_{q}^{∗}(x) for some x ∈ O_{q}(M_{n}^{≤t}). Now μ_{q}^{∗}(x) ∈ ⟨1 ⊗ d_{Z}^{s}⟩, and hence x ∈ P_{2}^{s}, by Proposition 6.3. In view of Remark 6.1(iii), we thus have x = ∑_{i=1}^{r} \overline{p}_{i}x_{i} for some x_{i} ∈ O_{q}(M_{n}^{≤t}) and some p_{i} ∈ O_{q}(M_{n}) of the form

p_{i} = [I_{i,1}|J̃][I_{i,2}|J̃] ⋯ [I_{i,s}|J̃].

Now by Remark 2.1(iii-v), each

μ_{q}^{∗}\overline{[I_{i,l}|J̃]} = [I_{i,l}|J̃] ⊗ [J̃|J̃] = [I_{i,l}|J̃] ⊗ d_{Z}.

Since μ_{q}^{∗} is an algebra morphism, it follows that each

μ_{q}^{∗}(\overline{p}_{i}x_{i}) = (1 ⊗ d_{Z}^{s})([I_{i,1}|J̃] ⊗ 1)([I_{i,2}|J̃] ⊗ 1) ⋯ ([I_{i,s}|J̃] ⊗ 1)μ_{q}^{∗}(x_{i}).

Consequently, (1 ⊗ d_{Z}^{s})w = μ_{q}^{∗}(x) = (1 ⊗ d_{Z}^{s})v for some v ∈ (A_{1} ⊗ 1) · Im μ_{q}^{∗}. Since 1 ⊗ d_{Z} is a nonzero element of the domain O_{q}(V), we conclude that w = v, and the proof of part (a) is complete.

(b) This is proved in a similar fashion. □

Theorem 6.6. Let m, n, t be arbitrary positive integers, and let A_{1} and A_{2} denote the respective subalgebras of O_{q}(M_{m,t}) and O_{q}(M_{t,n}) generated by all the t × t quantum minors. The set of Γ_{q}^{∗}-coinvariants in O_{q}(V) = O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}) is the subalgebra generated by A_{1} ⊗ A_{2} and Im μ_{q}^{∗}. More precisely,

(O_{q}(M_{m,t}) ⊗ O_{q}(M_{t,n}))^{co O_{q}(SL_{t})} = (A_{1} ⊗ A_{2}) · μ_{q}^{∗}(O_{q}(M_{m,n}^{≤t})).

Proof. As in Remark 2.4, it is enough to consider the case that m = n > t; we leave the details of the reduction to the reader. Let C denote the set of Γ_{q}^{∗}-coinvariants in O_{q}(V), and recall from Proposition 5.1 that C is a subalgebra of O_{q}(V) containing Im μ_{q}^{∗}.

If [I|J] is a t × t quantum minor in O_{q}(M_{n,t}) (necessarily, J = J̃), then as in Remark 2.1(iii)(iv), we see that ρ_{q}^{∗}[I|J] = [I|J] ⊗ d_{T}. It follows that γ_{q}^{∗}([I|J] ⊗ 1) = d_{T}^{−1} ⊗ [I|J] ⊗ 1, whence

Γ_{q}^{∗}([I|J] ⊗ 1) = (π ⊗ id)(d_{T}^{−1} ⊗ [I|J] ⊗ 1) = 1 ⊗ [I|J] ⊗ 1

and so [I|J]⊗1 ∈ C. Similarly, 1⊗[I|J] ∈ C for all t×t quantum minors [I|J] in O_{q}(M_{t,n}). Therefore C contains the subalgebra of O_{q}(V) generated by A_{1} ⊗ A_{2} and Im μ_{q}^{∗}.

On the other hand, it is immediate from Lemmas 5.2(b) and 6.5 that C is contained in (A_{1} ⊗ A_{2}) · Im μ_{q}^{∗}. The theorem follows. □

References

1. C. De Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math 21 (1976), 330-354.

2. K.R. Goodearl and T.H. Lenagan, Quantum determinantal ideals, Preprint, 1998 (available at http://www.math.ucsb.edu/∼goodearl/preprints.html).

3. D. A. Jordan, Notes, 1993.

4. T. Levasseur and J. T. Stafford, The quantum coordinate ring of the special linear group, J. Pure Appl. Algebra 86 (1993), 181-186.

5. S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conf. Series 82, Amer. Math. Soc., Providence, 1993.

6. 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.

7. B. Parshall and J.-P. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 89 (1991), no. 439.

8. C. Procesi, A Primer of Invariant Theory, Brandeis Lecture Notes 1, September 1982.

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

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 address: Laurent.Rigal@univ-st-etienne.fr