The First Fundamental Theorem of Coinvariant Theory for the Quantum General Linear Group

We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum groups Oq(GLt(K)) and Oq(SLt(K)) on the quantized algebra Oq(Mm,t(K)) ⊗ Oq(Mt,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 Oq(Mm,t(K))⊗Oq(Mt,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 GLt = GLt(K) on the variety V := Mm,t(K)×Mt,n(K) induces an action of GLt on O(V ), and the classical theorem determines the algebra of invariants, O(V )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 Oq(GLt) 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 GLt on V induces a coaction of O(GLt) on O(V ), under which O(V ) becomes an O(GLt)comodule algebra, and O(V )t equals the algebra of O(GLt)-coinvariants in O(V ). It is this situation which has a natural quantization: the coordinate ring O(V ) becomes the algebra Oq(V ) := Oq(Mm,t)⊗Oq(Mt,n), and the coaction of O(GLt) on O(V ) becomes a coaction of Oq(GLt) on Oq(V ). We prove the First Fundamental Theorem of Coinvariant Theory for this coaction, that is, we identify the set of coinvariants, Oq(V ) qt. There is a natural comultiplication map θ q : Oq(Mm,n) → Oq(Mm,t)⊗Oq(Mt,n), which is the quantum analog of matrix multiplication Mm,t ×Mt,n → Mm,n. We prove that the 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. Typeset by AMS-TEX 1 2 K. R. GOODEARL, T. H. LENAGAN, AND L. RIGAL 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 Oq(Mm,n): this is the Second Fundamental Theorem of Coinvariant Theory for this comodule action. Taken together, these two results give a complete description of Oq(V ) qt. Further, we investigate the coaction of Oq(SLt) on Oq(V ) induced by that of Oq(GLt) 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 Oq(GLt)-comodule Oq(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 Mu,v = Mu,v(K) for the set of u× v matrices with entries in K. We will be mainly interested in the general linear group GLt = GLt(K) and its standard action on the algebraic variety V = Vm,t,n := Mm,t × Mt,n. This action is given by: GLt × V γ // V (g, (A,B)) // (Ag, gB). Thus GLt acts on O(V ) ∼= O(Mm,t)⊗O(Mt,n). Classical invariant theory is interested in computing the subalgebra O(V )t of invariants for this action. The description of this algebra goes as follows. Consider the morphism of varieties Mm,t ×Mt,n θ // Mm,n (A,B) // AB and its associated comorphism θ : O(Mm,n) → O(Mm,t) ⊗ O(Mt,n). Let Xij (for 1 ≤ i ≤ m and 1 ≤ j ≤ n) stand for the usual coordinate functions on the variety Mm,n, and let It+1 denote the ideal of O(Mm,n) generated by all the (t+ 1) × (t+ 1) minors of the generic matrix (Xij) over O(Mm,n). (This ideal is zero if t ≥ min{m,n}.) Theorem 0.1. The ring of invariants O(V )t equals Im θ. Theorem 0.2. The kernel of θ is It+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 GLt). They give a complete description of O(V )t . We denote by M m,n the subvariety of Mm,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 Mm,t ×Mt,n μ −→ M m,n ⊆ −→ Mm,n. COINVARIANT THEORY FOR QUANTUM GLt 3 Since the restriction map r : O(Mm,n) → O(M ≤t m,n) 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 )t = Imμ. Further, Theorem 0.2 shows that ker(r) = ker(θ) = It+1, and therefore O(M m,n) = O(Mm,n)/It+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 the open subset of M m,n 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 GLt 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 = Mm,t ×Mt,n we write A and B in the forms A = [ A0 A1 ] and B = [B0 B1 ] with A0, B0 ∈ Mt while A1 ∈ Mm−t,t and B1 ∈ Mt,n−t. With these notations, we define a morphism of varieties: V ◦ i // M m,n ×GLt (A,B) // (AB,B0). 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 GLt on M m,n ×GLt induced by γ ◦ is just the natural one: GLt × (M ≤t,◦ m,n ×GLt) ξ // M m,n ×GLt (g, C, h) // (C, gh). The invariants for ξ are easy to compute, using for instance the coinvariants of the associated O(GLt)-comodule structure on O(M ≤t,◦ m,n ×GLt) (see the quantum case below). One then proves that O(M m,n ×GLt) GLt = O(M m,n ) ⊆ O(M ≤t,◦ m,n ×GLt). 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 given by multiplication of matrices, and denote by (μ) its comorphism: (μ) : O(M m,n ) −→ O(V ) = O(M m,t)⊗O(M ◦ t,n). 4 K. R. GOODEARL, T. H. LENAGAN, AND L. RIGAL Then we can describe i as the composition i : O(M m,n )⊗O(GLt) (μ◦)∗⊗id −−−−−−→ O(V )⊗O(GLt) id⊗⊆ −−−→ O(V )⊗O(M t,n) = −→ O(M m,t)⊗O(M ◦ t,n)⊗O(M ◦ t,n) id⊗m −−−→ O(V ), where 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 )⊗ 1) = (μ )(O(M m,n )). The ring of invariants for γ can thus be described as the localisation of Imμ with respect to the multiplicative set generated by dY ⊗ dZ where dY is the uppermost t × t minor of the generic matrix (Yij) of generators of O(Mm,t), and dZ is the leftmost t × t minor of the generic matrix (Zij) of generators of O(Mt,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 )dY ⊗dZ (for the action γ ). Hence, there is a non-negative integer s such that φ(dY ⊗ dZ) s ∈ Imμ. Thus, in order to establish that O(V )t = Imμ, it is enough to prove that for ψ ∈ O(V ), if there is a non-negative integer s such that ψ(dY ⊗ dZ) s ∈ Imμ, then ψ ∈ Imμ. 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: Oq(Mm,n), Oq(Mm,t), Oq(Mt,n) and Oq(GLt). In order to avoid confusion, we will denote their respective canonical generators by Xij , Yij , Zij and Tij . The definitions of these algebras will be as in [6, 2]; for instance, XijXik = qXikXij when j < k. Thus, each time we use results from [7] we must replace q by q. Finally, a convention concerning notation: each time that we have to deal with the multiplication map in an algebra, we will denote it by 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,m, η,∆, 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-comodule using the structure map A ρ∗ −→ A⊗H id⊗S −−−→ A⊗H τ12 −−→ H ⊗ A, where τ12 is the flip. COINVARIANT THEORY FOR QUANTUM GLt 5 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 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) a0 ⊗ a1 and λ (b) = ∑ (b) b−1 ⊗ b0 for a ∈ A and b ∈ B (cf. [5, p. 11]), one thus has γ(a⊗ b) = ∑ (a),(b) S(a1)b−1 ⊗ a0 ⊗ b0. 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 coH := {x ∈ M | ν(x) = 1⊗ x}. It is immediate that if (M, ν) is a comodule algebra then M coH 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) 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 = ∑r i=1 ai ⊗ bi. Since both ρ ∗ and λ are algebra morphisms, we have (ρ ⊗ λ)(vw) = r ∑