GROUP ACTIONS AND INVARIANTS IN ALGEBRAS OF GENERIC MATRICES

Z. REICHSTEIN AND N. VONESSEN

Abstract. We show that the fixed elements for the natural GL_{m}- action on the universal division algebra UD(m, n) of m generic n × nmatrices form a division subalgebra of degree n, assuming n ≥ 3 and 2 ≤ m ≤ n^{2} − 2. This allows us to describe the asymptotic behavior of the dimension of the space of SL_{m}-invariant homogeneous central polynomials p(X_{1}, …, X_{m}) for n × n-matrices. Here the base field is assumed to be of characteristic zero.

Contents

1. Introduction 1 2. Preliminaries 4 3. Geometric actions on division algebras 6 4. Dimension counting in the Grassmannian 9 5. Proof of Theorem 1.4 over an algebraically closed field 11 6. SL_{m}-invariant generic matrices 13 7. Proof of Theorems 1.2 and 1.4 14 8. The case m ≥ n^{2} − 1 15 9. Proof of Theorem 1.3 18 10. Standard polynomials 19 Acknowledgments 20 References 21

1. Introduction

Let k be a field of characteristic zero, m and n be integers ≥ 2, and G_{m,n} = k{X_{1}, …, X_{m}} be the k-algebra of m generic n × n-matrices. That

Date: September 2, 2005. 2000 Mathematics Subject Classification. 16R30, 14L30, 16K20, 16W22, 15A72. Key words and phrases. Generic matrices, universal division algebra, central polyno-

mial, PI-degree, group action, geometric action, invariants, concomitants, Gelfand-Kirillov dimension.

Z. Reichstein was supported in part by an NSERC research grant. N. Vonessen gratefully acknowledges the support of the University of Montana and the

hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

is, G_{m,n} is the k-subalgebra of M_{n}(k[x_{ij}^{(h)}]) generated by

X_{1} = (x_{ij}^{(1)}), …, X_{m} = (x_{ij}^{(m)}) ,

where the x_{ij}^{(h)} are mn^{2} independent commuting variables. By a theorem of Amitsur, G_{m,n} is a domain of PI-degree n. There is a natural action of the general linear group GL_{m} on G_{m,n} given by

(1.1)  g = (g_{ij}): X_{j} ⟼ ∑_{i=1}^{m} g_{ij}X_{i} .

In this paper we will prove the following theorem.

1.2. Theorem. For 2 ≤ m ≤ n^{2}−2, the domain (G_{m,n})^{SL^{m}} has PI-degree n.

The trace ring T_{m,n} of G_{m,n} is defined as the subring of M_{n}(k[x_{ij}^{(h)}]) generated by elements of the form Y and tr(Y), as Y ranges over G_{m,n}. The action (1.1) on G_{m,n} naturally extends to T_{m,n}. Note that the algebras G_{m,n} and T_{m,n}, and their centers Z(G_{m,n}) and Z(T_{m,n}) have a natural ℤ-grading inherited from M_{n}(k[x_{ij}^{(h)}]) (each variable x_{ij}^{(h)} has degree 1) and that this grading is preserved by the action (1.1). As a consequence of Theorem 1.2 we obtain the following result.

1.3. Theorem. Let 2 ≤ m ≤ n^{2} − 2, and let R be one of the rings G_{m,n}, T_{m,n}, Z(G_{m,n}), or Z(T_{m,n}). Denote the degree d homogeneous component of R by R[d]. Then

lim sup_{d→∞} \frac{dim_{k} R^{SL^{m}}[d]}{d^{(m−1)n^{2}−m^{2}+1}}

is a finite nonzero number.

One can think of the center of G_{m,n} as consisting of the m-variable central polynomials for n×n-matrices (over commutative k-algebras). Theorem 1.3 thus describes, for R = Z(G_{m,n}), the asymptotic behavior of the dimension of the space of SL_{m}-invariant homogeneous central polynomials p(X_{1}, …, X_{m}) for n × n-matrices.

The GL_{m}-representations on G_{m,n}, Z(G_{m,n}), T_{m,n} and Z(T_{m,n}) have been extensively studied; see, e.g., [1, 2, 7, 9, 17]. Once again, let R be one of these rings. Recall that the irreducible polynomial representations of GL_{m} are indexed by partitions λ = (λ_{1}, …, λ_{s}) with s ≤ m parts; cf., e.g., [9, Section 2]. Denote the multiplicity of the irreducible GL_{m}-representation corresponding to λ in R by mult_{λ}(R). If (r^{m}) is the partition (r, …, r) (m times) then it is easy to show that

dim R^{SL^{m}}[d] = \{ \begin{matrix}mult_{(r^{m})}(R) & if d = rm, \\ 0 & if d is not a multiple of m; \end{matrix}

see Remark 9.2. The conclusion of Theorem 1.3 can thus be rephrased by saying that

lim sup_{r→∞} \frac{mult_{(r^{m})}(R)}{r^{(m−1)n^{2}−m^{2}+1}}

is a finite nonzero number. We also note that by the Berele-Drensky-Formanek correspondence, mult_{(r^{m})}(R) equals the multiplicity of the S_{m}- character χ^{(d^{m})} in the cocharacter sequence of R; see [9, Section 4].

The division algebra of quotients of G_{m,n} (or equivalently, of T_{m,n}) is called the universal division algebra of m generic n × n-matrices; we shall denote it by UD(m, n). Note that the GL_{m}-action (1.1) on G_{m,n} naturally extends to UD(m, n). We shall deduce Theorem 1.2 from the following related result.

1.4. Theorem. If 2 ≤ m ≤ n^{2} − 2 and n ≥ 3, then UD(m, n)^{GL^{m}} is a division algebra of degree n.

For all other values of m, n ≥ 2, UD(m, n)^{GL^{m}} is a field; see Propositions 8.1 and 8.3. A brief summary of the properties of UD(m, n)^{GL^{m}} and UD(m, n)^{SL^{m}} is given in the two tables below.

Table 1. Properties of UD(m, n)^{GL^{m}}
Case  PI-Degree Transcendence Degree/k Central in UD(m, n)?
m ≤ n^{2} − 2, n ≥ 3  n  (m − 1)n^{2} − m^{2} + 1  No
m = n^{2} − 1  1  n − 1  No
m = n = 2  1  1  No
m ≥ n^{2}  1  0  Yes

Table 2. Properties of UD(m, n)^{SL^{m}}
Case  PI-Degree Transcendence Degree/k Central in UD(m, n)?
m ≤ n^{2} − 2  n  (m − 1)n^{2} − m^{2} + 2  No
m = n^{2} − 1  1  n  No
m = n^{2}  1  1  Yes
m ≥ n^{2} + 1  1  0  Yes

The assertions of the tables in the cases where m ≤ n^{2} − 2 and n ≥ 3 are based on Theorems 1.4 and 5.1, the case where m = n = 2 is considered in [20, Section 14], and the cases where m ≥ n^{2} − 1 are treated in detail in Section 8.

It appears likely that Theorems 1.2 – 1.4 remain valid in prime characteristic (perhaps, not dividing n); we have not attempted to extend them in this direction. Our arguments rely on the work of Richardson [21] and on our own prior papers [19, 20], all of which make the characteristic zero assumption.^{1}

Conventions and Terminology. All central simple algebras in this paper are assumed to be finite-dimensional over their centers. All algebraic varieties, algebraic groups, group actions, morphisms, rational maps, etc., are assumed to be defined over the base field k (which we always assume to be of characteristic zero). By a point of an algebraic variety X we shall always mean a k-point. Throughout, G will denote a linear algebraic group. We shall refer to an algebraic variety X endowed with a regular G-action as a G-variety. We will say that a G-variety X (or the G-action on X) is generically free if Stab_{G}(x) = {1} for x ∈ X in general position. Finally, unless otherwise specified, m and n are integers ≥ 2.

2. Preliminaries

Concomitants. Let Γ be an algebraic group and V and W be Γ-varieties. Then we shall denote the set of Γ-equivariant morphisms V ⟶ W (also known as concomitants) by Morph_{Γ}(V, W) and the set of Γ-equivariant rational maps V −−→ W (also known as rational concomitants) by RMaps_{Γ}(V, W).

In the case where W is a finite-dimensional linear representation of Γ, we also define a relative concomitant as a morphism f: V ⟶ W satisfying the following condition (which is slightly weaker than Γ-equivariance): there is a character χ: Γ ⟶ k^{∗} such that

f(g · v) = χ(g) (g · f(v))

for all v ∈ V and g ∈ Γ. For a rational map f: V −−→ W the notion of a relative rational concomitant is defined in a similar manner. If W = k, with trivial Γ-action, then the term “invariant” is used in place of “concomitant”. For future reference we record the following:

2.1. Lemma. Suppose V and W are finite-dimensional linear representations of Γ. Every rational concomitant f: V −−→ W can be written as \frac{a}{b}, where a is a relative concomitant and b is a relative invariant.

Proof. See the proof of [5, Chapter 1, Proposition 1]. Note that the characters associated to a and b are necessarily equal. □

If W is a k-algebra and Γ acts on W by k-algebra automorphisms, then the algebra structure of W induces algebra structures on Morph_{Γ}(V, W) and RMaps_{Γ}(V, W) in a natural way. Namely, given a, b: V ⟶ W (or a, b: V −−→ W), one defines a + b and ab by (a + b)(v) = a(v) + b(v) and ab(v) = a(v)b(v) for v ∈ V.

1We remark that Richardson [21] worked over k = ℂ, and his proofs are based on analytic techniques. The results we need (in particular, [21, Theorem 9.3.1]), remain valid over any algebraically closed field of characteristic zero by the Lefschetz principle.

Extending [21, Theorem 9.3.1] to prime characteristic is an open problem of independent interest.

2.2. Theorem. (Procesi [16, Theorem 2.1]; cf. also [9, Theorem 10], or [24, Theorem 14.16]) Let (M_{n})^{m} be the space of m-tuples of n × n-matrices; the group PGL_{n} acts on it by simultaneous conjugation. Then

(a) Morph_{PGL_{n}}((M_{n})^{m}, M_{n}) ≃ T_{m,n}

(b) RMaps_{PGL_{n}}((M_{n})^{m}, M_{n}) ≃ UD(m, n) Moreover, the two isomorphisms identify the i-th projection (M_{n})^{m} ⟶ M_{n} with the i-th generic matrix X_{i}. □

Here T_{m,n} and UD(m, n) are, respectively, the trace ring and the universal division algebra of m generic n × n-matrices, defined in the introduction. Note that part (b) of Theorem 2.2 follows from part (a) by Lemma 2.1, since the simple group PGL_{n} does not have nontrivial characters (so that relative concomitants and invariants are actually concomitants and invariants, respectively).

We also remark that the construction of T_{m,n} remains well-defined if m = 1. Theorem 2.2 also holds in this case, provided that one defines UD(1, n) to be the field of quotients of T_{1,n}, rather than G_{1,n}. (For m ≥ 2, T_{m,n} and G_{m,n} have the same division algebra of quotients, but this is not the case for m = 1.).

Geometric actions. For the rest of this section we will assume that k is algebraically closed. If X is a PGL_{n}-variety, then, as we mentioned above, RMaps_{PGL_{n}}(X, M_{n}) has an algebra structure naturally induced from M_{n}. If the PGL_{n}-action on X is generically free then RMaps_{PGL_{n}}(X, M_{n}) is a central simple algebra of degree n, with center k(X)^{PGL^{n}}; cf. [18, Lemmas 8.5 and 9.1].

Suppose that X is a G × PGL_{n}-variety, and that the PGL_{n}-action on X is generically free. Then the G-action on X naturally induces a G-action on RMaps_{PGL_{n}}(X, M_{n}). Following [20] we define the action of an algebraic group G on a central simple algebra A to be geometric if A is G-equivariantly isomorphic to RMaps_{PGL_{n}}(X, M_{n}) for some G × PGL_{n}-variety X as above. The G × PGL_{n}-variety X is then called the associated variety for the Gaction on A; this associated variety is unique (as a G × PGL_{n}-variety), up to birational isomorphism; cf. [20, Corollary 3.2].

Note that we defined geometric actions only if k is algebraically closed. Also note that if an algebraic group acts geometrically on a central simple algebra A, then the center of A is necessarily a finitely generated field extension of k.

Of particular interest to us will be the case where X = (M_{n})^{m} is the space of m-tuples of n × n-matrices. Here PGL_{n} acts on (M_{n})^{m} by simultaneous conjugation (since m ≥ 2, this action is generically free) and G = GL_{m} acts on (A_{1}, …, A_{m}) ∈ (M_{n})^{m} by sending (A_{1}, …, A_{m}) to (B_{1}, …, B_{m}) where B_{j} = ∑_{i=1}^{m} c_{ij}A_{i} and g^{−1} = (c_{ij}). The actions of GL_{m} and PGL_{n} commute, and the GL_{m}-action on (M_{n})^{m} induces the GL_{m}-action (1.1) on UD(m, n). So (M_{n})^{m} is the associated variety for the GL_{m}-action on UD(m, n); see Theorem 2.2 (cf. also [20, Example 3.4]).

We conclude this section with a simple result which we will use repeatedly.

2.3. Lemma. Assume k is algebraically closed. Let X be a G × PGL_{n}- variety which is PGL_{n}-generically free. Denote by π: X −−→ X/G the rational quotient map for the G-action. Then for x ∈ X in general position, the projection pr_{2} : G × PGL_{n} ⟶ PGL_{n} onto the second factor induces an isomorphism from Stab_{G×PGL_{n}}(x)/ Stab_{G}(x) onto Stab_{PGL_{n}}(π(x)). □

Proof. Recall that by a theorem of Rosenlicht, π^{−1}(\overline{x}) is a single G-orbit for \overline{x} ∈ X/G in general position; see [22, Theorem 2] or [14, Section 2.3]. Consequently, for x ∈ X in general position the projection pr_{2} restricts to a surjective morphism Stab_{G×PGL_{n}}(x) ⟶ Stab_{PGL_{n}}(π(x)) of algebraic groups. The kernel of this morphism is clearly Stab_{G}(x), and the lemma follows. □

3. Geometric actions on division algebras

Throughout this section we will assume that k is algebraically closed. The main result of this section is the following theorem; after its proof, we will deduce several corollaries.

3.1. Theorem. Assume k is algebraically closed. Let G be a linear algebraic group acting geometrically on a division algebra D of degree n. Let X be the associated G × PGL_{n}-variety. Then for x ∈ X in general position,

S_{x} := Stab_{G×PGL_{n}}(x)/ Stab_{G}(x)

is reductive.

Proof. Let X be the associated G × PGL_{n}-variety for the G-action on D. Recall that the PGL_{n}-action on X is generically free. We want to show that the group S_{x} = Stab_{G×PGL_{n}}(x)/ Stab_{G}(x) is reductive for x ∈ X in general position. Assume the contrary. Denoting the unipotent radical by R_{u}, this means that R_{u}(Stab_{G×PGL_{n}}(x)) is not contained in G. Since unipotent groups are connected, this is equivalent to

(3.2)  Lie(R_{u}(Stab_{G×PGL_{n}}(x))) ⊈ Lie(G)

for x ∈ X in general position. Here and in the sequel Lie stands for the Lie algebra. To simplify notation, set H = G × PGL_{n}, and for x ∈ X, set H_{x} = R_{u}(Stab_{H}(x)). Now define U_{X} ⊆ X × Lie(H) by

U_{X} = {(x, a) | x ∈ X and a ∈ Lie(H_{x})} .

We first show that U_{X} is a vector bundle over a dense open subset X_{0} ⊂ X. By [21, 6.2.1, 9.2.1, and 6.5.3], there is an H-stable dense open subset X_{0} of X such that {H_{x} | x ∈ X_{0}} is an algebraic family of algebraic subgroups of H. Moreover, dim(H_{x}) is constant for x ∈ X_{0}, say equal to d. Replacing X by X_{0}, we may assume that {H_{x} | x ∈ X} is an algebraic family of algebraic subgroups of H. By [21, 6.2.2], x ↦ Lie(H_{x}) defines a morphism of algebraic varieties from X to the Grassmannian of d-dimensional subspaces of Lie(H). Since the universal bundle over this Grassmannian is a vector bundle (see, e.g., [28, 3.3.1]), its pull-back U_{X} is a vector bundle over X.

Note also that U_{X} is, by definition, an H-invariant subbundle of the trivial bundle X × Lie(H) ⟶ X; here H acts on its Lie algebra by the adjoint action. Since the PGL_{n}-action on X is generically free, the no-name lemma tells us that there is a PGL_{n}-equivariant birational isomorphism U_{X} −−→ X × k^{d} such that the following diagram commutes

\xymatrix{U_{X} \ar[d] \ar@{-->}[r]^{≃} & X × k^{d} \ar[dl] \\ X}

(For a proof and a brief discussion of the history of the no-name lemma, see [4, Section 4.3].) In other words, the vector bundle U_{X} ⟶ X has d PGL_{n}-equivariant rational sections β_{1}, …, β_{d}: X −−→ U_{X} such that β_{1}(x), …, β_{d}(x) are linearly independent for x ∈ X in general position. We identify here β_{i}(x) with a if β_{i}(x) = (x, a) ∈ {x}×Lie(H_{x}). In view of (3.2), some klinear combination β = c_{1}β_{1}+⋯+c_{d}β_{d} has the property that β(x) ∉ Lie(G) for x ∈ X in general position.

Now recall that the natural projection SL_{n} ⟶ PGL_{n} induces a Lie algebra isomorphism sl_{n} ⟶ Lie(PGL_{n}), allowing us to identify the two Lie algebras. Hence

U_{X} ⊆ X × Lie(G) × sl_{n} .

Let f = pr ∘ β: X −−→ sl_{n}, where pr: U_{X} ⟶ sl_{n} denotes the natural projection. Note that sl_{n} ↪ gl_{n} = M_{n}, so that f may be viewed as a PGL_{n}- equivariant rational map X −−→ M_{n}, i.e., as an element of D. The condition that β(x) ∉ Lie(G) ensures that f ≠ 0. On the other hand, we will show below that for x ∈ X in general position, f(x) is a nilpotent n × n-matrix, so that f^{n} = 0. This means that D contains a non-zero nilpotent element f, contradicting our assumption that D is a division algebra.

It remains to be shown that for x ∈ X in general position, f(x) is a nilpotent matrix. The natural projection G × PGL_{n} ⟶ PGL_{n} maps the unipotent group H_{x} to a unipotent subgroup U of PGL_{n}. Denote by K the preimage of U in SL_{n}. It is a solvable group, so its subset K_{u} of unipotent elements is a closed subgroup. The surjection K_{u} ⟶ U is finite-to-one, so their Lie algebras are isomorphic. In particular, f(x) belongs to Lie(K_{u}) ⊂ sl_{n} ⊂ gl_{n} = M_{n}. Finally, since K_{u} is a unipotent subgroup of GL_{n}, its Lie algebra in M_{n} consists of nilpotent matrices, see, e.g., [3, I.4.8]. This completes the proof of Theorem 3.1. □

We now proceed with the corollaries. Recall that a subgroup S ⊂ Γ is said to be a stabilizer in general position for a Γ-variety X if there exists a dense Γ-invariant subset U ⊂ X such that Stab(x) is conjugate to S for any x ∈ U. For a detailed discussion of this notion, see [14, Section 7].

3.3. Corollary. Assume k is algebraically closed. Let G be a linear algebraic group acting geometrically on a division algebra D of degree n. Let X be the associated G × PGL_{n}-variety.

(a) The induced PGL_{n}-action on the rational quotient X/G has a stabilizer S in general position. Moreover, S is reductive, and S ≃ S_{x} = Stab_{G×PGL_{n}}(x)/ Stab_{G}(x) for x ∈ X in general position.

(b) If the G-action on X is generically free, then

trdeg_{k}(Z(D^{G})) = trdeg_{k}(Z(D)^{G})
= dim(X) − dim(G) + dim(S) − n^{2} + 1 .

Proof. (a) It follows from Theorem 3.1 and Lemma 2.3 that points in X/G in general position have a reductive stabilizer. A theorem of Richardson (see [21, Theorem 9.3.1] or [14, Theorem 7.1]) now implies that the PGL_{n}- action on X/G has a stabilizer S in general position. By Lemma 2.3, S ≃ S_{x} = Stab_{G×PGL_{n}}(x)/ Stab_{G}(x) for x ∈ X in general position.

(b) The first equality follows from the fact that Z(D^{G}) is an algebraic extension of Z(D)^{G}. Indeed, the minimal polynomial of any element of D^{G} over Z(D) is unique and must therefore have coefficients in Z(D)^{G}.

To prove the second equality, note that, Z(D) = k(X/PGL_{n}) = k(X)^{PGL^{n}} and thus

Z(D)^{G} ≃ (k(X)^{PGL^{n}})^{G} = k(X)^{G×PGL_{n}} = k(X/(G × PGL_{n})) .

Since we are assuming that G acts generically freely on X, part (a) implies that S ≃ Stab_{G×PGL_{n}}(x) for x ∈ X in general position. Hence the dimension of the general fiber of the rational quotient map X −−→ X/(G × PGL_{n}) is equal to the dimension of (G × PGL_{n})/S. The fiber dimension theorem now tells us that the transcendence degree of Z(D)^{G} is

dim X/(G × PGL_{n}) = dim(X) − dim(G) − dim(PGL_{n}) + dim(S) . □

3.4. Corollary. Assume k is algebraically closed. Let G be a unipotent linear algebraic group acting geometrically on a division algebra D of degree n. Then D^{G} is a division algebra of degree n.

This was proved for algebraic actions in [20, Proposition 12.1].

Proof. By [20, Lemma 7.1], for x ∈ X in general position, Stab_{G×PGL_{n}}(x) is isomorphic to a subgroup of G, so is unipotent. On the other hand, by Theorem 3.1, the projection S_{x} of this group to PGL_{n} is reductive. Thus S_{x} is both unipotent and reductive, which is only possible if S_{x} = {1}. In other words,

Stab_{G×PGL_{n}}(x) ⊂ G × {1} .

The desired conclusion now follows from [20, Theorem 1.4]. □

4. Dimension counting in the Grassmannian

In preparation for the proof of Theorem 1.4 in the next section, we will now establish the following:

4.1. Proposition. Assume k is algebraically closed. Let V be an N-dimen-sional k-vector space and let V = V_{1} ⊕⋯⊕V_{r}, where dim(V_{i}) = N_{i} ≥ 1. Let Z be the subset of the Grassmannian Gr(m, N) consisting of m-dimensional subspaces W of V of the form

W = W_{1} ⊕ ⋯ ⊕ W_{r} ,

where W_{i} ⊆ V_{i}. (Here we allow W_{i} = (0) for some i.) Then Z is a closed subvariety of Gr(m, N). If 2 ≤ m ≤ N − 2, then each irreducible component of Z has codimension ≥ N−max_{i=1,...,r}(N_{i}) in Gr(m, N). Moreover, equality holds (for some irreducible component of Z) only if (i) r = 1 or (ii) r = 2, m = 2 and N = 4.

Proof. Let m_{1}, … m_{r} be non-negative integers such that m_{1} + ⋯ + m_{r} = m and such that m_{i} ≤ N_{i} for all i. Let Z_{m_{1},...,m_{r}} be the image of the map

ϕ_{m_{1},...,m_{r}}: Gr(m_{1}, N_{1}) × ⋯ × Gr(m_{r}, N_{r}) ⟶ Gr(m, N)

given by (W_{1}, …, W_{r}) ↦ W_{1} ⊕⋯⊕W_{r}. (Here Gr(m_{i}, N_{i}) is the Grassmannian of m_{i}-dimensional vector subspaces of V_{i}.) Since the domain of the map ϕ_{m_{1},...,m_{r}} is projective, its image is closed in Gr(m, N). Thus each Z_{m_{1},...,m_{r}} is a closed irreducible subvariety of Gr(m, N) birationally isomorphic to

Gr(m_{1}, N_{1}) × ⋯ × Gr(m_{r}, N_{r})

and Z is the union of the Z_{m_{1},...,m_{r}}. It remains to show that

(4.2)  dim Gr(m, N) − ∑_{i=1}^{r} dim Gr(m_{i}, N_{i}) ≥ N − max_{i=1,...,r}(N_{i}) ,

and that equality is only possible if r = 1 or r = 2, N_{1} = N_{2} = 2 and m_{1} = m_{2} = 1 (and thus N = N_{1} + N_{2} = 4 and m = m_{1} + m_{2} = 2). Recall that dim Gr(m, N) = (N − m)m. Letting l_{i} = N_{i} − m_{i} and l = N − m = l_{1} + ⋯ + l_{r}, we can rewrite (4.2) as

lm − ∑_{i=1}^{r} l_{i}m_{i} ≥ l + m − max_{i=1,...,r} (l_{i} + m_{i})

or, equivalently,

(l − 1)(m − 1) − 1 ≥ ∑_{i=1}^{r} l_{i}m_{i} − max_{i=1,...,r}(l_{i} + m_{i}) .

Proposition 4.1 is thus a consequence of the following elementary lemma. □

4.3. Lemma. Let (l_{1}, m_{1}), …, (l_{r}, m_{r}) be r pairs of non-negative integers and let l = ∑_{i=1}^{r} l_{i} and m = ∑_{i=1}^{r} m_{i}. Assume that l_{i} + m_{i} ≥ 1 for every i = 1, …, r and l, m ≥ 2. Then

(4.4)  (l − 1)(m − 1) − 1 ≥ ∑_{i=1}^{r} l_{i}m_{i} − max_{i=1,...,r}(l_{i} + m_{i}) .

Moreover, equality holds if and only if either (i) r = 1 or (ii) r = 2 and (l_{1}, m_{1}) = (l_{2}, m_{2}) = (1, 1).

Proof. We consider two cases.

Case 1: Suppose that for every i = 1, …, r, either l_{i} = 0 or m_{i} = 0. Since l, m ≥ 2, we have (l − 1)(m − 1) − 1 ≥ 0. On the other hand, ∑_{i=1}^{r} l_{i}m_{i} − max_{i=1,...,r}(l_{i} + m_{i}) = − max_{i=1,...,r}(l_{i} + m_{i}) < 0. Hence, in this case (4.4) holds and is a strict inequality.

Case 2: Now suppose that l_{i}, m_{i} ≥ 1 for some i ≥ 1, …, r. After renumbering the pairs (l_{1}, m_{1}), …, (l_{r}, m_{r}), we may assume i = 1. Now set

l′_{j} = \begin{cases}l_{1} − 1, if j = 1, \\ l_{j}, otherwise; \end{cases} and m′_{j} = \begin{cases}m_{1} − 1, if j = 1 \\ m_{j}, otherwise. \end{cases}

Note that l′_{j}, m′_{j} ≥ 0 for every j = 1, …, r. Thus

(l − 1)(m − 1) − 1 = (∑_{i=1}^{r} l′_{i})(∑_{j=1}^{r} m′_{j}) − 1 = ∑_{i=1}^{r} l′_{i}m′_{i} + ∑_{i≠j} l′_{i}m′_{j} − 1

≥ ∑_{i=1}^{r} l′_{i}m′_{i} − 1 = ∑_{i=1}^{r} l_{i}m_{i} − (l_{1} + m_{1})

≥ ∑_{i=1}^{r} l_{i}m_{i} − max_{i=1,...,r}(l_{i} + m_{i}) .

This completes the proof of the inequality (4.4).

It is easy to see that equality holds in cases (i) and (ii). It remains to show that the inequality (4.4) is strict for all other choices of (l_{1}, m_{1}), …, (l_{r}, m_{r}). Indeed, a closer look at the above argument shows that equality in (4.4) can hold if and only if we are in Case 2 and

(a) l′_{i}m′_{j} = 0 whenever i ≠ j and

(b) l_{1} + m_{1} = max_{i=1,...,r}(l_{i} + m_{i}).

Assume that conditions (a) and (b) are satisfied. Since ∑_{i=1}^{r} l′_{i} = l−1 ≥ 1, we cannot have l′_{i} = 0 for all i = 1, …, r. In other words, l′_{i_{0}} ≥ 1 for some i_{0} ∈ {1, …, r}. Then condition (a) says that m′_{j} = 0 for every j ≠ i_{0}. On the other hand, m′_{i_{0}} = ∑_{j=1}^{r} m′_{j} = m − 1 ≥ 1, and applying condition (a) once again, we conclude that that l′_{i} = 0 for every i ≠ i_{0}. To sum up, there exists an i_{0} ∈ {1, …, r} such that l′_{i_{0}} = l − 1, m_{i_{0}} = m − 1 and l′_{i} = m′_{i} = 0 for every i ≠ i_{0}.

In particular, for every i ≠ 1, i_{0}, we have l_{i} = l′_{i} = 0 and m_{i} = m′_{i} = 0, contradicting our assumption that l_{i} + m_{i} ≥ 1. Thus i ∈ {1, i_{0}} for every i = 1, …, r. In other words, either i_{0} = 1 and r = 1 (in which case (i) holds, and we are done) or i_{0} = 2 and r = 2. In the latter case l′_{1} = m′_{1} = 0, l′_{2} = l −1 and m′_{2} = m−1, i.e., (l_{1}, m_{1}) = (1, 1) and (l_{2}, m_{2}) = (l −1, m−1). Condition (b) now tells us that l = m = 2, so that (ii) holds.

This completes the proof of Lemma 4.3 and thus of Proposition 4.1. □

5. Proof of Theorem 1.4 over an algebraically closed field

Recall from Section 2 that X = (M_{n})^{m} is the associated GL_{m} × PGL_{n}- variety for the GL_{m}-action on UD(m, n). Here PGL_{n} acts on (M_{n})^{m} by simultaneous conjugation (since m ≥ 2, this action is generically free) and GL_{m} acts on (M_{n})^{m} by sending (A_{1}, …, A_{m}) to (B_{1}, …, B_{m}), where B_{j} = ∑_{i=1}^{m} c_{ij}A_{i} and g^{−1} = (c_{ij}).

We shall assume throughout this section that k is an algebraically closed field of characteristic zero. Our goal is to prove Theorem 1.4 over such k. In view of [20, Theorem 1.4(a)], we only need to establish the following.

5.1. Theorem. Assume k is algebraically closed. If n ≥ 3 and 2 ≤ m ≤ n^{2} − 2, then the GL_{m} × PGL_{n}-action on (M_{n})^{m} is generically free.

Proof. The linear action of GL_{m} on (M_{n})^{m} is easily seen to be the direct sum of n^{2} copies of the natural m-dimensional representation of GL_{m}, i.e., to be isomorphic to the GL_{m}-action on n^{2}-tuples of vectors in k^{m}. Since n^{2} > m, this action is generically free. Corollary 3.3(a) with G = GL_{m} and X = (M_{n})^{m} tells us that the PGL_{n}-action on (M_{n})^{m}/GL_{m} has a reductive stabilizer S in general position, and that S ≃ Stab_{GL_{m}×PGL_{n}}(x)/ Stab_{GL_{m}}(x) = Stab_{GL_{m}×PGL_{n}}(x) for x ∈ X in general position.

Recall that (M_{n})^{m}/GL_{m} is PGL_{n}-equivariantly birationally isomorphic to the Grassmannian Gr(m, n^{2}) of m-dimensional subspaces of M_{n}. Thus the PGL_{n}-action on Gr(m, n^{2}) has a stabilizer S in general position, where S is a reductive subgroup of PGL_{n}. (Recall that S is only well-defined up to conjugacy in PGL_{n}). To prove Theorem 5.1, it suffices to show that S is trivial.

Assume the contrary. Since S is reductive, it contains a non-trivial element g of finite order. Then every L ∈ Gr(m, n^{2}) in general position is fixed by some conjugate of g. In other words, the map

(5.2)  \begin{matrix}PGL_{n} × Gr(m, n^{2})^{g} & ⟶ & Gr(m, n^{2}) \\ (h, L) & ↦ & h(L) \end{matrix}

is dominant; here Gr(m, n^{2})^{g} denotes the fixed points of g in Gr(m, n^{2}). Denote by C(g) the centralizer of g in PGL_{n}. Note that Gr(m, n^{2})^{g} is C(g)- stable. Hence the fiber of (5.2) over h(L) contains (hc, c^{−1}(L)) for every c ∈ C(g). So by the fiber dimension theorem,

(5.3)  dim Gr(m, n^{2}) + dim C(g) ≤ dim PGL_{n} + dim Gr(m, n^{2})^{g} .

Since g has finite order, it is diagonalizable. So we may assume that

g = diag(\underbrace{λ_{1}, …, λ_{1}}_{l_{1} times}, …, \underbrace{λ_{s}, …, λ_{s}}_{l_{s} times}) = diag(α_{1}, …, α_{n}) ,

where λ_{1}, …, λ_{s} are the (distinct) eigenvalues of g. Note that s ≥ 2, because g ≠ 1 in PGL_{n} and that g acts on the matrix units E_{ij} by g·E_{ij} = α_{i}α_{j}^{−1}E_{ij}. The matrix algebra M_{n} naturally decomposes as a direct sum of character spaces

V_{μ} = Span(E_{ij} | α_{i}α_{j}^{−1} = μ) .

In particular, V_{1} is the commutator of g in M_{n}. Now (5.3) implies

dim Gr(m, n^{2}) − dim Gr(m, n^{2})^{g} ≤ dim(PGL_{n}) − dim C(g)
= n^{2} − dim(V_{1}) .

So if n ≥ 3, part (b) of the following lemma gives the desired contradiction, which completes the proof of Theorems 5.1, and thus of Theorem 1.4 in the case that k is algebraically closed. □

5.4. Lemma. Let n ≥ 2, and 2 ≤ m ≤ n^{2} − 2.

(a) dim V_{1} ≥ dim V_{μ} for any μ ≠ 1.

(b) If n ≥ 3 (or n = 2 and there are more than two distinct nonzero V_{μ}), then dim Gr(m, n^{2}) − dim Gr(m, n^{2})^{g} > n^{2} − dim(V_{1}).

Proof. (a) Note that dim V_{1} = l_{1}^{2} + ⋯ + l_{s}^{2} and

dim V_{μ} = ∑_{λ_{i}λ_{j}^{−1}=μ} l_{i}l_{j}

Since the eigenvalues λ_{1}, …, λ_{s} of g are distinct, the last sum has at most one term for each i = 1, …, s. Thus there is a permutation σ of {1, …, s} such that

dim V_{μ} ≤ l_{1}l_{σ(1)} + ⋯ + l_{s}l_{σ(s)} .

So for v = (l_{1}, …, l_{s}) and w = (l_{σ(1)}, …, l_{σ(s)}), dim V_{μ} ≤ v · w. Hence by the Cauchy-Schwarz inequality,

dim V_{μ} ≤ v · w ≤ |v| |w| = |v|^{2} = l_{1}^{2} + ⋯ + l_{s}^{2} = dim(V_{1}) .

(b) Since g is semisimple, every L ∈ Gr(m, n^{2})^{g} is a direct sum of its character subspaces spaces, i.e., a direct sum of vector subspaces of the V_{μ}. Part (b) now follows from Proposition 4.1 with V = M_{n}, N = n^{2}, N_{μ} = dim(V_{μ}), Z = Gr(m, n^{2})^{g}, and r the number of distinct nonzero V_{μ}. □

5.5. Remark. We assumed throughout this section that n ≥ 3. If n = 2 then the above argument still goes through provided there are more than two distinct non-zero character subspaces V_{μ}; see Lemma 5.4(b). In particular, this will always be the case if g^{2} ≠ 1 in PGL_{n}; indeed, in this case g = (λ_{1}, λ_{2}), where μ = λ_{1}/λ_{2} ≠ ±1 and the three spaces V_{1}, V_{μ} and V_{μ^{−1}} are distinct. Thus the above argument also shows that for n = m = 2, either |S| = 1 or S has exponent 2. It turns out that, in fact, in this case |S| = 2; see [20, Lemma 14.2].

5.6. Remark. An alternative approach to proving Theorem 5.1 would be to appeal to the classification, due to A. G. Elashvili [8] and A. M. Popov [13], of pairs (G, ϕ), where G is a semisimple algebraic group and ϕ: G ↪ GL(V) is an irreducible linear representation of G such that the G-action on V has a non-trivial stabilizer in general position. Since this classification is rather involved, and since additional work would be required to apply it in our situation (note that the group GL_{m} × PGL_{n} is not semisimple, and that its representation on (M_{n})^{m} is not irreducible), we opted instead for the self-contained direct proof presented in this section.

6. SL_{m}-invariant generic matrices

The goal of this section is to relate the rings of SL_{m}-invariants in G_{m,n} and UD(m, n).

6.1. Lemma. (a) Every element of UD(m, n)^{GL^{m}} can be written in the form \frac{a}{b}, where a is a homogeneous element of (T_{m,n})^{SL^{m}}, and b is a non-zero homogeneous element of Z(T_{m,n})^{SL^{m}} of the same degree as a.

(b) Assume that a subgroup G of GL_{m} has no non-trivial characters. Then every element of UD(m, n)^{G} can be written as \frac{a}{b} where a ∈ (T_{m,n})^{G} and 0 ≠ b ∈ Z(T_{m,n})^{G}.

Proof. Both parts follows from Lemma 2.1. In part (a), we take Γ = GL_{m} × PGL_{n}, V = (M_{n})^{m} (with the Γ-action defined in the beginning of Section 5) and W = M_{n} (where GL_{m} acts trivially on W and PGL_{n} acts by conjugation). Here the relative concomitants (M_{n})^{m} ⟶ M_{n} are the homogeneous elements of (T_{m,n})^{SL^{m}} and the relative invariants (M_{n})^{m} ⟶ k are the homogeneous elements of Z(T_{m,n})^{SL^{m}}; cf. Theorem 2.2.

If G has no non-trivial characters then relative concomitants are (absolute) concomitants, i.e., elements of (T_{m,n})^{G}. Similarly, relative invariants are elements of Z(T_{m,n})^{G}, and part (b) is thus simply a restatement of Lemma 2.1 in this special case. □

6.2. Proposition. Let G be a subgroup of GL_{m} such that G has no nontrivial characters. Then the following conditions are equivalent:

(a) UD(m, n)^{G} has PI-degree n.

(b) (T_{m,n})^{G} has PI-degree n.

(c) (G_{m,n})^{G} has PI-degree n.

Proof. The equivalence of (a) and (b) follows from Lemma 6.1(b). The implication (c) ⟹ (b) is obvious, since G_{m,n} ⊂ T_{m,n}. It thus remains to prove that (b) ⟹ (c).

Let g_{n} be the multilinear central polynomial for n × n-matrices in [12, 13.5.11] (or [23, p. 26]). If R is a prime PI-algebra of PI-degree n, denote by g_{n}(R) the set of all evaluations of g_{n} in R, and denote by R g_{n}(R) the nonzero ideal of R generated by g_{n}(R). Denote by T the trace ring of R. (Since we are working in characteristic zero, the (noncommutative) trace ring in [12, 13.9.2] is the same as the one we are using, see [12, 13.9.4].) Then R g_{n}(R) is a common ideal of R and T, see [12, 13.9.6] (or [23, 4.3.1]).

Now let R = G_{m,n}. Then its trace ring is T = T_{m,n}. Recall that we are assuming (b) holds, i.e., T^{G} has PI-degree n. Let s be a non-zero evaluation of g_{n} on T^{G}. Then s is a nonzero G-invariant, and a central element of T (since it is also an evaluation of g_{n} on T). Since g_{n} is multilinear, and since T is generated as an R-module by central elements, s belongs to the ideal of T generated by g_{n}(R), so that s T ⊆ R g_{n}(R) ⊆ R. Since s is a G-invariant, it follows that s T^{G} ⊆ R^{G}. Consider the central localization R^{G}[s^{−1}] ⊆ UD(m, n). Since it contains T^{G}, R^{G}[s^{−1}] must have PI-degree n, implying that also R^{G} must have PI-degree n. This completes the proof of the implication (b) ⟹ (c) and thus of Proposition 6.2. □

6.3. Remark. The same argument also shows that if the three equivalent conditions in Proposition 6.2 are true, then the division algebras of fractions of (G_{m,n})^{G} and (T_{m,n})^{G} are both equal to UD(m, n)^{G}.

7. Proof of Theorems 1.2 and 1.4

Proof of Theorem 1.2. Proposition 6.2 tells us that (G_{m,n})^{SL^{m}} has PIdegree n if and only if so does UD(m, n)^{SL^{m}}. Thus in order to prove Theorem 1.2 it suffices to show that UD(m, n)^{SL^{m}} has PI-degree n whenever 2 ≤ m ≤ n^{2} − 2.

For n = m = 2 we showed this in [20, Remark 14.4] (in fact, the argument we gave there remains valid over any base field k of characteristic ≠ 2). For n ≥ 3 and 2 ≤ m ≤ n^{2} − 2, Theorem 1.4 tells us that UD(m, n)^{GL^{m}} has PIdegree n (and consequently, so does UD(m, n)^{SL^{m}}). In summary, we have shown that Theorem 1.2 follows from Theorem 1.4. □

Proof of Theorem 1.4. We have already proved Theorem 1.4 in the case where the base field k is algebraically closed; see Section 5. We will now reduce the general case to this one by using Lemma 6.1(a).

We begin with a simple lemma.

7.1. Lemma. Let K be an extension field of k, let V be a finite-dimensional k-vector space, and V_{K} = V ⊗_{k} K. Given a linear representation of SL_{m}(k) on V, we have

(V_{K})^{SL^{m}(K)} = V^{SL^{m}(k)} ⊗_{k} K .

Proof. Since SL_{m}(k) is dense in SL_{m}(K), the subspace (V_{K})^{SL^{m}(K)} is defined inside V_{K} by a system of homogeneous linear equations with coefficients in k. Clearly finitely many of these equations suffice. Since the dimension of the solution space of such a system is the rank of the corresponding matrix (which has coefficients in k), (V_{K})^{SL^{m}(K)} has a K-basis consisting of elements of V^{SL^{m}(k)}. □

For the remainder of this section, we will write G_{m,n}(K), T_{m,n}(K) and UD(m, n)(K) to denote the generic matrix algebra, trace ring and universal division algebra defined over the field K. Denote the algebraic closure of k by \overline{k}. Since the SL_{m}-action on UD(m, n) preserves degree, Lemma 7.1 immediately implies the following fact, which we record for later use.

7.2. Corollary. G_{m,n}(\overline{k})^{SL^{m}(\overline{k})} = G_{m,n}(k)^{SL^{m}(k)} ⊗_{k} \overline{k}, and T_{m,n}(\overline{k})^{SL^{m}(\overline{k})} = T_{m,n}(k)^{SL^{m}(k)} ⊗_{k} \overline{k}. □

We are now ready to complete the proof of Theorem 1.4 over an arbitrary field k of characteristic zero. In Section 5 we showed that Theorem 1.4 holds over the algebraic closure \overline{k} of k. That is, if 2 ≤ m ≤ n^{2} − 2 then there exist elements c_{1}, …, c_{r} ∈ UD(m, n)(\overline{k})^{GL^{m}(\overline{k})} which span UD(m, n)(\overline{k}) as a vector space over its center. By Lemma 6.1 we can write c_{i} = a_{i}/b_{i}, where a_{i} ∈ T_{m,n}(\overline{k})[d_{i}]^{SL^{m}} and 0 ≠ b_{i} ∈ Z(T_{m,n}(\overline{k}))[d_{i}]^{SL^{m}} for some d_{i} ≥ 0, i = 1, …, r. By Lemma 7.1, with K = \overline{k} and V = Z(T_{m,n}(k))[d_{i}], we have Z(T_{m,n}(k))[d_{i}]^{SL^{m}} ≠ 0. We may now replace b_{i} by a non-zero element of Z(T_{m,n}(k))[d_{i}]^{SL^{m}}. The new c_{i} = a_{i}/b_{i} are still GL_{m}-invariant elements of UD(m, n)(\overline{k}), and they still generate UD(m, n)(\overline{k}) as a vector space over its center.

We now apply Lemma 7.1 once again (this time with V = T_{m,n}(k)[d_{i}]) to write each a_{i} as a finite sum ∑ γ_{ij}a_{ij}, where each γ_{ij} ∈ \overline{k} and each a_{ij} ∈ T_{m,n}(k)[d_{i}]^{SL^{m}}. Now replace our collection of GL_{m}-invariant elements \{c_{i} = a_{i}/b_{i}\} in UD(m, n)(\overline{k}) by \{c_{ij} = a_{ij}/b_{i}\}. By construction, the elements c_{ij} lie in UD(m, n)(k)^{GL^{m}} and span UD(m, n)(\overline{k}) over its center. Hence, these elements generate a k-subalgebra of UD(m, n)(k)^{GL^{m}} of PI-degree n. Consequently, UD(m, n)(k)^{GL^{m}} itself has PI-degree n. This completes the proof of Theorem 1.4 (and of Theorem 1.2). □

8. The case m ≥ n^{2} − 1

Theorems 1.2 and 1.4 assume that m ≤ n^{2} − 2. We will now describe UD(m, n)^{GL^{m}} and UD(m, n)^{SL^{m}} for m ≥ n^{2} − 1.

Recall the definition of the discriminant of n^{2} matrices of size n × n, say A_{1}, …, A_{n^{2}}: it is the determinant of the n^{2} × n^{2}-matrix whose i-th row consists of the entries of A_{i}, cf. (8.5). When viewed as a function (M_{n})^{n^{2}} ⟶ k, Δ is the unique multilinear alternating function such that Δ(e_{11}, e_{12}, …, e_{nn}) = 1; cf., e.g., [10, Lemma 3]. Here the e_{ij} are the matrix units.

8.1. Proposition. (a) If m > n^{2}, then UD(m, n)^{SL^{m}} = UD(m, n)^{GL^{m}} = k. Now let m = n^{2}, and denote by Δ the discriminant of the generic matrices X_{1}, …, X_{m}.

(b) UD(m, n)^{GL^{m}} = k.

(c) (T_{m,n})^{SL^{m}} = k[Δ].

(d) UD(m, n)^{SL^{m}} = k(Δ).

Proof. (a) We may clearly assume that k is algebraically closed. In this case SL_{m} has a dense orbit in the associated variety X = (M_{n})^{m}. Thus the rational quotient X/SL_{m} is a single point (with trivial PGL_{n}-action), and

UD(m, n)^{SL^{m}} = RMaps_{PGL_{n}}(pt, M_{n}) = k .

Now suppose m = n^{2}. Then GL_{m} has a dense orbit in X = (M_{n})^{m}. Arguing as in part (a), we prove (b); cf. [20, Proposition 13.1(a)]. (c) is proved in [9, p. 210], and (d) follows from (c) by Lemma 6.1(b). □

8.2. Remark. Let m = n^{2}. Formanek showed that Δ ∉ (G_{m,n})^{SL^{m}} ([9, p. 214]) but Δ^{i} ∈ G_{m,n} for every integer i ≥ 2 (this follows from [10, Theorem 16]). Consequently for m = n^{2},

(G_{m,n})^{SL^{m}} = k[Δ^{2}, Δ^{3}] .

8.3. Proposition. Suppose m = n^{2} − 1, and let

Y = ∑_{i,j=1}^{n} Δ(X_{1}, …, X_{m}, e_{ji}) e_{ij} ,

where the e_{ij} are the matrix units.

(a) Y ∈ (T_{m,n})^{SL^{m}}.

(b) The eigenvalues of Y are algebraically independent over k (and, in particular, distinct).

(c) (T_{m,n})^{SL^{m}} = k[c_{1}, …, c_{n−1}, Y] is a polynomial ring in n independent variables over k. Here c_{1} = − tr(Y), …, c_{n} = (−1)^{n} det(Y).

(d) UD(m, n)^{SL^{m}} = k(c_{1}, …, c_{n−1}, Y ).

(e) UD(m, n)^{GL^{m}} = k(\frac{c_{2}}{(c_{1})^{2}} , …, \frac{c_{n−1}}{(c_{1})^{n−1}} , \frac{1}{c_{1}} Y ).

Proof. For the proof of (a)—(c), we may assume that k is algebraically closed (cf. Corollary 7.2). (a) We view Y as a regular map (M_{n})^{m} ⟶ M_{n}. We want to show that this map is PGL_{n}-equivariant, i.e., Y ∈ T_{m,n}. Since Y is clearly SL_{m}-equivariant (recall that SL_{m} acts trivially on M_{n}), this will imply part (a).

We begin by observing that for any (A_{1}, …, A_{m}) ∈ (M_{n})^{m}, and Z ∈ M_{n},

(8.4)  tr(Y (A_{1}, …, A_{m})Z) = Δ(A_{1}, …, A_{m}, Z) .

Indeed, both sides are linear in Z, so we only need to check (8.4) for the elementary matrices Z = e_{ij}, where it is easy to do directly from the definition of Y.

Fix an m-tuple (A_{1}, …, A_{m}) ∈ (M_{n})^{m} of n × n-matrices. Since the trace form on M_{n} is non-singular, Y (A_{1}, …, A_{m}) is the unique matrix satisfying (8.4) for every Z ∈ M_{n}. The PGL_{n}-equivariance of Y : (M_{n})^{m} ⟶ M_{n} is an easy consequence of this and the fact that Δ is PGL_{n}-invariant (see [9, p. 209]). This concludes the proof of (a).

Our proof of parts (b) and (c) relies on the following claim: Y : (M_{n})^{m} ⟶ M_{n} is the categorical quotient map for the SL_{m}-action on (M_{n})^{m}. In other words, we claim that the n^{2} elements Δ(X_{1}, …, X_{m}, e_{ij}) (i, j = 1, …, n) generate k[(M_{n})^{m}]^{SL^{m}} as a k-algebra. To prove this claim we will temporarily write (A_{1}, …, A_{m}) ∈ (M_{n})^{m} as an m × n^{2}-matrix

(8.5)  A = (\begin{matrix}a_{11}^{(1)} & a_{12}^{(1)} & … & a_{ij}^{(1)} & … & a_{nn}^{(1)} \\ ⋮ & ⋮ & & ⋮ & & ⋮ \\ a_{11}^{(m)} & a_{12}^{(m)} & … & a_{ij}^{(m)} & … & a_{nn}^{(m)} \end{matrix}) .

That is, we write each n × n matrix A_{h} = (a_{ij}^{(h)}) as a single row of A. In this notation, g ∈ SL_{m} acts on (M_{n})^{m} by multiplication by the transpose of g^{−1} on the left; that is, g(A) = (g^{−1})^{transpose} · A for every g ∈ SL_{m}. Let δ_{ij}(A_{1}, …, A_{m}) be the m × m-minor of this matrix obtained by removing the ij-column from A and taking the determinant of the resulting m × mmatrix. The first theorem of classical invariant theory (see [27, Theorem II.6.A] or [6, Theorem 2.1]) says that the elements δ_{ij}(X_{1}, …, X_{m}) generate k[(M_{n})^{m}]^{SL^{m}} as a k-algebra. On the other hand, it is easy to see that δ_{ij}(X_{1}, …, X_{m}) = ±Δ(X_{1}, …, X_{m}, e_{ij}). This proves the claim.

Now observe that since m = n^{2} − 1,

dim((M_{n})^{m} // SL_{m}) = mn^{2} − (m^{2} − 1) = n^{2} = dim(M_{n}) .

This means that the n^{2} SL_{m}-invariant functions

Δ(X_{1}, …, X_{m}, e_{ji}): (M_{n})^{m} ⟶ k

are algebraically independent over k. In other words, Y (viewed as a matrix in T_{m,n} ⊂ M_{n}(k[x_{ij}^{(h)}]) has algebraically independent entries. Part (b) easily follows from this assertion; cf. [15, Lemma II.1.4].

Furthermore,

(T_{m,n})^{SL^{m}} = Morph_{SL_{m}×PGL_{n}}((M_{n})^{m}, M_{n})
≃ Morph_{PGL_{n}}((M_{n})^{m} // SL_{m}, M_{n})
≃ Morph_{PGL_{n}}(M_{n}, M_{n}) = T_{1,n} ,

where T_{1,n} is the trace ring of one generic n × n-matrix. Here the last equality is a special case of Procesi’s Theorem 2.2(a) (with m = 1). Since the chain of isomorphisms identifies Y with the identity map M_{n} ⟶ M_{n}, we conclude that

(T_{m,n})^{SL^{m}} = k[c_{1}, …, c_{n}, Y] .

Since Y^{n} + c_{1}Y^{n−1} + ⋯ + c_{n} = 0, k[c_{1}, …, c_{n−1}, Y] = k[c_{1}, …, c_{n}, Y]. This proves the first assertion in (c).

To show that c_{1}, …, c_{n−1}, Y are algebraically independent over k, denote the eigenvalues of Y by λ_{1}, …, λ_{n}. By part (b), λ_{1}, …, λ_{n} are algebraically independent over k. Since Y is algebraic over k(c_{1}, …, c_{n}), we have

trdeg_{k} k(c_{1}, …, c_{n−1}, Y) = trdeg_{k} k(c_{1}, …, c_{n}, Y)
= trdeg_{k} k(c_{1}, …, c_{n}) = trdeg_{k} k(λ_{1}, …, λ_{n}) = n .

This shows that c_{1}, …, c_{n−1}, Y are algebraically independent over k, thus completing the proof of (c).

(d) is an immediate consequence of (c) and Lemma 6.1(b). To prove (e), denote the central torus of GL_{m} by 𝔾_{m}. Then

UD(m, n)^{GL^{m}} = (UD(m, n)^{SL^{m}})^{𝔾_{m}} = k(c_{1}, …, c_{n−1}, Y)^{𝔾_{m}} ,

where 𝔾_{m} acts on the purely transcendental extension k(c_{1}, …, c_{n−1}, Y) as follows: t · c_{i} ↦ t^{im}c_{i} for i = 1, …, n − 1, and and t · Y ↦ t^{m}Y. Part (e) easily follows from this description. □

8.6. Remark. Note that c_{1} = −Δ(X_{1}, …, X_{n^{2}−1}, I_{n}), where I_{n} is the n × n identity matrix. By a theorem of Formanek, (c_{1})^{2} is an element of Z(G_{m,n})^{SL^{m}} for m = n^{2} − 1, see [10, Theorem 16].

9. Proof of Theorem 1.3

By Corollary 7.2, we may assume that k is algebraically closed. Set A = (G_{m,n})^{SL^{m}} and B = (T_{m,n})^{SL^{m}}. By Theorem 1.2, A and B both have PI-degree n. Thus Z(A) = (Z(G_{m,n}))^{SL^{m}} and Z(B) = (Z(T_{m,n}))^{SL^{m}}. Since SL_{m} is a reductive group, and since T_{m,n} is a finitely generated k-algebra and a finite module over its center, B is a finite Z(B)-module, and both B and Z(B) are finitely generated Noetherian k-algebras, cf. [26, Proposition 4.2]. Moreover, B is an FBN ring, cf. [12, 13.6.6].

By Corollary 3.3(b) and Remark 6.3, the transcendence degrees of both B and Z(B) are t = (m − 1)n^{2} − m^{2} + 2. For notational simplicity, set

μ(S) = lim sup_{d→∞} \frac{dim_{k} S[d]}{d^{t−1}}

for any graded k-algebra S = ⊕_{d≥0}S[d]. By [25, Lemma 6.1] (cf. also [11, §12.6]), f(d) = dim_{k} B[d] is eventually periodically polynomial, i.e., there are polynomials f_{1}, …, f_{s} with rational coefficients such that f(d) = f_{i}(d) whenever d is large enough and congruent to i modulo s; moreover, the maximum of the degrees of the f_{i} is t − 1. Consequently μ(B) exists and is equal to the largest among the leading coefficients of those f_{i} of degree t−1. A similar argument shows that μ(Z(B)) exists and is a nonzero number.

Consider the multilinear central polynomial g_{n} for n × n matrices used in the proof of Proposition 6.2. Since it is multilinear and nonzero on A, we can find a nonzero evaluation c of g_{n} at homogeneous elements of A; this c is homogeneous. Since c is also an evaluation of g_{n} on G_{m,n}, cT_{m,n} ⊂ G_{m,n}, so that cB ⊂ A and c Z(B) ⊂ Z(A). Then for all integers d ≥ j, cB[d − j] ⊆ A[d] ⊆ B[d], where j = deg c. Replacing c by c^{s} if necessary, we may assume that s divides j. Consequently, whenever d is large enough and congruent to i modulo s,

f_{i}(d − j) ≤ dim_{k} A[d] ≤ f_{i}(d) .

It follows easily that μ(A) exists and is equal to the largest among the leading coefficients of those f_{i} of degree t − 1. A similar argument shows that μ(Z(A)) exists and is a nonzero number. □

9.1. Remark. The above proof shows that μ((G_{m,n})^{SL^{m}}) = μ((T_{m,n})^{SL^{m}}), and that μ(Z((G_{m,n})^{SL^{m}})) = μ(Z((T_{m,n})^{SL^{m}})).

9.2. Remark. Consider the GL_{m}-representation on R, where R = G_{m,n}, T_{m,n}, Z(G_{m,n}) or Z(T_{m,n}). Recall that irreducible polynomial representations of GL_{m} are indexed by partitions λ = (λ_{1}, …, λ_{s}) with s ≤ m parts; cf. [9, Section 2]. Denote the multiplicity of the irreducible representation corresponding to λ in R by mult_{λ}(R). Writing (r^{m}) for the partition λ = (r, ..., r) (m times), we have

(a) dim_{k} R^{SL^{m}}[d] = 0 if d is not a multiple of m, and

(b) dim_{k} R^{SL^{m}}[rm] = mult_{(r^{m})}(R) for any integer r ≥ 1.

Proof. (a) Assume R^{SL^{m}}[d] is nonzero. Then it is a direct sum of onedimensional representations of GL_{m} of the form M = Span_{k}(v). Moreover, any such representation is given by g(v) = det(g)^{r}v for some integer r; cf., e.g., [9, Theorem 3(a)]. On the other hand, substituting g = tI_{m}, where t ∈ k and I_{m} is the m × m identity matrix, we obtain, g(v) = t^{d}v. Since det(tI_{m}) = t^{m}, we see that d = rm, as claimed.

(b) If d = rm and 0 ≠ v ∈ R^{SL^{m}}[rm] then the partition associated to the 1-dimensional irreducible GL_{m}-module M = Span(v) is (r^{m}); cf. , e.g., [9, Theorem 2]. Now consider the direct sum decomposition R = ⊕R_{λ}, where R_{λ} is the sum of all irreducible GL_{m}-submodules of R with associated partition λ. The argument of part (a) shows that R_{(r^{m})} = R^{SL^{m}}[rm]. Moreover, since dim(M) = 1, we have

dim_{k} R^{SL^{m}}[rm] = dim_{k} R_{(r^{m})} = mult_{(r^{m})}(R) ,

as claimed. □

10. Standard polynomials

Let G_{m,n} be the ring of m generic n × n-matrices. By Theorem 1.2, (G_{m,n})^{SL^{m}} is a PI domain of degree n, whenever 2 ≤ m ≤ n^{2} − 2. We will now describe one particular element of this ring. Let

F_{m}(x_{1}, …, x_{m}) = ∑_{σ∈S_{n}}(−1)^{σ}x_{σ(1)} … x_{σ(n)} ∈ k{x_{1}, …, x_{m}}

be the standard polynomial. Since F_{m} is multilinear and alternating, one checks easily that for g ∈ GL_{m},

(10.1)  g(F_{m}) = det(g) · F_{m};

see, e.g., [23, 1.4.12]. Substituting m generic n×n-matrices X_{1}, …, X_{m} into F_{m}, we obtain

f_{m,n} = F_{m}(X_{1}, …, X_{m}) = ∑_{σ∈S_{n}}(−1)^{σ}X_{σ(1)} … X_{σ(n)} ∈ G_{m,n} .

From (10.1), we see that f_{m,n} ∈ (G_{m,n})^{SL^{m}}. By the Amitsur-Levitzki Theorem, f_{m,n} = 0 iff m ≥ 2n.

Fix m, n ≥ 2 and let K be the center of UD(m, n).

10.2. Proposition. For 2 ≤ m < 2n, K(f_{m,n}) generates a GL_{m}-stable maximal subfield of UD(m, n).

The proof is algebraic in nature and works in characteristic ≠ 2.

Proof. The fact that K(f_{m,n}) is a GL_{m}-stable subfield follows from (10.1). In order to prove that this subfield is maximal, it suffices to verify that f_{m,n} has an eigenvalue of multiplicity 1. (Indeed, if, say, n = d · [K(f_{m,n}) : K], then the characteristic polynomial p(t) of f_{m,n} in UD(m, n) has the form p(t) = q(t)^{d}, where q(t) is the minimal polynomial of f_{m,n} over K. This shows that the multiplicity of each eigenvalue of f_{m,n} is divisible by d.)

Since the multiplicity of eigenvalues cannot decrease when evaluating f_{m,n} in M_{n}, it suffices now to show that f_{m,n} (or equivalently, F_{m}) has some evaluation in M_{n} with an eigenvalue of multiplicity one. We now proceed to construct such an evaluation. Since

F_{m}(1, x_{2}, …, x_{m}) = F_{m−1}(x_{2}, …, x_{m})

for m odd (cf. [23, Exercise 1.2.3]), we may assume that m is even, say m = 2r −2, with 1 < r ≤ n. In M_{n}, consider the sequence of m matrix units

e_{1,2}, e_{2,2}, e_{2,3}, e_{3,3}, …, e_{r−2,r−1}, e_{r−1,r−1}, e_{r−1,r}, e_{r,1} .

When permuting these matrix units cyclically, their product is nonzero; for any other permutation, their product is zero. Since an m-cycle is odd, it follows that F_{m} evaluated at these matrix units is

e_{1,1} − e_{2,2} + e_{2,2} − + ⋯ − e_{r−1,r−1} + e_{r−1,r−1} − e_{r,r} = e_{1,1} − e_{r,r} ,

which has 1 as an eigenvalue of multiplicity one (since char(k) ≠ 2). □

We do not know an explicit expression for any non-constant element of (G_{m,n})^{SL^{m}} (as a polynomial in the generic n × n-matrices X_{1}, …, X_{m}) in the case where 2n ≤ m ≤ n^{2} − 2; we leave this as an open question. Note that for m = n^{2} and m = n^{2} −1, such elements are exhibited in Remarks 8.2 and 8.6.

Acknowledgments

We are grateful to A. Berele and E. Formanek for helpful comments.

References

[1] A. Berele, Homogeneous polynomial identities, Israel J. Math. 42 (1982), no. 3, 258– 272.

[2] A. Berele, Colength sequences for matrices, J. Algebra 283 (2005), no. 2, 700–710.

[3] A. Borel, Linear Algebraic Groups, second enlarged edition, Springer-Verlag, New York, 1991.

[4] V. Chernousov, Ph. Gille, Z. Reichstein, Resolving G-torsors by abelian base extensions, Journal of Algebra, to appear. Preprint available at http://www.math.ubc.ca/^{∼}reichst/torsor-ind.html.

[5] J. A. Dieudonné, J. B. Carrell, Invariant Theory, Old and New, Academic Press, New York-London, 1971.

[6] I. Dolgachev, Lectures on Invariant Theory, Cambridge Univ. Press, Cambridge, 2003.

[7] V. S. Drensky, Codimensions of T -ideals and Hilbert series of relatively free algebras, C. R. Acad. Bulgare Sci. 34 (1981), no. 9, 1201–1204.

[8] A. G. Elashvili, Stationary subalgebras of points of general position for irreducible linear Lie groups (in Russian), Funkcional. Anal. i Priložen. 6 (1972), no. 2, 65–78; English translation in Functional Anal. Appl. 6 (1972), 139–148.

[9] E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), no. 1, 178–223.

[10] E. Formanek, A conjecture of Regev about the Capelli polynomial, J. Algebra 109 (1987), 93–114.

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

[12] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, Chichester, 1987.

[13] A. M. Popov, Finite isotropy subgroups in general position of irreducible semisimple linear Lie groups (in Russian), Trudy Moskov. Mat. Obshch. 50 (1987), 209–248; English translation in Trans. Moscow Math. Soc. 1988, 205–249.

[14] V. L. Popov and E. B. Vinberg, Invariant Theory, Algebraic Geometry IV, Encyclopedia of Mathematical Sciences 55, Springer, 1994, 123–284.

[15] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967), 237–255.

[16] C. Procesi, The invariant theory of n × n matrices, Advances in Math. 19 (1976), no. 3, 306–381.

[17] A. Regev, Codimensions and trace codimensions of matrices are asymptotically equal, Israel J. Math. 47 (1984), no. 2-3, 246–250.

[18] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

[19] Z. Reichstein and N. Vonessen, Polynomial identity rings as rings of functions, J. Algebra, to appear. Preprint available at www.arxiv.org/math.RA/0407152.

[20] Z. Reichstein and N. Vonessen, Group actions on central simple algebras: a geometric approach, www.arxiv.org/math.RA/0408420.

[21] R. W. Richardson Jr., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35–73.

[22] M. Rosenlicht, Some basic theorems on algebraic groups, American Journal of Math. 78 (1956), 401–443.

[23] L. H. Rowen, Polynomial Identities in Ring Theory, Academic Press, 1980.

[24] D. J. Saltman, Lectures on Division Algebras, CBMS Regional Conference Series in Mathematics, No. 94, Amer. Math. Soc., Providence, 1999.

[25] J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, J. Algebra 168 (1994), 988–1026.

[26] N. Vonessen, Actions of linearly reductive groups on affine PI-algebras, Mem. Amer. Math. Soc. 414, 1989.

[27] H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.

[28] J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge University Press, Cambridge, UK, 2003.

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichste@math.ubc.ca
URL: www.math.ubc.ca/^{∼}reichst

Department of Mathematical sciences, University of Montana, Missoula, MT 59812-0864, USA

E-mail address: Nikolaus.Vonessen@umontana.edu
URL: www.math.umt.edu/vonessen