The Ring of Generic Matrices

Journal of Algebra 258 (2002) 310–320 www.elsevier.com/locate/jalgebra

The ring of generic matrices

Edward Formanek

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA Received 14 November 2001 Dedicated to Claudio Procesi on the occasion of his 60th birthday

1. Introduction

Let K be a ﬁeld of characteristic zero, let n be a positive integer, and let { k | ∈ N} xij 1 i, j n, k be independent commuting indeterminates over K. Then = k ∈ k Xk xij Mn K xij × [ k ] is called an n n generic matrix over K,andtheK-subalgebra of Mn(K xij ) generated by the Xk is called a ring of generic matrices and will be denoted K{X}. The ring of generic matrices is a basic object in the study of the polynomial identities and invariants of n×n matrices. My goal is to give a brief history of this topic, describe some of the decisive contributions of Claudio Procesi, and draw attention to my favorite problems in the area, where only slight progress has been made in the past ten years.

2. Ancient history (1857–1965)

The single most basic result is the Cayley–Hamilton Theorem, which was ﬁrst stated by Cayley in a letter to Sylvester in 1857. The connection between the Cayley–Hamilton Theorem and the polynomial identities satisﬁed by n × n matrices was not realized until the work of Kostant [Ko], Procesi [P4], and Razmyslov [R2].

E-mail address: [email protected].

In the latter part of the nineteenth century there were many papers concerned with a single matrix and its invariants, but the corresponding questions for sets of matrices—which are far from being resolved today—were mostly untouched before the twentieth century. However, there were two interesting cases in the nineteenth century where sets of 2 × 2 matrices were studied. The first of these is a little known paper of Sylvester [Sy] in which he studied apairof2× 2 matrices X and Y . This paper was written at Johns Hopkins University in 1883, his last year in the United States. Sylvester proceeds by taking the determinant of the matrix αI + βX + γY, which today we would note is equal to α2 + tr(X)αβ + tr(Y )αγ + det(X)β2 + det(Y )γ 2 + tr(X) tr(Y ) − tr(XY ) βγ. His main result (in modern language) is that any element in the K-algebra generated by X and Y can be expressed as a linear combination of I,X,Y, and XY over the commutative K-algebra generated by the final five coefficients tr(X),tr(Y ),det(X),det(Y ),tr(X) tr(Y ) − tr(XY ) in the above equation. He also determines when I,X,Y,XY are linearly independent—i.e., when their discriminant is nonzero. By the way, the “involution of two matrices” in the title of his paper means this discriminant. These same computations were used by Procesi [P1] to show that the center of the generic division ring of two 2 × 2 generic matrices is a rational function field in five variables. A further result of Sylvester was rediscovered in the determination of the commutator ideal of the generic matrix ring generated by a pair of 2 × 2 generic matrices by Formanek et al. [FHL]. I will quote him, since it is an example of his entertaining habit of reviewing his own work as he writes: “Hence follows this new and striking theorem concerning matrices of the second order: If f(X,Y) and φ(X,Y) are any rational functions whatever of X, Y , the determinant to the matrix XY − YX is contained as a factor in the determinant to the matrix fφ− φf .” The second instance where sets of 2×2 matrices are studied is the work of Vogt [Vo], and Fricke and Klein [FK] around 1890. Unlike the nearly forgotten paper of Sylvester, their work has been the basis of much subsequent research. The idea was to study SL(2, C)-representations of groups and their traces or characters. The survey of Magnus [M] contains several applications of this method to combinatorial group theory. The underlying structure was made precise and extended to arbitrary n via the introduction of the variety of degree n semisimple representations by M. Artin [Ar] and Procesi [P3]. However, for n>2, there are few cases where this variety has been described, and no applications to group theory. This is partly because the ring of n×n generic matrices is well understood only for n 2, and partly because much more is known about finitely generated subgroups of GL(n, C) for n = 2 than for n>2 (see Bass [B1,B2]). 312 E. Formanek / Journal of Algebra 258 (2002) 310–320

In 1935 Dubnov [D] gave the multilinear form of the Cayley–Hamilton Theorem for 2 × 2 matrices, and the candidate thesis of Ivanov (see [DI]) in the next decade gave a minimal generating set for the invariants of triples of 2 × 2 matrices. Also worth mentioning are many computations of invariants of sets of 2 × 2and3× 3 matrices in the period 1948–1965 by Rivlin, Spencer, and others (see [Sp] for a compilation of this work). The first and second fundamental theorems of vector invariants are highly relevant, but it will take a better historian than me to trace their history. All I will say is that they appear in the books of Weyl [We] and Gurevich [G]. The first polynomial identities satisfied by Mn(K) were given by Wagner in 1937 [W], and the Amitsur–Levitzki Theorem [AL], which gives the polynomial identity of minimal degree satisfied by Mn(K), was proved in 1950. In the same decade Amitsur proved that the ring of n × n generic matrices is a domain [A2] and that it has a ring of fractions which is central simple of dimension n2 over its center [A1]. The preceding is everything that was known about the ring of generic matrices before Procesi was a graduate student at the University of Chicago in the mid- 1960s.

3. Rings related to the ring of n × n generic matrices

There are several rings closely associated with the ring of n × n generic matrices, and most of what we know about it comes via these rings. They are all k k K-subalgebras of Mn(K(xij )), and their centers are all K-subalgebras of K(xij ), ∈ k · where α K(xij ) is identiﬁed with the scalar matrix α I. These rings are:

• R = K{X}, the ring of generic matrices; • C(R), the center of R; • C(R), the set of traces of elements of R, called the ring of invariants; • R, the ring generated by R and C(R), called the trace ring; • Q(R), the generic division ring; • C(Q(R)), the center of Q(R).

Figure 1 presents the diagram of inclusions, where the rings in the upper half of the diagram are the centers of the rings in the lower half. This diagram and proofs of some of its properties can be found in [F5, pp. 195–199] and [F6, Chapter 8]. It took approximately twenty years for this diagram to come into existence, and I will now give a sequential account of its discovery. As noted above, Amitsur had proved in 1955 that R was a domain and that it had a division ring of fractions Q(R) of dimension n2 over its center C(Q(R)). He obtained Q(R) from R by noncommutative localization using the Ore condition. However, the “ring of generic matrices” in the form that I have deﬁned it did not appear in his work. E. Formanek / Journal of Algebra 258 (2002) 310–320 313

k K xij

k C(R) C(R) C(Q(R)) K xij

k R R Q(R) Mn K xij

k Mn K xij

Fig. 1.

For Amitsur, R was the relatively free algebra K X/T(n),whereK X is a free associative algebra in countably many variables over K and T(n) is the T -ideal of polynomial identities satisfied by Mn(K). It was Procesi who, as a graduate student (see [P5, p. 275]), made the simple but crucial observation that K X/T(n) is isomorphic to R = K{X}, the ring of n × n generic matrices. This replaced an abstract relatively free algebra with a concrete object and made calculations possible. Indeed, all subsequent work on the ring of generic matrices depends on this observation. It also brought k k Mn(K(xij )) and its center K(xij ) into the diagram. To the best of my knowledge, the first appearance of the term “ring of generic matrices” occurs in the seminal paper [P1] of Procesi—see Section 4. The next relevant work was M. Artin’s paper [Ar] on Azumaya algebras in 1969, where he asked the following question: If GL(n, K) acts by simultaneous conjugation on a set of n × n matrices, is the ring of invariants generated by the traces of products of the matrices? It seems that the Russians already knew this (see [Ki]). Working on this question led Procesi to describe R and C(R) as rings of GL(n, K)-invariants—see Section 5. In the meantime C(R) was added to the diagram. More precisely, central polynomials were constructed in 1972 by Formanek [F1] and Razmyslov [R1], giving a positive answer to a problem of Kaplansky [K]. Central polynomials are elements of the free associative algebra K X, without constant term, whose evaluations on Mn(K) are scalar matrices. It is easy to see that C(R), the center of R, is the direct sum of K and the evaluations of the central polynomials on R. Thus the existence of central polynomials is equivalent to the fact that C(R) is strictly larger than K. Two useful consequences of the existence of central polynomials are that Q(R), the generic division ring, is obtained from R by inverting the nonzero elements of C(R),andC(Q(R)) is the field of quotients of C(R). 314 E. Formanek / Journal of Algebra 258 (2002) 310–320

A priori, the principal objects in the diagram are R and C(R). Although R is a relatively free ring, the analogue of a commutative polynomial ring, R and C(R) fail to be Noetherian. (That is, for n 2 they are not Noetherian even if the number of generic matrices is taken to be finite and at least two. They are not Noetherian for trivial reasons if the number of matrices is infinite.) They are also graded rings which can be studied quantitatively by counting the dimensions of their graded parts. This has been the theme of Regev’s work [Re], not only for these rings, but for other T -ideals and relatively free algebras. For the most part only asymptotic results have been obtained. Exact formulas are known only in a handful of cases. The final two rings to appear in the diagram, R and C(R), largely overcome these difficulties, while not being too different from R and C(R). They had been introduced and shown to be Noetherian (assuming, of course, a finite number of generic matrices) in Procesi’s book [P2] and mostly expository article [P3], but were only described later as rings of GL(n, K)-invariants in [P4], another seminal paper. Since they are rings of GL(n, K)-invariants, the dimensions of their graded parts are given by explicit formulas. Admittedly, evaluating these formulas presents further problems.

4. The center of the generic division ring

There is no doubt that the most important open problem about the ring of n× n generic matrices is:

Problem 1. Is C(Q(R)), the center of the generic division ring, (stably) rational over K?

AﬁeldL is rational over K if it is a pure transcendental extension of K.It is stably rational over K if a pure transcendental extension L(t1,...,ts ) of L is rational over K.Itisunirational over K if it is contained in a rational extension of K. This problem was raised in the Procesi’s ﬁrst seminal paper about generic matrices [P1], where he introduced them and also:

• exhibited R, Q(R) and C(Q(R)) as concrete objects, as already noted above; • reduced Problem 1 to the generic division ring generated by a pair of n × n generic matrices; • gave an affirmative answer to Problem 1 for 2 × 2 generic matrices; • showed that for a pair of n × n generic matrices C(Q(R)) the center of the generic division ring is the fixed field of Sn, the symmetric group on n letters, acting on a rational function field in n2 + 1 variables over K. E. Formanek / Journal of Algebra 258 (2002) 310–320 315

To avoid introducing more notation, I use the same notation for the generic matrix ring generated by a pair of n × n generic matrices as for that generated by an infinite number. Procesi’s description of C(Q(R)) as a fixed field of Sn has been the basis for all subsequent positive results in Problem 1. Since they are few in number, I can easily mention all of them. The first was my 1978 paper giving an affirmative answer for 3 × 3 generic matrices [F2]. The starting point was the observation that the action of Sn in Procesi’s description of C(Q(R)) is a lattice action on a relatively simple lattice. If G is a group, G-lattice is a finitely generated free abelian group A = a1,...,at (written multiplicatively) on which G acts. This induces an action of [ ]= [ ±1 ±1] G on the group ring or Laurent polynomial ring K A K a1 ,...,at ,and hence on its quotient field. Such an action is called a lattice action. In the problem at hand, G acts trivially on K, but the definition permits nontrivial actions. As an aside, let me mention that my realization that Procesi’s description of C(Q(R)) was via a lattice action came indirectly from Amitsur’s proof that Q(R) is not a crossed product when n is divisible by 8 or the square of an odd prime [A3]. The starting point of his proof is the observation that Q(R) is a “universal object” among central simple algebras of dimension n2 over their center, where the center contains K. Over the years, many other kinds of universal division algebras were defined. One such, due to Snider [Sn], is universal for division algebras having a maximal subfield Galois over the center of the division algebra, with Galois group G,whereG is a given finite group. In his construction, the center is the fixed field of a lattice action of G. This suggested looking at Procesi’s description of the center of the generic division ring as a fixed field of Sn from this point of view. For more details, see [F4]. For n = 3, the lattice is small enough for the field problem to be solved by hand [F2], but for n 4 this has not been done. In any event, my solution for n = 4 [F3] was accomplished by using part of the arsenal of methods devised to study lattice actions (my survey [F4] gives some of them) to reduce the problem from a lattice of rank 17 to a lattice of rank 3. The same strategy was used by Bessenrodt and Le Bruyn [BL] to establish the stable rationality of C(Q(R)) over K for n = 5andn = 7. In both cases they were able to reduce the problem from a lattice of rank n2 + 1 to a lattice of rank n − 1. This required massive calculations with Sn-lattices. Furthermore, they revealed the limitations of this approach by showing that a reduction to a lattice of rank n − 1 is not possible if n is a prime larger than 7. Beneish was able to give a proof for n = 5andn = 7 of much less complexity by showing how to reduce the problem from an action of Sn to an action of the normalizer of a Sylow n-subgroup in these two cases [Be]. This is the only new idea in ten years, and offers some hope for a positive solution for prime n.Onthe other hand, if Problem 1 has a positive solution for all n, it looks like a new idea is needed. 316 E. Formanek / Journal of Algebra 258 (2002) 310–320

Katsylo [Ka] and Schofield [S1] have shown that the question of stable rationality can be reduced to the prime power case. This implies, in conjunction with the above results, that the center is stably rational if n is a divisor of 3 · 4 · 5 · 7 = 420. I recommend the survey of Le Bruyn [L] for a complete (at the time it was written) discussion of this problem, including approaches aimed at a negative answer, as well as the implications of a solution to the problem. In particular, he notes that there was a series of major open problems which would have been solved by a solution to Problem 1, thus providing extra motivation for solving it. These were solved in turn, so each motivation would vanish and be replaced by a new motivation. This process has likely come to an end, since at present the motivation is a result of Schofield [S2] that C(Q(R)) is stably birational to the function fields associated with many moduli spaces. (Two fields F1 and F2 are stably birational if some finite pure transcendental extension of F1 is isomorphic to some finite pure transcendental extension of F2.) Thus, if the rationality problem for these moduli spaces is resolved, Problem 1 will also be resolved.

5. The invariants of n × n matrices

We now turn to the final two rings in the diagram, C(R) and R.Theyfirst appeared as I have defined them: C(R) as the set of traces of R elements, and R as the ring generated by R and C(R). It was in this form that Procesi proved they were Noetherian [P2,P3]. Of greater importance are two further descriptions of C(R) and R in the second seminal paper of Procesi [P4], the first as rings of GL(n, K)-invariants, the second as factor rings of certain “free trace rings” by explicit ideals. There is considerable overlap between [P4] and Razmyslov’s paper [R2]: both introduce trace identities and show that all of them can be derived from the fundamental trace identity, which is a multilinear form of the Cayley–Hamilton polynomial, and both show how ordinary polynomial identities for n × n matrices can be interpreted as trace identities. The fundamental trace identity and the passage between ordinary polynomial identities and trace identities had first been used by Kostant [Ko], to give a proof of the Amitsur–Levitzki Theorem. The first description of C(R) shows that it is a ring of GL(n, K)-invariants ∈ = k in the classical sense. For P GL(n, K) and a generic matrix Xk (xij ), −1 = k P k = k let P XkP (yij ) and define ϕ (xij ) yij . This extends by K-linearity to a homogeneous action P k → k ϕ : K xij K xij . The First Fundamental Theorem of Matrix Invariants is that the fixed ring of [ k ] K xij under this action of GL(n, K) is C(R). In other words, the fixed ring is generated by all tr(Xi1 ...Xit ), the traces of products of the generic matrices, E. Formanek / Journal of Algebra 258 (2002) 310–320 317 which explains why C(R) is called the ring of invariants. This consequence of the theory of vector invariants was rediscovered by Procesi [P4]. (It was already known to the Russians, see [Ki].) Procesi also showed that R is the fixed [ k ] ∈ → ring of GL(n, K) acting on Mn(K xij ),whereP GL(n, K) acts by (uij ) P −1 k P(ϕ (uij ))P . Note that this action of GL(n, K) extends to Mn(K(xij )),and then its fixed ring is Q(R), the generic division ring. Because C(R) and R are rings of GL(n, K)-invariants, classical invariant theory gives formulas for their Poincaré series. This gives rise to estimates for the Poincaré series of R and C. Much more information would be given by a solution to the next problem, but this seems at present out of reach, except for n = 2, where almost everything is known.

Problem 2. Give presentations of R,C(R),R,andC(R) as K-algebras, or of R and R as modules over their centers.

The second description of C(R) starts with the pure free trace ring K[tr], which is a commutative polynomial ring with one independent generator tr(xi1 ...xik ), for each conjugacy class of cyclic words (omitting the empty word, 1) in a countable set of noncommuting variables x1,x2,....Inotherwords, = tr(xi1 ...xik ) tr(xj1 ...xjk ) if and only if (i1,...,ik) is a cyclic permutation of (j1,...,jk). It can also be deﬁned in terms of the Hattori–Stallings trace,which for a ring S is the canonical map tr : S → S/[S,S],where[S,S] is the abelian group generated by commutators of elements of S.IfK0 X is the free associative algebra (without unit) in countably many variables x1,x2,..., then the pure free trace ring is the symmetric algebra on K0 X/[K0 X,K0 X]. → The map tr(xi1 ...xit ) tr(Xi1 ...Xit ) extends to a K-algebra homomor- phism Θ : K[tr]→C(R), which is onto by the ﬁrst fundamental theorem. The Second Fundamental Theorem of Matrix Invariants ([P4, Theorem 4.5] and [R2, Theorem 2]) says that the kernel of Θ is generated by all substitutions in the fundamental trace identity, which is a multilinear form of the Cayley–Hamilton Theorem. For example, the fundamental trace identity for n = 2is

tr(x1) tr(x2) tr(x3) + tr(x1x2x3) + tr(x1x3x2) − tr(x1) tr(x2x3)

− tr(x2) tr(x3x1) − tr(x3) tr(x1x2), and by a substitution is meant any polynomial obtained by replacing x1,x2,x3 by any monomials in x1,x2,.... There is a similar description for R. It is obtained by replacing K[tr] by K[tr]⊗ K X and extending Θ by deﬁning K ⊗ = Θ tr(xi1 ...xis ) xj1 ...xjt tr(Xi1 ...Xis )Xj1 ...Xjt . See [F6, Chapter 8] for more details. 318 E. Formanek / Journal of Algebra 258 (2002) 310–320

The Second Fundamental Theorem allows many questions about the ring of n × n generic matrices to be translated into combinatorial questions about the group algebra of the symmetric group. The most interesting of these concerns d(n), the least integer such that C(R) is generated by elements of degree d(n). Razmyslov [R2] showed by an elegant argument that d(n) n2. The key point was that if a Young tableau has n2 + 1 boxes, then either its first row or its first column has at least n + 1 boxes. Procesi related d(n) to the Nagata–Higman Theorem, which (we now know) was first proved by Dubnov and Ivanov in 1943 [DI]. The Nagata–Higman Theorem says that if xn = 0 for every element x in a K-algebra A, then there is an integer e such that Ae = 0 (of course, A does not have a unit, and it is essential that K be of characteristic zero). Let e(n) denote the least integer n which works for all A. Kuzmin [Ku] showed that e(n) n(n + 1)/2, and conjectured that e(n) = n(n + 1)/2. Using the fact that xn is the leading term of the Cayley– Hamilton polynomial, Procesi [P4] showed that d(n)= e(n).

Problem 3. Find d(n), the least integer such that C(R) is generated by elements of degree d(n).

Combining the above results shows that n(n + 1)/2 d(n) n2. It is only known that d(n) = n(n + 1)/2forn 4, and the proof for n = 4wasadifﬁcult computer-aided calculation (Vaughan–Lee [V]).

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