Let K be a field of characteristic zero and let \(R(n,r)\) be the K-algebra generated by r generic \(n\times n\) matrices. The algebra \(\bar C(\)n,r) generated by the traces of elements of R(n,r) and the trace ring \(\bar R(n,r)=R(n,r)\bar C(n,r)\) play important roles in ring theory. The investigation of their numerical invariants is one of the main problems in the theory of PI-algebras. In particular, the Hilbert (or Poincaré) series \(\chi (A)(x_ 1,...,x_ r)\) of the multigraded vector spaces \(A=\bar C(n,r)\) and \(A=\bar R(n,r)\) are actively studied. These series are rational symmetric functions and their explicit form is known for \(n=2\) only.
This paper is motivated by a recent result of L. Le Bruyn [Isr. J. Math. 52, 355-360 (1985; Zbl 0587.16014)] which claims that \(\chi(\bar R(2,r))(x_ 1,...,x_ r)\) satisfies a functional equation. In the paper under review many interesting properties of \(\bar C(n,r)\) and \(\bar R(n,r)\) are established. In particular, the ring \(\bar C(n,r)\) is a Cohen-Macaulay unique factorization domain, hence also Gorenstein. Using a result of R. P. Stanley [Adv. Math. 28, 57-83 (1978; Zbl 0384.13012)] and his previous results [E. Formanek, J. Algebra 89, 178-223 (1984; Zbl 0549.16008)] the author proves that \(\chi(A)(x_ 1,...,x_ r)\) satisfies the functional equation \[ \chi(A)(x_ 1^{- 1},...,x_ r^{-1})= (-1)^ d(x_ 1,...,x_ r)^{n^ 2}\chi(A)(x_ 1,...,x_ r). \] Here \(r\geq n^ 2-n\) for \(A=\bar C(n,r)\), \(r\geq n^ 2\) for \(A=\bar R(n,r)\) and \(d=(r-1)n^ 2+1\) is the Krull dimension of A.
Recently Y. Teranishi [The ring of invariants of matrices; Linear diophantine equations and invariant theory of matrices (preprints)] has obtained an independent proof for the functional equation for \(\chi(\bar C(n,r))(x_ 1,...,x_ r)\) for \(r\geq 2\) and has found \(\chi(C(3,2))\) and \(\chi(C(4,2))\) in an explicit form. His method is entirely different and is to apply Cauchy’s integral formula to the Molien-Weyl expression for \(\chi(\bar C)\) as a multiple integral.