The functional equation for Poincaré series of trace rings of generic 2\(\times 2\) matrices (Q1072624)

Let \(G_{m,n}\) be the algebra generated by m generic \(n\times n\) matrices. By adjoining to \(G_{m,n}\) the traces of all its elements one obtains the trace ring \(\Pi_{m,n}\); \(G_{m,n}\) and \(\Pi_{m,n}\) play important roles in the theory of the algebras with polynomial identities. In the paper under review the trace ring of \(2\times 2\) matrices is studied. Using some 60-year-old results due to I. Schur and H. Weyl the author gives a rational expression of the Poincaré (or Hilbert) series \({\mathcal P}(\Pi_{m,2},t_ 1,...,t_ m)\) of the graded vector space \(\Pi_{m,2}\). Up till now the explicit form for these series has been known for \(m\leq 4\) only [\textit{E. Formanek}, J. Algebra 89, 178-223 (1984; Zbl 0549.16008)]. As a consequence the author establishes that the Poincaré series of \(\Pi_{m,2}\) satisfies the functional equation \[ {\mathcal P}(\Pi_{m,2},1/t)=-t^{4m}{\mathcal P}(\Pi_{m,2},t),\quad m>2. \] Recently \textit{E. Formanek} [Trans. Am. Math. Soc. 294, 647-663 (1986)] and \textit{Y. Teranishi} [Nagoya Math. J. 104 (1986) (to appear)] independently have generalized this result. They have obtained that the trace ring of \(n\times n\) generic matrices satisfies a similar functional equation. A complete exposition on various properties of \(\Pi_{m,2}\) can be found in [\textit{L. Le Bruyn}, Trace rings of generic 2 by 2 matrices, Mem. Am. Math. Soc. (to appear)].