Functional equations for character series associated with 𝑛×𝑛 matrices

Formanek, Edward

doi:10.1090/S0002-9947-1986-0825728-8

Functional equations for character series associated with $n\times n$ matrices
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by Edward Formanek PDF

Trans. Amer. Math. Soc. 294 (1986), 647-663 Request permission

Abstract:

Let $A$ be either the ring of invariants or the trace ring of $r$ generic $n \times n$ matrices. Then $A$ has a character series $\chi (A)$ which is a symmetric rational function of commuting variables ${x_1}, \ldots ,{x_r}$. The main result is that if $r \geq {n^2}$, then $\chi (A)$ satisfies the functional equation \[ \chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})\], where $d$ is the Krull dimension of $A$.

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