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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Functional equations for character series associated with $ n\times n$ matrices


Author: Edward Formanek
Journal: Trans. Amer. Math. Soc. 294 (1986), 647-663
MSC: Primary 15A72; Secondary 16A38
MathSciNet review: 825728
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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

$\displaystyle \chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})$

, where $ d$ is the Krull dimension of $ A$.

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DOI: https://doi.org/10.1090/S0002-9947-1986-0825728-8
Article copyright: © Copyright 1986 American Mathematical Society