Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table

Abstract

Let $R$ be the regular representation of a finite abelian group $G$ and let $C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the Poincare series of all $C_n$-isotypic components in $S^{\cdot }R\otimes {\wedge }^{\cdot }R$ (the symmetric tensor exterior algebra of $R$). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley table, $M_G$, of $G$ and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of $M_G$ equals $(S^{n}R{)}^G$, where $n$ is the order of $G$.

arXiv comment15 pages, to appear in Journal of Algebraic Combinatorics