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

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

2010-07-11

arxiv.org/abs/1007.1791v1

Mathematics

Representation Theory

15 pages, to appear in Journal of Algebraic Combinatorics

Scientific paper

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 $G$.

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