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

TL;DR

This paper computes the Poincare series for cyclic group representations, derives a reciprocity and number-theoretic identities, and relates the permanent of the Cayley table to invariants of abelian groups, generalizing previous results.

Contribution

It introduces a new approach to compute Poincare series for abelian groups and links the permanent of Cayley tables to group invariants, extending prior work by Fredman and others.

Findings

Computed Poincare series for cyclic groups' isotypic components.

Derived a general reciprocity and number-theoretic identities.

Linked the permanent of Cayley tables to invariants of abelian groups.

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 $G$.