# A reciprocity on finite abelian groups involving zero-sum sequences

## TL;DR

This paper establishes a reciprocity relation between zero-sum sequences over finite abelian groups with coprime orders, providing combinatorial and invariant theory insights, and partially answering a related open question.

## Contribution

It introduces a new reciprocity law for zero-sum sequences on finite abelian groups and offers combinatorial and invariant theory interpretations, extending previous understanding.

## Key findings

- Proves the reciprocity $| ext{M}(G,|H|)|=| ext{M}(H,|G|)|$ for coprime groups.
- Provides a combinatorial interpretation using rational Catalan combinatorics.
- Partially answers a question posed by Panyushev.

## Abstract

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.