A reciprocity on finite abelian groups involving zero-sum sequences

2019/05/28 by Dongchun Han, Han, Dongchun, Hanbin Zhang +1
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1905.11949

openalex publication_date 2019/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On a zero-sum problem arising from factorization theory

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