A Poset Structure on the Alternating Group Generated by 3-Cycles

TL;DR

This paper explores the poset structure of the alternating group generated by 3-cycles, providing enumerative results and analyzing the Hurwitz action on maximal chains, with comparisons to the symmetric group case.

Contribution

It introduces a detailed study of the poset structure on the alternating group generated by 3-cycles, including orbit descriptions and enumerative properties, extending known results from the symmetric group.

Findings

Enumerative results for the poset intervals

Description of Hurwitz action orbits on maximal chains

Comparison with the symmetric group's absolute order

Abstract

We investigate the poset structure on the alternating group that arises when the latter is generated by 3-cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the Hurwitz action on maximal chains. Our motivating example is the well-studied absolute order arising when the symmetric group is generated by transpositions, i.e. 2-cycles, and we compare our results to this case along the way. In particular, noncrossing partitions arise naturally in both settings.