An embedding of the skein action on set partitions into the skein action on matchings

TL;DR

This paper demonstrates how the skein action on noncrossing set partitions can be embedded into the skein action on matchings, revealing that the generalized relations originate from the Ptolemy relation, with implications for algebraic structures.

Contribution

It provides a novel embedding of Rhoades' skein action on set partitions into the matching skein module, linking generalized relations to the Ptolemy relation.

Findings

Rhoades' skein action on set partitions embeds into the matching skein module.

Generalized skein relations derive from the Ptolemy relation.

Connections established between set partition actions and matching actions.

Abstract

Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.