Commutative Schur Rings Over Symmetric Groups II: The Case n=6

TL;DR

This paper classifies all commutative Schur rings over the symmetric group S_6 that include all transpositions, identifying eight types with four having the transpositions as a principal set.

Contribution

It provides a complete classification of certain commutative Schur rings over S_6, expanding understanding of their structure and properties.

Findings

Identified eight conjugacy classes of commutative Schur rings over S_6.

Four of these rings have the set of all transpositions as a principal set.

The classification advances the theory of Schur rings over symmetric groups.

Abstract

We determine the commutative Schur rings over $S_6$ that contain the sum of all the transpositions in $S_6$. There are eight such types (up to conjugacy), of which four have the set of all the transpositions as a principal set of the Schur ring.