On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$

TL;DR

This paper investigates the structure of the Terwilliger algebra associated with the conjugacy class scheme of the symmetric group sym(7), providing detailed algebraic decompositions and insights into permutation codes.

Contribution

It extends the understanding of Terwilliger algebras to sym(7), including dimension, Wedderburn decomposition, and block structure, which were previously known only for smaller groups.

Findings

Determined the dimension of the Terwilliger algebra for sym(7).

Provided the Wedderburn decomposition of the algebra.

Analyzed the block dimension decomposition.

Abstract

Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined.