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

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.