On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group Sym ( 7 )

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 $\text{Sym}\left(n\right)$, for $3\le n\le 6$, have been studied and completely determined. The case for $\text{Sym}\left(7\right)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\text{Sym}\left(7\right)$ 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 $\text{Sym}\left(7\right)$ are determined.