Representing alternating groups as self-dual string C-groups of high rank

Abstract

The highest rank of a string C-group representation of the alternating group $A_n$ is known for each $n$, but no self-dual representations attaining this highest rank are known when $n > 12$. Motivated by computational results for alternating groups of small degree, we examine a vertex-gluing construction for permutation representation graphs. We establish conditions under which gluing two string C-groups produces another string C-group, and use this construction to obtain infinite families of self-dual representations of alternating groups. In particular, for every $n = 4m+3 \geq 15$, we construct $\left \lfloor \frac{n+9}{8} \right \rfloor$ distinct self-dual string C-groups of rank $2m$ isomorphic to $A_{n}$. These representations have rank one below the maximum possible rank of string C-group representations for $A_n$, and to the authors' knowledge are the highest-rank self-dual representations currently known for alternating groups.