Character theoretic techniques for nonabelian partial difference sets

Abstract

A $(v,k,λ, μ)$-partial difference set (PDS) is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element $g$ of $G$ can be expressed in either $λ$ or $μ$ different ways as a product $xy^{-1}$, $x, y \in D$, depending on whether or not $g$ is in $D$. If $D$ is inverse closed and $1 \notin D$, then the Cayley graph ${\rm Cay}(G,D)$ is a $(v,k,λ, μ)$-strongly regular graph (SRG). PDSs have been studied extensively over the years, especially in abelian groups, where techniques from character theory have proven to be particularly effective. Recently, there has been considerable interest in studying PDSs in nonabelian groups, and the purpose of this paper is develop character theoretic techniques that apply in the nonabelian setting. We prove that analogues of character theoretic results of Ott about generalized quadrangles of order $s$ also hold in the general PDS setting, and we are able to use these techniques to compute the intersection of a putative PDS with the conjugacy classes of the parent group in many instances. With these techniques, we are able to prove the nonexistence of PDSs in numerous instances and provide severe restrictions in cases when such PDSs may still exist. Furthermore, we are able to use these techniques constructively, computing several examples of PDSs in nonabelian groups not previously recognized in the literature, including an infinite family of genuinely nonabelian PDSs associated to the block-regular Steiner triple systems originally studied by Clapham and related infinite families of genuinely nonabelian PDSs associated to the block-regular Steiner $2$-designs first studied by Wilson.