Non-degeneracy of Killing forms on real conjugacy classes of finite groups

Abstract

Killing forms on finite groups arise as special cases of braided Killing forms on braided Lie algebras. If $\mathcal{C}$ is a conjugation-stable subset of a finite group $G$, the Killing form on $\mathbb{C}\mathcal{C}$ is given by $K_\mathcal{C}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b \in \mathcal{C}$. It is conjectured in previous work by L\'opez Pe\~na, Majid and Rietsch that $K_\mathcal{C}$ is non-degenerate for any real conjugacy class $\mathcal{C}$ in a finite simple group. In this article, we reformulate the conjecture and introduce combinatorial conditions - the $\textit{1-element condition}$ and the $\textit{2-element condition}$ - that are sufficient for non-degeneracy to hold. This allows us to prove the conjecture for simple groups of the form $\mathrm{PSL}_2(q)$ and certain conjugacy classes in the alternating and symmetric groups. Moreover, we verify computationally that every real conjugacy class in a simple group of order $\leq 10^9$ fulfills at least one of these two conditions, thereby significantly extending the computational evidence for the conjecture. This raises the question whether these conditions are satisfied by all conjugacy classes in finite simple groups.