Spin representations of twisted central products of double covering finite groups and the case of permutation groups

Abstract

Let $S$ be a finite group with a character, $\rm sgn$, of order 2, and $S'$ its central extension by a group $Z=\langle z\rangle$ of order 2. A representation $\pi$ of $S'$ is called {\it spin} if $\pi(z\sigma')=-\pi(\sigma')$ $(\sigma'\in S')$, and the set of all equivalence classes of spin irreducible representations (= IRs) of $S'$ is called the {\it spin dual} of $S'$. Take a finite number of such triplets $(S'_j,z_j,{\rm sgn}_j)$ $(1\le j\le m)$. We define twisted central product $S'=S'_1\hat{*}S'_2\hat{*}\cdots\hat{*}S'_m$ as a double covering of $S=S_1\times\cdots \times S_m$, $S_j=S'_j/\langle z_j\rangle$, and for spin IRs $\pi_j$ of $S'_j$, define twisted central product $\pi=\pi_1\hat{*}\pi_2\hat{*}\cdots\hat{*}\pi_m$ as a spin IR of $S'$. We study their characters and prove that the set of spin IRs $\pi$ of this type gives a complete set of representatives of the spin dual of $S'$. These results are applied to the case of representation groups $S'$ for $S={\mathfrak S}_n$ and ${\mathfrak A}_n$, and their (Frobenius-)Young type subgroups.