## Journal of the Mathematical Society of Japan

### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 4 (2014), 1191-1226.

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.jmsj/1414090240

Digital Object Identifier
doi:10.2969/jmsj/06641191

Mathematical Reviews number (MathSciNet)
MR3272597

Zentralblatt MATH identifier
1329.20010

#### Citation

HIRAI, Takeshi; HORA, Akihito. Spin representations of twisted central products of double covering finite groups and the case of permutation groups. J. Math. Soc. Japan 66 (2014), no. 4, 1191--1226. doi:10.2969/jmsj/06641191. https://projecteuclid.org/euclid.jmsj/1414090240

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