P Amrutha: On the determinant of representations of generalized symmetric groups Zr≀Sn.
Channel: Amritanshu Prasad
Date: 09/11/2020

Seminar website: https://www.imsc.res.in/~amri/algcomb

The problem of enumeration of the number of irreducible representations of the symmetric group with a non trivial determinant was first considered by L. Solomon and later posed by Stanley in his book. Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, we are interested in the study of the determinant of irreducible representations of the generalized symmetric groups, $Z_r \wr S_n$. We give an explicit formula to compute the determinant of an irreducible representation of $Z_r \wr S_n$. Also, for a given integer $n$, and a prime number $r$ and $\zeta$ a nontrivial multiplicative character of $Z_r \wr S_n$ with $n\lt r$, we obtain an explicit formula to compute $N_\zeta(n)$, the number of irreducible representations of $Z_r \wr S_n$ whose determinant is $\zeta$.