#### IMSc Webinar

#### On the determinant of representations of generalized symmetric groups $Z_r\wr S_n$

#### Amrutha P

##### IISER Thiruvananthapuram

*Zoom link: us02web.zoom.us/j/84181041409pwd=dHJVdTU0TVBpWTdLcFVVcEJBUEQ5*

Meeting ID: 841 8104 1409; Password: Littlewood.

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 $\zeta$.

Done