On the determinant of representations of generalized symmetric groups ℤ_r≀S_n

Abstract

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups ${\mathbb{Z}}_r\wr S_n$. We give an explicit formula to compute the determinant of an irreducible representation of ${\mathbb{Z}}_r\wr S_n$. Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of ${\mathbb{Z}}_r\wr S_n$ with n < r, we obtain an explicit formula to compute $N_{\zeta }(n)$, the number of irreducible representations of ${\mathbb{Z}}_r\wr S_n$ whose determinant is ζ.

KEYWORDS:

• Chiral partitions
• hyperoctahedral group
• symmetric group
• wreath products

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05E10
• 05E15
• 20B30
• 20C30
• 20C15

Acknowledgments

The authors are very grateful to Amritanshu Prasad and Steven Spallone for their help and encouragement. We thank the referee for his or her useful comments and suggestions for increasing the article’s readability. This research was driven by computer exploration using the open-source mathematical software Sage [15]. The authors acknowledge the center for high performance computing of IISER TVM for the use of the Padmanabha cluster.

Additional information

Funding

AP is supported by the Indian Institute of Science Education and Research Thiruvananthapuram institute fellowship. GT is supported by DST-SERB MATRIX grant MTR/2017/000424.