An Indicator Formula for the Hopf Algebra $k^{S_{n-1}}\#kC_n$

TL;DR

This paper introduces a new Froebnius-Schur indicator formula for a class of Hopf algebras constructed from symmetric groups and cyclic groups, revealing properties of their irreducible representations and how indicators behave as the group size increases.

Contribution

It provides a novel indicator formula for the irreducible representations of the semisimple bismash product Hopf algebra $J_n$, including detailed counting formulas and asymptotic behavior analysis.

Findings

All indicators are nonnegative for odd n.

Explicit formulas for indicators of 2D and odd-dimensional irreps.

Nonzero indicators become rare as n grows large.

Abstract

The semisimple bismash product Hopf algebra $J_n=k^{S_{n-1}}\#kC_n$ for an algebraically closed field $k$ is constructed using the matched pair actions of $C_n$ and $S_{n-1}$ on each other. In this work, we reinterpret these actions and use an understanding of the involutions of $S_{n-1}$ to derive a new Froebnius-Schur indicator formula for irreps of $J_n$ and show that for $n$ odd, all indicators of $J_n$ are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all $2$-dimensional irreps of $J_n$ and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of $J_n$ and use these formulas to show that nonzero indicators become rare for large $n$.