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

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.