Kayla Orlinsky: The indicators of some representations of a bismash product Hopf algebra


The bismash product $H_n = \mathbb C^{C_n}\#\mathbb CS_{n−1}$ and its dual $J_n = \mathbb C^{S_{n−1}}\#\mathbb CC_n$ are semisimple Hopf algebras whose structure is induced from a factorization of the symmetric group $S_n = C_n S_{n−1}$ . It is known that the indicators of all simple modules of $H_n$ are +1 [JM]. In this talk, we describe the representation 
theory of $J_n$ and how permutations in the symmetric group were used to prove that the algebraic dual $J_n$ can have negative indicators. Specifically we list the conditions under which simple modules of $J_n$ can have negative indicators and describe how we are able to count them.