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

mercredi 30 nov. 2022, 16:15 → 17:15 Europe/Paris

The bismash product $H_n={\mathbb{C}}^{C_n}\mathrm{\#}\mathbb{C}{S}_{n-1}$ and its dual $J_n={\mathbb{C}}^{S_{n-1}}\mathrm{\#}\mathbb{C}{C}_n$ 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.