# OIST Representation Theory Seminar

### Date

Tuesday, January 12, 2021 - 16:30

On Zoom

### Title: The Mullineux map

Abstract: In characteristic p, the simple modules for the symmetric group $$S_n$$ are the James modules $$D^\lambda$$, labelled by p-regular partitions of n. If we let $$sgn$$ denote the 1-dimensional sign module, then for any p-regular $$\lambda$$, the module $$D^\lambda\otimes sgn$$ is also a simple module. So there is an involutory bijection $$m_p$$ on the set of p-regular partitions such that $$D^\lambda\otimes sgn=D^{m_p(\lambda)}$$. The map $$m_p$$ is called the Mullineux map, and an important problem is to describe $$m_p$$ combinatorially. There are now several known solutions to this problem. I will describe the history of this problem and explain the known combinatorial solutions, and then give a new solution based on crystals and regularisation.

Zoom info can be found on the seminar webpage.

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