# OIST Representation Theory Seminar

### Date

### Location

### Description

### Speaker: Matthew Fayers, Queen Mary University of London

### 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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