In characteristic p, the simple modules for the symmetric group Sn are the James modules Dλ, labelled by p-regular partitions of n. If we let sgn denote the 1-dimensional sign module, then for any p-regular λ, the module Dλ ⊗ sgn is also a simple module. So there is an involutory bijection mp on the set of p-regular partitions such that Dλ ⊗ sgn=Dmp(λ). The map mp is called the Mullineux map, and an important problem is to describe mp 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 regularization.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
