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30 June 2021

Diego Millan Berdasco - On the computation of decomposition numbers of the symmetric group.

The most important open problem in the modular representation theory of the symmetric group is finding the multiplicity of the simple modules as composition factors of the Specht modules. In characteristic 0 the Specht modules are just the simple modules of the symmetric group algebra, but in positive characteristic they may no longer be simple. We will survey the rich interplay between representation theory and combinatorics of integer partitions, review a number of results in the literature which allow us to compute composition series for certain infinite families of Specht modules from a finite subsetof them, and discuss the extension of these techniques to other Specht modules.