Determining the characters of indecomposable tilting modules for reductive groups is one of the most fundamental

open problems in modular representation theory. A solution for GL_n would answer the question of the dimensions

of the simple modules for the symmetric group in characteristic p.

It is related to (but almost certainly harder than) the determination of the simple characters. I will describe a new

algorithm (based on a long and ongoing series of work with Elias, Riche, Libedinsky, Achar-Makisumi-Riche)

which allows one to calculate much further. There are many ideas involved, including from geometric representation

theory and the representation theory of monoidal categories. Long calculations using this algorithm on a super

computer at the MPI led Lusztig and I to a conjecture for SL_3 which I find quite startling, and I hope to explain

in detail. If time permits I will also discuss Sp_4 (joint work in progress with Jensen and Lusztig), which looks

even more interesting!

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