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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