New incompressible symmetric tensor categories in positive characteristic

Abstract: Let k be an algebraically closed field of characteristic p>0. The category of tilting modules for SL2(k) has a tensor ideal In generated by the n-th Steinberg module. I will explain that the quotient of the tilting category by In admits an abelian envelope, a finite symmetric tensor category Verpn, which is not semisimple for n>1. This is a reduction to characteristic p of the semisimplification of the category of tilting modules for the quantum group at a root of unity of order pn. These categories are incompressible, i.e. do not admit fiber functors to smaller categories. For p=1, these categories were defined by S. Gelfand and D. Kazhdan and by G. Georgiev and O. Mathieu in early 1990s, but for n>1 they are new. I will describe these categories in detail and explain a conjectural formulation of Deligne's theorem in characteristic p in which they appear. This is joint work with D. Benson and V. Ostrik.

Mathematics

Audience: researchers in the topic

T-Rep: A midsummer night's session on representation theory and tensor categories

Series comments: The mini workshop will be run via Zoom. To register for the event please go to www.math.uni-bonn.de/people/thorsten/t-rep.htmpl

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We will have a discussion round after the first talk at 21:30 CEST.