Deligne’s interpolation categories and modular symmetric functions

by Robert Laugwitz (Nottingham)

Friday Sep 13, 2024, 10:30 AM → 11:30 AM Europe/Paris

E2 1180 (Tours)

In a seminal paper from 2004, Deligne introduced tensor
categories that interpolate the classical representation categories of
symmetric groups. These categories are described combinatorially using
partitions and are indexed by a complex number _t_. Deligne (for generic
_t_) and Comes-Ostrik (for general _t_) showed that indecomposable
objects of Deligne’s categories are parametrized by partitions of
arbitrary length. Moreover, the graded Grothendieck ring is isomorphic
to the ring of symmetric functions. In this talk, I will introduce and
motivate Deligne’s interpolation categories and explain techniques
used to classify indecomposable objects in these and more general
families of interpolation categories due to Khovanov-Sazdanov. In
particular, these techniques show that the graded Grothendieck ring of
characteristic _p_ analogues of Deligne’s categories is given by a
ring of modular symmetric functions. This talk is based on joint work
with Johannes Flake (Bonn) and Sebastian Posur (Münster).