Categorical diagonalization and $p$-cells

TL;DR

This paper explores the action of twists on the Hecke category and $p$-canonical basis in characteristic $p$, proving conjectures in type $C_2$ and providing computational evidence in low ranks.

Contribution

It formulates and proves conjectures about the categorified action of twists on the $p$-canonical basis in characteristic $p$, extending previous results beyond type A.

Findings

Proves the categorified conjecture in type C2.

Computational verification of the decategorified conjecture in ranks up to 6.

Provides new insights into the structure of the Hecke category in characteristic p.

Abstract

In the Iwahori-Hecke algebra, the full twist acts on cell modules by a scalar, and the half twist acts by a scalar and an involution. A categorification of this statement, describing the action of the half and full twist Rouquier complexes on the Hecke category, was conjectured by Elias-Hogancamp, and proven in type $A$. In this paper we make analogous conjectures for the $p$-canonical basis, and the Hecke category in characteristic $p$. We prove the categorified conjecture in type $C_2$, where the situation is already interesting. The decategorified conjecture is confirmed by computer in rank at most 6; information is found in the appendix, written by Joel Gibson.