Categorical diagonalization and $p$-cells

Ben Elias, Joel Gibson, Lars Thorge Jensen

Publication date: 23 November 2021

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.

Has companion code repository: https://github.com/joelgibson/asloc

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