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Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

Hazi, A. ORCID: 0000-0001-7264-2211 (2020). Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups. .

Abstract

Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic $p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a categorification of the Hecke algebra for $W$ with rich connections to modular representation theory. We explicitly construct a functor from $\mathcal{D}$ to a matrix category which categorifies a recursive representation $\xi : \mathbb{Z}W \rightarrow M_{p^r}(\mathbb{Z}W)$, where $r$ is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are "smaller" by a factor of $p$. It also explains the presence of self-similarity in the $p$-canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the $p$-canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent $p$-canonical tilting character formula due to Achar-Makisumi-Riche-Williamson.

Publication Type: Monograph (Working Paper)
Additional Information: 62 pages, many figures, best viewed in color
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
Related URLs:
Date Deposited: 27 May 2020 08:59
URI: https://openaccess.city.ac.uk/id/eprint/23593
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