The $p$-Canonical Basis for Hecke Algebras

TL;DR

This paper introduces a positive characteristic analogue of the Kazhdan-Lusztig basis for Hecke algebras, providing an algorithm for its computation and exploring its properties and applications in modular representation theory.

Contribution

It develops a new p-canonical basis for Hecke algebras, extending classical theory to positive characteristic and offering computational tools via Soergel calculus.

Findings

Describes the p-canonical basis for Hecke algebras in positive characteristic.

Provides an algorithm for calculating this basis.

Discusses potential applications in modular representation theory.

Abstract

We describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. Using Soergel calculus we describe an algorithm to calculate this basis. We outline some known or expected applications in modular representation theory. We conclude by giving several examples.