# The ABC of p-Cells

## TL;DR

This paper develops a theory of p-cells in Hecke algebras for positive characteristic, extending Kazhdan-Lusztig cell theory, with explicit descriptions in type A and decompositions in types B and C.

## Contribution

It introduces a p-cell theory parallel to characteristic zero, utilizing star-operations, and provides explicit classifications in finite types A, B, and C.

## Key findings

- Explicit description of p-cells in type A using Robinson-Schensted correspondence
- Decomposition of Kazhdan-Lusztig cells into p-cells for p > 2 in types B and C
- Numerical consequences of star-operations for p-canonical bases

## Abstract

Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic $0$, we try to build a similar theory in positive characteristic. We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556(2)). Our main technical tool are the star-operations introduced by Kazhdan-Lusztig which have interesting numerical consequences for the $p$-canonical basis. As an application, we explicitely describe $p$-cells in finite type $A$ (i.e. for symmetric groups) using the Robinson-Schensted correspondence. Moreover, we show that Kazhdan-Lusztig cells in finite types $B$ and $C$ decompose into $p$-cells for $p > 2$.