On the Induction of p-Cells

TL;DR

This paper investigates the structure of p-cells in Coxeter groups, demonstrating how they decompose upon induction and revealing new positivity properties of the p-canonical basis.

Contribution

It introduces a decomposition result for p-cell modules under induction and establishes new positivity properties of the p-canonical basis.

Findings

p-cell modules decompose as direct sums upon induction

New positivity properties of the p-canonical basis are established

Compatibility of p-cells with standard parabolic subgroups is analyzed

Abstract

We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.