On canonical bases and induction of $W$-graphs

TL;DR

This paper generalizes the concept of canonical bases beyond free modules and demonstrates that the induction process for W-graphs is functorial, transitive, and satisfies Mackey-type properties.

Contribution

It extends canonical basis theory to non-free modules and establishes the functoriality and desirable properties of W-graph induction.

Findings

Canonical bases can be defined for non-free modules.

W-graph induction is a well-behaved functor between module categories.

The induction process satisfies transitivity and Mackey-type theorems.

Abstract

A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed "standard basis" through a triangular base change matrix with polynomial entries whose constant terms equal the identity matrix. Among the better known examples of canonical bases are the Kazhdan-Lusztig basis of Iwahori-Hecke algebras, Lusztig's canonical basis of quantum groups and the Howlett-Yin basis of induced $W$-graph modules. This paper has two major theoretical goals: First to show that having bases is superfluous in the sense that canonicalisation can be generalized to non-free modules. This construction is functorial in the appropriate sense. The second goal is to show that Howlett-Yin induction of $W$-graphs is well-behaved a functor between module categories of $W$-graph-algebras that satisfies various properties one hopes for when a functor is called "induction", for example transitivity and a Mackey theorem.