Automata and cells in affine Weyl groups

TL;DR

This paper proves that the set of reduced expressions of elements in any Kazhdan--Lusztig cell of an affine Weyl group forms a regular language, enabling enumeration via finite automata.

Contribution

It establishes that reduced expressions in affine Weyl group cells are regular languages, connecting algebraic structures with formal language theory.

Findings

Reduced(C) is a regular language

Automata can enumerate elements in Kazhdan--Lusztig cells

Bridges algebraic and computational perspectives

Abstract

Let W~ be an affine Weyl group, and let C be a left, right, or two-sided Kazhdan--Lusztig cell in W~. Let Reduced (C) be the set of all reduced expressions of elements of C, regarded as a formal language in the sense of the theory of computation. We show that Reduced (C) is a regular language. Hence the reduced expressions of the elements in any Kazhdan--Lusztig cell can be enumerated by a finite state automaton.