Quantum cluster mutations and reduced word graphs

TL;DR

This paper proves the independence of Coxeter moves in positive representations of split-real quantum groups using algebraic methods, introducing a new quantum cluster algebra approach and explicit calculations for rank 3 cycles.

Contribution

It introduces a new quantized version of Lusztig's Injectivity Lemma within quantum cluster algebra and provides a constructive proof of Tits' Lemma with explicit mutation computations.

Findings

Proved independence of Coxeter moves in quantum group representations.

Developed a new quantum cluster algebra framework for this proof.

Explicitly computed quantum cluster mutations for rank 3 cycles.

Abstract

We give an algebraic proof of the independence of Coxeter moves involved in the construction of positive representations of split-real quantum groups, thus completing a gap in the original construction. To do this, we propose a new quantized version of Lusztig's Injectivity Lemma in the language of quantum cluster algebra, the proof of which by Tits' Lemma reduces to calculations involving sequences of Coxeter moves forming rank 3 cycles. We give a new, constructive proof of Tits' Lemma, and provide the required explicit computation of the quantum cluster mutations under these rank 3 cycles using certain cluster algebraic tricks via universally Laurent polynomials.