Algorithm for computing canonical bases and foldings of quantum groups

TL;DR

This paper extends an algorithm for computing canonical bases from symmetric to non-symmetric quantum groups, utilizing folding theory to relate different types.

Contribution

It generalizes Antor's algorithm to non-symmetric quantum groups using folding theory, establishing connections between algorithms for different types.

Findings

Extended the algorithm to non-symmetric quantum groups.

Linked algorithms for symmetric and non-symmetric cases via folding.

Provided a framework for computing canonical bases in broader settings.

Abstract

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. Let $P$ be the transition matrix between the canonical basis and a PBW basis of ${\mathbf U}_q^-$. In the case ${\mathbf U}_q^-$ is symmetric, Antor gave a simple algorithm of computing $P$ by making use of monomial bases. By the folding theory, ${\mathbf U}_q^-$ (symmetric, with a certain automorphism) is related to a quantum group $\underline{{\mathbf U}}_q^-$ of non-symmetric type. In this paper, we extend the results of Antor to the non-symmetric case, and discuss the relationship between the algorithms for ${\mathbf U}_q^-$ and for $\underline{\mathbf U}_q^-$.