iCanonical basis arising from quasi-split rank one iquantum group

TL;DR

This paper explicitly computes the iCanonical basis for a specific quasi-split rank one iquantum group, revealing detailed transition matrices among various bases and their categorifications.

Contribution

It introduces explicit transition matrices among the iCanonical, monomial, and canonical bases for the quasi-split rank one iquantum group, connecting them to quantum rak{sl}_3 modules.

Findings

Explicit transition matrices among bases are obtained.

All bases can be naturally categorified.

Results connect to quantum rak{sl}_3} modules.

Abstract

We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among the icanonical basis, monomial basis, and canonical basis on simple finite-dimensional modules of quantum $\mathfrak{sl}_3$.