Canonical bases of quantum Schubert cells and their symmetries

TL;DR

This paper constructs canonical bases for quantum Schubert cells and proves their invariance under Lusztig's symmetries, providing a clearer understanding of their structure and symmetries.

Contribution

It offers an elementary construction of the canonical basis in quantum Schubert cells and demonstrates its invariance under Lusztig's symmetries.

Findings

Constructed canonical bases explicitly for quantum Schubert cells.

Proved invariance of these bases under Lusztig's symmetries.

Characterized the upper global basis using a bilinear form.

Abstract

The goal of this work is to provide an elementary construction of the canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a direct characterization of the upper global basis $\mathbf B^{up}$ in terms of a suitable bilinear form and show that $\mathbf B(w)$ is contained in $\mathbf B^{up}$ and its large part is preserved by modified Lusztig's symmetries.