Canonical bases of quantum Schubert cells and their symmetries

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.

AMSLaTeX, 32 pages,typos corrected

References in corpus (5)

• Quantum cluster characters of Hall algebras
• Double canonical bases
• Remarks on quantum unipotent subgroup and dual canonical basis
• Modules over quantized coordinate algebras and PBW-bases
• Quantum twist maps and dual canonical bases

Cited by in corpus (1)

• Double canonical bases