Quantum twist maps and dual canonical bases

TL;DR

This paper demonstrates how quantum twist maps create bijections between dual canonical bases and unipotent quantum minors, revealing structural properties of quantum nilpotent subalgebras.

Contribution

It establishes the bijective nature of quantum twist maps on dual canonical bases and unipotent quantum minors, and explores their unitriangular relations with PBW bases.

Findings

Quantum twist maps induce bijections between dual canonical bases.

Quantum twist maps relate to unipotent quantum minors.

Unitriangular property between dual canonical and PBW bases is shown.

Abstract

In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincar\'e-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.