An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras

Abstract

Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the 𝘵-analogs of 𝑞-characters of simple modules of quantum affine algebras.