Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 122, 22 pages      arXiv:2004.12466      https://doi.org/10.3842/SIGMA.2020.122
Contribution to the Special Issue on Cluster Algebras

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

Fan Qin
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China

Received May 14, 2020, in final form November 13, 2020; Published online November 27, 2020

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.

Key words: dual canonical bases; cluster algebras; Leclerc's conjecture.

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