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.

References in corpus (13)

• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Kac-Moody groups and cluster algebras
• A diagrammatic approach to categorification of quantum groups III
• Quiver varieties and cluster algebras
• Cluster structures on quantum coordinate rings
• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Monoidal categorification and quantum affine algebras
• Disordered skyrmion phase stabilized by magnetic frustration in a chiral magnet
• Dual canonical bases and quantum cluster algebras
• Integral quantum cluster structures
• Monoidal categorification of cluster algebras (merged version)

Cited by in corpus (2)

• On support $\tau$-tilting graphs of gentle algebras
• Cluster algebras and their bases