Compare triangular bases of acyclic quantum cluster algebras

Abstract

Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with acyclic quivers. This result implies that the Berenstein-Zelevinsky's basis contains all the quantum cluster monomials.We also give an easy proof that the two bases are the same for the seeds associated with bipartite skew-symmetrizable matrices.

Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases

“…Berenstein-Rupel [BR15] studied the quantum unipotent cells via the Hall algebra technique and they constructed quantum analogue of the twist maps under the conjecture concerning the quantum cluster algebra structure and they showed that the quantum twist automorphisms preserve the triangular bases (in the sense of Berenstein-Zelevinsky [BZ14]) of the quantum unipotent cells when the Weyl group element w is the square of an acyclic Coxeter element c with ℓ (w) = 2ℓ (c). We note that Qin [Qin16] proved that the triangular bases (=localized dual canonical bases) in the sense of [Qin17] coincide with the triangular bases in the sense of Berenstein-Zelevinsky [BZ14] when g is symmetric.…”

Canonical bases of quantum Schubert cells and their symmetries

“…) and E 2132 = Y (2,1,3,2) = E 2 E 213 − q −1 E 21 E 23 as defined in Theorem 5.5. The following was essentially proved in [4], although with a slightly different definition of· and hence with different powers of q (see also Theorems 1.4.1 and 3.1.3 in a recent work [18]). Since all elements w ∈ W with ℓ(w) ≤ 4 are either repetition free or with a single repetition, all remaining Schubert cells have already been described in §5.1 and §5.2.…”