Adapted algebras for the Berenstein-Zelevinsky conjecture

Abstract

Let G be a simply connected semisimple complex Lie group and fix a maximal unipotent subgroup U^- of G. Let q be an indeterminate and let B* denote the dual canonical basis (cf. [19]) of the quantized algebra Cq[U^-] of regular functions on U^-. Following [20], fix a Z^N _≧0-parametrization of this basis, where N = dim U^-. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B* q-commute if and only if they are multiplicative, i.e., their product is an element of B* up to a power of q. To any reduced decomposition w₀ of the longest element of the Weyl group of g, we associate a subalgebra A_w0, called adapted algebra, of Cq[U^-] such that (1) A_w0 is a q-polynomial algebra which equals Cq[U^-] up to localization, (2) A_w0 is spanned by a subset of B*, (3) the Berenstein–Zelevinsky conjecture is true on A_w0. Then we test the conjecture when one element belongs to the q-center of Cq[U^-].