Adapted algebras for the Berenstein-Zelevinsky conjecture
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- Caldero, P. Transformation Groups (2003) 8: 37. doi:10.1007/s00031-003-1121-3
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. ) of the quantized algebra Cq[U-] of regular functions on U-. Following , fix a ZN≧0-parametrization of this basis, where N = dim U-. In , 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 Aw0, called adapted algebra, of Cq[U-] such that (1) Aw0 is a q-polynomial algebra which equals Cq[U-] up to localization, (2) Aw0 is spanned by a subset of B*, (3) the Berenstein–Zelevinsky conjecture is true on Aw0. Then we test the conjecture when one element belongs to the q-center of Cq[U-].