Adapted algebras for the Berenstein-Zelevinsky conjecture

Abstract

Let G be a simply connected semi-simple complex Lie group and fix a maximal unipotent subgroup U − of G. Let q be an indeterminate and let's denote by B * the dual canonical basis, [18], of the quantized algebra C q [U − ] of regular functions on U − . Following [19], fix a Z N ≥0 -parametrization of this basis, where N =dimU − . 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. For all reduced decomposition w0 of the longest element of the Weyl group of g, we associate a subalgebra A w0 , called adapted algebra, of C q [U − ] such that 1) A w0 is a q-polynomial algebra which equals C q [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 C q [U − ].

Adapted algebras and standard monomials

“…Theorem 3 (Caldero [3]). The subalgebra A Ä w0 is adapted with the elements of S Ä w0 (up to a rescaling by a power of q) as spanning set.…”

“…These elements have been studied in great detail by the ÿrst author in [3], where he shows that they have the following remarkable properties:…”

“…An example for g = sl 4 is the element b ∈ B * whose class in B(∞) is u b = f 2 f 1 f 3 u ∞ . It is easily seen with the help of parametrizations, see [3], that b * ∈ Ä w0 S Ä w0 . Nevertheless, by Theorem 20, the other elements of B(∞) of the same weight…”

“…Before tackling Examples 24 and 25, we give the explicit adapted algebras for g=sl 3 , see [3]. Let us introduce some notation.…”

The Lusztig Cones of a Quantized Enveloping Algebra of Type A

“…, i is a reduced expression for the longest 1 We call the set of points associated to i satisfying these linear equalities w x the Lusztig cone associated to i. This result was extended to type A in 12 4 w x w x but has been shown to fail for larger n by Reineke 16 and Xi 17 . However, in the crystal limit of Kashiwara, this cone plays an important w x role. This role will be explored in 13 and in the joint paper with Carter w x 5 , where a link is made with the regions of linearity of Lusztig's reparametrization functions.…”