Realisation of Lusztig cones

Abstract

Let Uq(g) be the quantised enveloping algebra associated to a simple Lie algebra g over C. The negative part U − of Uq(g) possesses a canonical basis B with favourable properties. Lusztig has associated a cone to a reduced expression i for the longest element w 0 in the Weyl group of g, with good properties with respect to monomial elements of B. The first author has associated a subalgebra A i of U − , compatible with the dual basis B * , to each reduced expression i. We show that, after a certain twisting, the string parametrisation of the adapted basis of this subalgebra coincides with the corresponding Lusztig cone. As an application, we give explicit expressions for the generators of the Lusztig cones.

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

“…We generalize these results (Corollary 2.14 and Corollary 2.17) to any group G and to any choice of a reduced decomposition. A part of our results was already announced in [14], and applied in [7].…”

“…Proposition 2.21 ( [7]). Let i be a reduced word and let λ = λ 1 1 + · · · + λ n n be a dominant weight.…”

On the interplay of the parametrizations of canonical bases by Lusztig and string data

“…The fact that G i (λ) is an affine unimodular isomorphism is proved in Proposition 8.3 of [GKS17] and can also be deduced from [CMM04].…”

Cones from quantum groups to tropical flag varieties

“…In [13], Lusztig defined a polyhedral cone in the simply laced cases in order to study monomials in the canonical basis. The natural generalizations to all Lie types are given in [5].…”

“…It is shown in [5,Theorem 8.10] that the cone L w 0 (g) is simplicial. We will denote L − w 0 (g) ⊂ R N , termed a negative tight monomial cone, the polyhedral cone defined by (i)' for any two indices 1 ≤ p < p ≤ N with p = p [1], we have…”