On Mirković-Vilonen cycles and crystal combinatorics

Abstract

Abstract. Let G be a complex connected reductive group and let G ∨ be its Langlands dual. Let us choose a triangular decomposition n −,∨ ⊕ h ∨ ⊕ n +,∨ of the Lie algebra of G ∨ . Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannianis a crystal isomorphic to the crystal of the canonical basis of U (n +,∨ ). Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.

Dimensions of Affine Deligne–Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators

“…The relationship between the two notions of dimension given by Definitions 4.4 and 4.7 is delicate, essentially due to the fact that if γ is a minimal alcove-to-alcove gallery then the canonically associated vertex-to-vertex gallery γ need not be minimal. These subtleties are known to experts and have been considered in other contexts; see for instance [BG08] and Remark 4.3 or 4.9 in [Sch06]. For our purposes, it will suffice to show that for certain positively folded alcove-to-alcove galleries γ, the dimension of γ is equal to the dimension of the canonically associated vertex-to-vertex gallery γ .…”

Tensor products and Minkowski sums of Mirković–Vilonen polytopes

“…In view of the coincidence of the LBZ and BFG crystal structures on the set of MV polytopes (see Remark 2.2.2), we deduce the following fact from [BaG,Prop. 4.2]; notice the convention in [BaG] that the roots in B are the positive ones, which is opposite to ours.…”

The symplectic plactic monoid, crystals, and MV cycles

“…We obtain results similar to those obtained in [6] concerning the defining relations of the symplectic plactic monoid, described explicitly by Lecouvey in [13], as well as words of readable galleries. These results together with the work of Gaussent-Littelmann [4], [5], and Baumann-Gaussent [1] allow us to show in Theorem 6.2 that given a readable gallery δ ∈ Γ(γ λ ) R there is an associated dominant coweight ν δ ≤ λ such that:…”

“…These subgroups are generated by the torus T and the root subgroups U (α,n) such that F ⊂ H − (α,n) , the root subgroups U (α,n) ⊂ P F such that α ∈ Φ + , and those affine reflections s (α,n) ∈ W aff such that F ⊂ H (α,n) , respectively. See [4], Section 3.3, Example 3, and [1], Proposition 5.1 (ii).…”