Affine Mirković-Vilonen polytopes

A quiver variety approach to root multiplicities

“…Stability conditions. The following loosely follows [BKT14], and also draws on notation from [TW16].…”

“…Fix a stability condition c so that, for any root β, if arg c(α) = arg c(β) then β and α are parallel. The following can be extracted from [BKT14], and the proof below can be found in [TW16, Corollary 2.12] in a somewhat different context.…”

“…In [BKT14] the category of representations of the preprojective algebra is studied using Harder-Narasimhan filtrations. Irreducible components where this filtration generically has only one step are called stable, and it is shown that the number of stable irreducible components of a given weight is a restricted Kostant partition function.…”

Cluster algebras and singular supports of perverse sheaves

“…In view of §1(iv) it is natural to look for a characterization of B(w) in terms of Lusztig's lagrangian subvarieties Λ V , or the representation theory of the preprojective algebra. It turns out to be related to the subcategory introduced by Buan-Iyama-Reiten-Scott [4], and further studied by Geiss-Leclerc-Schröer [8] and Baumann-Kamnitzer-Tingley [1].…”

“…2.1]. The upper semicontinuity also shows Ext1 (T /T k , Z n ) = 0, hence Z n ∈ add(T /T k ).As the inequality above must be an equality, we get dim Hom(T * k ⊕ T k , Z n ) = dim Hom(Z n , T * k ⊕ T k ) = dim Hom(T * k ⊕ T k , T * k ⊕ T k ) − 1. Therefore dim Hom(T * k , Z n ) + dim Hom(T k , Z n ) = dim Hom(T * k , T * k ⊕ T k ) + dim Hom(T k , T * k ⊕ T k ) − 1.…”