Related papers: Mirkovic-Vilonen cycles and polytopes in type A

Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they…

We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope…

We describe how Mirkovic-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations…

Let Gr be the affine Grassmannian for a connected complex reductive group G. Let C_G be the complex vector space spanned by (equivalence classes of) Mirkovic-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a…

Mirkovi\'c-Vilonen (MV) polytopes are a class of generalized permutahedra originating from geometric representation theory. In this paper we study MV polytopes coming from matroid polytopes, flag matroid polytopes, Bruhat interval…

Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine…

Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\sigma$ be a Dynkin automorphism on $G$. In this paper, we get a bijection between the set of $\st$-invariant MV cycles (polytopes) for $G$ and the set of…

We realize affine Mirkovi{\'c}-Vilonen polytopes using Littelmann's path model in the framework of masures. We are also able to read the decorations on the paths in the case of sl2.

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the…

We give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent…

We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by…

We study cells in generalised Bott-Samelson varieties for type C. These cells are parametrised by certain galleries in the affine building. We define a set of readable galleries - we show that the closure in the affine Grassmannian…

In an earlier work, we proved that MV polytopes parameterize both Lusztig's canonical basis and the Mirkovic-Vilonen cycles on the Affine Grassmannian. Each of these sets has a crystal structure (due to Kashiwara-Lusztig on the canonical…

Mirkovi\'c--Vilonen (MV) polytopes play a key role in the representation theory of reductive algebraic groups, while the geometric behavior of prime MV polytopes under Minkowski addition remains a subtle open problem. This paper focuses on…

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated…

This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirkovi\'c-Vilonen…

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two…

Let $G$ be a complex reductive group and let $G^\vee$ be its Langlands dual. Let us choose a triangular decomposition $\mathfrak g^\vee=\mathfrak n^\vee_-\oplus\mathfrak h^\vee\oplus\mathfrak n^\vee_+$ of the Lie algebra $G^\vee$.…

We study the algebraic geometry and combinatorics of the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) in type A. More specifically, we elucidate the central…

We study the description of the crystal structure on the set of Mirkovi\'c-Vilonen polytopes. Anderson and Mirkovi\'c defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for…