Related papers: The crystal structure on the set of Mirkovic-Vilon…

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…

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…

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 $G^\vee$.…

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…

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…

In the current paper, we give a quiver theoretical interpretation of Mirkovi\'c-Vilonen polytopes in type $A_n$. As a by-product, we give a new proof of the Anderson-Mirkovi\'c conjecture which describes the explicit forms of the actions of…

We give a realization of the infinity crystal for affine sl(2) using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of the infinity crystal in terms of the *…

In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a…

Mirkovi\'c-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal $B(\infty)$ of a semisimple group $G$. We retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr O(N)$ of…

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…

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 study, in type A, the algebraic cycles (MV-cycles) discovered by I. Mirkovi\'c and K. Vilonen [MV]. In particular, we partition the loop Grassmannian into smooth pieces such that the MV-cycles are their closures. We explicitly describe…

Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra g has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given g, there is a limit crystal, usually denoted by…

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 an explicit description of the (lowering) Kashiwara operators on Mirkovi\'c-Vilonen polytopes in types $B$ and $C$, which provides a simple method for generating Mirkovi\'c-Vilonen polytopes inductively. This description can be…

We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the…

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula"…

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.

Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of…

We study a class of varieties which generalize the classical orbital varieties of Joseph. We show that our generalized orbital varieties are the irreducible components of a Mirkovic-Vybornov slice to a nilpotent orbit, and can be labeled by…