Pierre Baumann

A Solomon descent theory for the wreath products ≀_{}

We propose an analogue of Solomons descent theory for the case of a wreath product G/G n , where G is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Spechts theory for the representations of wreath products, Okadas extension to wreath products of the Robinson-Schensted correspondence, and Poiriers quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.

On Mirković-Vilonen cycles and crystal combinatorics

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 Mirkovic-Vilonen cycles in the affine Grassmannian G ( C((t)) ) /G ( C[[t]] ) is 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.

Rank 2 affine MV polytopes

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 Saitos characterization of the infinity crystal in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the analogues in this case of the Mirkovic-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwaras similarity of crystals we also give MV polytopes for

Weyl group action and semicanonical bases

A_2^{(2)}

Réflexions dans un cristal

, the only other rank two affine Kac-Moody algebra.

Affine Mirković-Vilonen polytopes

Abstract Let U be the enveloping algebra of a symmetric Kac–Moody algebra. The Weyl group acts on U, up to a sign. In addition, the positive subalgebra U + contains a so-called semicanonical basis, with remarkable properties. The aim of this paper is to show that these two structures are as compatible as possible.

Preprojective algebras and MV polytopes

Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let \lambda be a dominant integral weight, let T_\lambda = {t_\lambda} be the crystal with one element of weight \lambda, and let B(\lambda) \subset B(\infty) \otimes T_\lambda be the crystal of the integrable representation of highest weight \lambda. We compute the descending string parameters of an element b \otimes t_\lambda in B(\lambda) in terms of the Lusztig parameters of b.