Related papers: Tilings defined by affine Weyl groups

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…

Let $W$ be an irreducible Weyl group and $W_a$ its affine Weyl group. In a previous work the author introduced an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. The…

Let $G$ be a simple algebraic group of type $E_6$ over an algebraically closed field of characteristic $p>0$. We determine the submodule structure of the Weyl modul es with highest weight $r\omega_1$ for $0\leq r\leq p-1$, where $\omega_1$…

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $\Gamma < G$ an irreducible lattice. Consider the group measure space von Neumann algebra…

We give several necessary and sufficient conditions for the existence of {\it the presentation by conjugation} for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence…

In the present paper we further the study of the compression cone of a real spherical homogeneous space $Z=G/H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Kr\"otz. Our…

The reflections in a Coxeter group are defined as conjugates of a single generator, and thus admit palindromic expressions as products of generators. Our main result gives closed formulas providing a palindromic reduced expression for each…

We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$,…

Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that…

Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to…

Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such…

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is…

For affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an affine type. We compare our construction…

We show that positive elements with respect to the twisted convolutions, belonging to some ultra-test function space of certain order at origin, belong to the ultra-test function space of the same order everywhere. We apply the result to…

In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated.…

In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some…

We provide a proof of the Alpern multi-tower theorem for Z^d actions. We reformulate the theorem as a problem of measurably tiling orbits of a Z^d action by a collection of rectangles whose corresponding sides have no non-trivial common…

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on…

We construct families of representations for quantum groups over $\mathbb{Z}[v,v^{-1}]$-algebras that interpolate between Weyl modules and tilting modules. These families might be candidates for objects with characters satisfying the {\em…

We study the diagrammatic Hecke category associated with the affine Weyl group of type $\tilde{A}_2$. More precisely we find a (surprisingly simple) basis for the Hom spaces between indecomposable objects, that we call indecomposable double…