Related papers: Coxeter groups and quiver representations
This is the first of a series of papers which define and study structures called rootoids, which are groupoids equipped with a representation in the category of Boolean rings and with an associated 1-cocycle. The axioms for rootoids are…
The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil…
We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group…
The new approach to the theory of complex representrations of the finite symmetric groups which based on the notions of Coxeter generators., Gelfand-Zetlin algebras, Hecke algebra, Young-Jucys-Murphi generators and which hardly used…
We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin's Frobenius manifold structure which is lifted…
Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of…
This paper introduces and studies a class of Weyl-type algebras \(A_{p,t,\cA} = \Weyl{e^{\pm x^{p} e^{t x}},\; e^{\cA x},\; x^{\cA}}\) constructed over exponential-polynomial rings, where \(\FF\) is a field of characteristic zero, \(\cA\)…
It is well-known that a quiver Q of type A_n is representation-finite, and that its indecomposable representations are thin (all Jordan-Hoelder multiplicities are 0 or 1). By now, various methods of proof are known. The aim of this note is…
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between…
The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of…
It was conjectured at the end of the book "Representation theory of Artin algebras" by M. Auslander, I. Reiten and S. Smalo that an Artin algebra with the property that its finitely generated indecomposable modules are up to isomorphism…
Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine…
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction…
We study a class of two-generator two-relator groups, denoted $J_n(m,k)$, that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature…
We investigate the faces and the face lattices of arbitrary Coxeter group invariant convex subcones of the Tits cone of a linear Coxeter system as introduced by E. B. Vinberg. Particular examples are given by certain Weyl group invariant…
We present a worked example for the new extensions of the multi-particle Calogero model endowed with infinite Weyl group symmetry of affine and hyperbolic type. Building upon the hyperbolic extension of the $A_3$-Kac-Moody algebra, we…
We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well…
We consider the class of finitely generated groups whose relators are powers of commutators of the generators. This class contains as a small subclass graph groups (also called RAAGs), namely if all powers are one. Graph groups are the only…
A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…