Related papers: The combinatorics of Coxeter categories

We show that the structure of blocks outside the critical hyperplanes of category O over any symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules by…

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…

We continue the study of extended Weyl groups $W$, which are reflection groups. Further we recall the definition of a hyperbolic cover of an extended Weyl group, and show that the hyperbolic covers of the extended Weyl groups are extended…

Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category ("Soergel bimodules"). We also outline…

We give a gentle introduction to the concept of folding. That is, we provide an elementary discussion of equivariant categories, their weighted Grothendieck groups, and the technical aspects of computing with them. We then perform the…

We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an…

We prove that the monoidal 2-category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain…

For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in…

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category…

Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective…

We investigate invertible projective representations and their 2-categorical analogues using the language of TQFTs with defects. The main result is a freeness property for invertible projective representatios. While trivial in the…

Given a finite dimensional algebra $A$, we consider certain sets of idempotents of $A$, called self-injective cores, to which we associate 2-subcategories of the 2-category of projective bimodules over $A$. We classify the simple transitive…

We give a unifying description of all inequivalent vector bundles over the 2-dimensional sphere $S^2$ by constructing suitable global projectors $p$ via equivariant maps. Each projector determines the projective module of finite type of…

In this paper, we study Coxeter systems with two-dimensional Davis-Vinberg complexes. We show that for a Coxeter group $W$, if $(W,S)$ and $(W,S')$ are Coxeter systems with two-dimensional Davis-Vinberg complexes, then there exists…

The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\C)$; $ (b) $ simple modules over deformed preprojective algebras $…

A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of…

We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of all non exceptional irreducible complex…

We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of…

We classify simple weight modules over infinite dimensional Weyl algebras and realize them using the action on certain localizations of the polynomial ring. We describe indecomposable projective and injective weight modules and deduce from…

We construct a functor from the Hecke category to a groupoid built from the underlying Coxeter group. This fixes a gap in an earlier work of the authors. This functor provides an abstract realization of the localization of the Hecke…