Related papers: A singular Coxeter presentation

We consider the $[q+1,q-3,5]_q3$ generalized doubly-extended Reed-Solomon code of codimension $4$ as the code associated with the twisted cubic in the projective space $\mathrm{PG}(3,q)$. Basing on the point-plane incidence matrix of…

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…

We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded…

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point…

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order)…

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to…

In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…

For infinite-dimensional groups $G\supset K$ the double cosets $K\setminus G/K$ quite often admit a structure of a semigroup; these semigroups act in $K$-fixed vectors of unitary representations of $G$. We show that such semigroups can be…

This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these…

We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in…

Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting…

We show how the classification of simple singularities of functions can be reduced directly, not using the normal forms, to the classification of irreducible Weyl groups. We also prove that the class of a singularity in its local algebra…

We construct a "Koszul duality" equivalence relating the (diagrammatic) Hecke category attached to a Coxeter system and a given realization to the Hecke category attached to the same Coxeter system and the dual realization. This extends a…

We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ commutation classes of reduced expressions, where $N(w)$ is a certain statistic defined in…

Given a function defined over a parabolic subgroup of a Coxeter group, equidistributed with the length, we give a procedure to construct a function over the entire group, equidistributed with the length. Such a procedure permits to define…

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we tackle different topics around the field of definition of a hyperbolic…

The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic…

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to…

We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group…