Related papers: The Coxeter transformation on Cominuscule Posets

We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the…

We derive the explicit formula for the inverse of zeta matrix for any graded posets with the finite set of minimal elements . The combinatorial interpretation of this result is given. For that to do special number theoretic code triangles…

Let $G({\mathbb F}_{q})$ be a finite Chevalley group defined over the field of $q=p^{r}$ elements, and $k$ be an algebraically closed field of characteristic $p>0$. A fundamental open and elusive problem has been the computation of the…

For an exact category $\mathcal{E}$, we study the Butler's condition "AR=Ex": the relation of the Grothendieck group of $\mathcal{E}$ is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to…

We establish the existence of an IC basis for the generalized Temperley--Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra…

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…

In contrast to the classical and semiclassical settings, the Coxeter element (12...n) which cycles the columns of an mxn matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained…

This paper has been withdrawn by the authors due to a crucial computational error. In this paper we deal with the finite case. We prove that a finite bounded ordered set can be represented as the order of principal congruences of a finite…

We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any…

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies…

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group…

We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver's sense.…

For a multidegree t in N^n, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A…

We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The…

We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of…

Interchanging character and co-character groups of a torus $T$ over a field $k$ introduces a contravariant functor $T \rightarrow \widehat{T}$. Interpreting $\rho:T\rightarrow {\mathbb C}^\times$, half the sum of positive roots for $T$ a…

Over 50 years ago, Rota posted the following celebrated `Research Problem': prove or disprove that the partial order of partitions on an $n$-set (i.e., the refinement order) is Sperner. A counterexample was eventually discovered by Canfield…

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite…

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B…

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice…