Related papers: A Coxeter type classification of Dynkin type $\mat…

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of…

We classify finite posets with a particular sorting property, generalizing a result for rectangular arrays. Each poset is covered by two sets of disjoint saturated chains such that, for any original labeling, after sorting the labels along…

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this…

We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver cycle is determined by quiver…

In this paper we prove a series of matching theorems for two sets of Coxeter generators of a finitely generated Coxeter group that identify common features of the two sets of generators. As an application, we describe an algorithm for…

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive matrices strictly contains the cone of completely positive matrices, i.e., all matrices of the…

These graphs generalize both affine and finite type and provide an explanation for some quantum properties of roots of unity.

In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it.…

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group.…

We study $3d$ $\mathcal{N}=4$ quiver gauge theories with gauge nodes forming a $D_n$ Dynkin diagram. The class of good $D_n$ Dynkin quivers is completely characterised and the moduli space singularity structure fully determined for all such…

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for…

In this paper we show that for a simply-laced root system a choice of $C$ gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to $C$-orbits in $R$; moreover, it gives an identification of…

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally…

Let $P$ be a finite poset and $L$ the associated distributive lattice of order ideals of $P$. Let $\rho$ denote the rowmotion bijection of the order ideals of $P$ viewed as a permutation matrix and $C$ the Coxeter matrix for the incidence…

Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

Acyclic categories were introduced by Kozlov and can be viewed as generalised posets. Similar to posets, one can define their incidence algebras and a related topological complex. We consider the incidence algebra of either a poset or…

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial…

Let Q be a Dynkin quiver. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We study maximal Hom-free sets in the…

Given a family $\F$ of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category $\C_{\F}$ called the \emph{incidence category of $\F$}. This category is "nearly abelian" in the sense that all…