Related papers: On dominance and minuscule Weyl group elements

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the…

We introduce the notion of a strong minuscule element, and prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root. In addition, we enumerate the strong…

If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$ is a negative…

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide…

Let $G$ be a reductive group with Borel $B$ and Weyl group $W$. Then $B$-double cosets in $G$ are indexed by the Weyl group, say $O(w)$ for $w\in W$. Then we prove the minimal $B$-double coset in the convolution $O(w_1)*O(w_2)$ is…

We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in…

In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and…

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the…

In a recent paper, Barot and Marsh presented an explicit construction of presentation of a finite Weyl group by any seed of corresponding cluster algebra, i.e. by any diagram mutation-equivalent to an orientation of a Dynkin diagram with…

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$…

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y…

The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an…

Given a Coxeter group $W$ with Coxeter system $(W,S)$, where $S$ is finite. We provide a complete characterization of Boolean intervals in the weak order of $W$ uniformly for all Coxeter groups in terms of independent sets of the Coxeter…

Let $G$ be a simple complex Lie group with Weyl group $W$. We give a formula for the character of $W$ on the zero weight space of any finite dimensional representation of $G$. The formula involves partition functions, generalizing Kostant's…

In this article, we prove that for any indecomposable dominant character of a maximal torus $T$ of a simple adjoint group $G$ such that there is a Coxeter element $w \in W$ for which $X(w)^{ss}_T(\mathcal L_\chi) \neq \emptyset$. If…

For affine Weyl groups and elements associated to dominant coweights, we present a convex geometry formula for the size of the corresponding lower Bruhat intervals. Extensive computer calculations for these groups have led us to believe…

Let W be a Weyl group and let w be a Coxeter elememt of minimal length of W. In the early 1970's I.G.Macdonald stated that the trace of w on an irreducible representation of W is 0,1 or -1. In this paper we give a proof of this statement…

Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s \in S$. If there exists $t\in S$ such that $s$ and $t$ do not…

This note is a complement of a recent paper about low elements in affine Coxeter groups. We explain in terms of ad-nilpotent ideals of a Borel subalgebra why the minimal elements of dominant Shi regions are low. We also give a survey of the…

Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $\chi$-linear bicrystals. In this paper, we…