Low elements in dominant Shi regions

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 bijections involved in the study of dominant Shi regions in affine Weyl groups.Comment: 8 page

A symmetric group action on the irreducible components of the Shi variety associated to W(A~n)W(\widetilde{A}_n)

Let $W_a$ be an affine Weyl group with corresponding finite root system $\Phi$. In \cite{JYS1} Jian-Yi Shi characterized each element $w \in W_a$ by a $\Phi^+$-tuple of integers $(k(w,\alpha))_{\alpha \in \Phi^+}$ subject to certain conditions. In \cite{NC1} a new interpretation of the coefficients $k(w,\alpha)$ is given. This description led us to define an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. It turns out that this variety has more than one irreducible component, and the set of these components, denoted $H^0(\widehat{X}_{W_a})$, admits many interesting properties. In particular the group $W_a$ acts on it. In this article we show that the set of irreducible components of $\widehat{X}_{W(\widetilde{A}_n)}$ is in bijection with the conjugacy class of $(1~2~\cdots~n+1) \in W(A_n) = S_{n+1}$. We also compute the action of $W(A_n)$ on $H^0(\widehat{X}_{W(\widetilde{A}_n)})$.Comment: 18 pages, 5figures, 1 tabl

Affine twisted length function

Let $W_a$ be an affine Weyl group. In 1987 Jian Yi Shi gave a characterization of the elements $w \in W_a$ in terms of $\Phi ^+$-tuples $(k(w,\alpha ))_{\alpha \,\in \,\Phi ^+}$ called the Shi vectors. Using these coefficients, a formula is provided to compute the standard length of $W_a$. In this note we express the twisted affine length function of $W_a$ in terms of the Shi coefficients

Affine twisted length function

Let $W_a$ be an affine Weyl group. In 1987 Jian Yi Shi gave a characterization of the elements $w \in W_a$ in terms of $\Phi ^+$-tuples $(k(w,\alpha ))_{\alpha \,\in \,\Phi ^+}$ called the Shi vectors. Using these coefficients, a formula is provided to compute the standard length of $W_a$. In this note we express the twisted affine length function of $W_a$ in terms of the Shi coefficients

Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups

We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^\theta$, where $x$ is an element of $W$ and $W_L^\theta$ the subgroup of fixed points of an automorphism $\theta$ of order at most two of a standard parabolic subgroup $W_L$ of $W$. When $\theta\neq\mathrm{id}$, there is in general more than one element of minimal length in a given coset, and we explain how to relate elements of minimal length. We also show that elements of minimal length in cosets are exactly those elements which are minimal for the restriction of the Bruhat order.Comment: 8 pages, 2 figures. Comments welcome

Shi arrangements and low elements in affine Coxeter groups

Given an affine Coxeter group $W$, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan-Lusztig cells for $W$. In particular, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$. Low elements in $W$ were introduced to study the word problem of the corresponding Artin-Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in $W$. In this article, we show in the case of an affine Coxeter group that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a byproduct of our proof, we show that the descent-walls -- the walls that separate a region from the fundamental alcove -- of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.Comment: 38 pages, 6 figure

Atomic length in Weyl groups

We define a new statistic on Weyl groups called the atomic length and investigate its combinatorial and representation-theoretic properties. In finite types, we show a number of properties of the atomic length which are reminiscent of the properties of the usual length. Moreover, we prove that, with the exception of rank two, this statistic describes an interval. In affine types, our results shed some light on classical enumeration problems, such asthe celebrated Granville-Ono theorem on the existence of core partitions, by relating the atomic length to the theory of crystals

Atomic length in Weyl groups

We define a new statistic on Weyl groups called the atomic length and investigate its combinatorial and representation-theoretic properties. In finite types, we show a number of properties of the atomic length which are reminiscent of the properties of the usual length. Moreover, we prove that, with the exception of rank two, this statistic describes an interval. In affine types, our results shed some light on classical enumeration problems, such asthe celebrated Granville-Ono theorem on the existence of core partitions, by relating the atomic length to the theory of crystals

Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups

8 pages, 2 figures. Comments welcome!We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^\theta$, where $x$ is an element of $W$ and $W_L^\theta$ the subgroup of fixed points of an automorphism $\theta$ of order at most two of a standard parabolic subgroup $W_L$ of $W$. When $\theta\neq\mathrm{id}$, there is in general more than one element of minimal length in a given coset, and we explain how to relate elements of minimal length. We also show that elements of minimal length in cosets are exactly those elements which are minimal for the restriction of the Bruhat order