Reduced expression of minimal infinite reduced words of affine Weyl groups

Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the gr…

Admissible sequences, preprojective modules, and reduced words in the Weyl group of a quiver

This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following app…

Tilings defined by affine Weyl groups

Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint, and their union is the closed cone spanned by the positive roots. We show that similarly, if A is the Weyl alcove, the images (id-w)(A), for affine Weyl group elements w, are all disjoint, and their union is V.

取值于Hecke代数的庞加莱级数与仿射外尔群的几何分解

Hecke Algebra-valued Poincaré Series and Geometric Factorization of Affine Weyl Groups

本文通过测地线管及其双曲稳定子的几何研究仿射外尔群，引入关联框架证明相关猜想，为单连通群建立Hecke代数值ζ函数恒等式的部分结果，凸显三者间相互作用。

Commuting Involution Graphs in Classical Affine Weyl Groups

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde W$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde W$, then the commuting involution graph $\mathcal{C}(\tilde W, X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types $\tilde A_n$ and $\tilde C_n$, so the main work of this paper is to prove the theorem for types $\tilde B_n$ and $\tilde D_n$.

Reflection subgroups of finite and affine Weyl groups

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where finer classifications of reflection subgroups are given, and combinatorial aspects of root systems are shown to appear. Various parameter sets for certain types of subsets of roots are inte…

Minimal elements for the limit weak order on affine Weyl groups

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 with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule a…

Full heaps and representations of affine Weyl groups

We use the author's combinatorial theory of full heaps (defined in math.QA/0605768) to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of ${\Bbb Z}$ and geometrical examples such as the realization of the Weyl group of type $E_6$ as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the l…

Limiting directions for random walks in classical affine Weyl groups

Let $W$ be a finite Weyl group and $\widetilde W$ the corresponding affine Weyl group. A random element of $\widetilde W$ can be obtained as a reduced random walk on the alcoves of $\widetilde W$. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of $|W|$ many directions. We compute these directions when $W$ is $B_n$, $C_n$ and $D_n$ and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types $B$ and $C$ in the affirmative and for type $D$ in the negative. The main tool is a combinatorial two row model for a totally asymmetri…

仿射Weyl群中的平移及其在离散可积系统中的应用

Translations in affine Weyl groups and their applications in discrete integrable systems

该研究将离散可积系统领域的Weyl群相关研究从单laced的$ADE$型拓展到非单laced型，重点推导仿射Weyl群平移元公式，并用以阐明两类新型可积系统的性质。

Cluster algebras from surfaces and extended affine Weyl groups

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with every triangulation a basis in $V$, such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type $A$, which is invariant under flips. The construction is also extended to exceptional skew-symmet…

Weyl Groups with Coxeter Presentation and Presentation by Conjugation

We investigate which Weyl groups have a Coxeter presentation and which of them at least have the presentation by conjugation with respect to their root system. For most concepts of root systems the Weyl group has both. In the context of extended affine root systems (EARS) there is a small subclass allowing a Coxeter presentation of the Weyl group and a larger subclass allowing the presentation by conjugation. We give necessary and sufficient conditions for both classes. Our results entail that every extended affine Weyl group (EAWeG) has the presentation by conjugation with respect to a suitab…

Permutation Weights and Modular Poincare Polynomials for Affine Lie Algebras

Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying the Weyl orbit $W(ρ)$ of its Weyl vector $ρ$. A remarkable fact for Affine Lie algebras is that the number of elements of $W(ρ)$ is finite at each and every depth level though totally it has infinite number of elements. This allows us to look at $W(ρ)$ as a manifold graded by depths of its elements and hence a new kind of Poincare Polynomial is defined. We give these polynomials for all Affine Kac-Moody Lie algebras, non-twisted or twisted. The remarkable fact is however that, on the contrary to the ones which are cl…