Nathan Williams
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes using similar invariants we call reflexponents. Our verifications are case-by-case.
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The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m, p, n)$ of complex ref…
On generating functions and automata associated to reflections in Coxeter systems
本文研究Coxeter系统反射的生成函数与自动机，证明反射前缀语言的正则性，并推导出仿射Coxeter群反射Poincaré级数的简单公式。
In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple formula that computes the reflection length of any element in any affine Coxeter group and we provide a simple uniform proof.
Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.
In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on $\R^n$ is bounded above by $2n$ and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups …
In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants
本文证明有限Coxeter群中抛物型拟Coxeter元素的反射分解的Hurwitz轨道由生成子群和共轭类多重集完全区分，并分类了满足极小生成集为最小大小的有限Coxeter群。
The functions satisfying the mean value property for an n-dimensional cube are determined explicitly. This problem is related to invariant theory for a finite reflection group, especially to a system of invariant differential equations. Solving this problem is reduced to showing that a certain set of invariant polynomials forms an invariant basis. After establishing a certain summation formula over Young diagrams, the latter problem is settled by considering a recursion formula involving Bernoulli numbers. Keywords: polyhedral harmonics; cube; reflection groups; invariant theory; invariant dif…
In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.
For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.
We are building a theory of simple Hurwitz numbers for the reflection groups B and D parallel to the classical theory for the symmetric group. We also study analogs of the cut-and-join operators. An algebraic description of Hurwitz numbers and an explicit formula for them in terms of Schur polynomials are provided. We also relate Hurwitz numbers for B and D to ribbon decomposition of surfaces with boundary -- a similar result for the symmetric group was proved earlier by Yu.Burman and the author. Finally, the generating function of B-Hurwitz numbers is shown to give rise to two independent tau…
We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partition…
Walks avoiding a quadrant and the reflection principle
本文研究避开负象限的小步长平面格点游走计数，解决了允许八个最近邻步的国王模型，证明其生成函数为D-有限级数与代数级数之和，并推进了七个Weyl步集的代数性现象验证。
Hurwitz numbers for reflection groups $G(m,1,n)$
本文将简单 Hurwitz 数理论推广到反射群 $G(m,1,n)$，研究其 cut-and-join 算子、Schur 多项式显式公式、生成函数对应的 KP $\tau$ 函数及 ELSV 型公式。
Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
本文在良生成复反射群中定义并刻画抛物拟 Coxeter 元素，计算其约化反射分解与相对生成集的基数，并将后者与 Frobenius 流形几何不变量关联，为后续统一公式奠定基础。
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