Exponential formulas for models of complex reflection groups

Weyl群中反射的既约字

Reduced words for reflections in Weyl groups

研究Coxeter群中反射的回文既约表达式问题，为任意有限Weyl群中的每个反射给出了提供回文既约表达式的闭公式，填补了特殊情形之外缺乏明确公式的空白。

On the sign representations for the complex reflection groups $G(r,p,n)$

We present a formula for the values of the sign representations of the complex reflection groups $G(r,p,n)$ in terms of its image under a generalized Robinson-Schensted map.

Coxeter 群中的反射与商

Reflections and quotients in Coxeter groups

提出了一个公式，将 Coxeter 群元素的左下降集与其在极大商投影的左下降集联系起来，这是涉及商与下降的一般性结果的特例。

非相交路径与自由边界菱形镶嵌的反射原理

A reflection principle for nonintersecting paths and lozenge tilings with free boundaries

本文提出一个补充 Okada–Stembridge Pfaffian 公式的新公式，建立非相交路径的反射原理，将自由边界菱形镶嵌计数归结为无自由边界情形，并给出乘积公式、分解定理及行列式公式等应用。

一般线性群和仿射群中的反射长度

Reflection length in the general linear and affine groups

本文提出判断一般线性群反射分解约化的内在准则，并给出计算一般仿射群反射长度的公式。

Expected length of a product of random reflections

We present a simple formula for the expected number of inversions in a permutation of size $n$ obtained by applying $t$ random (not necessarily adjacent) transpositions to the identity permutation. More general, for any finite irreducible Coxeter group belonging to one of the infinite families (type A, B, D, and I), an exact expression is obtained for the expected length of a product of $t$ random reflections.

Excedance number for involutions in complex reflection groups

We define the excedance number on the complex reflection groups and compute its multidistribution with the number of fixed points on the set of involutions in these groups. We use some recurrence formulas and generating functions manipulations to obtain our results.

关于Coxeter系统反射的生成函数与自动机

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。

Catalan numbers for complex reflection groups

We construct (q,t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik algebras of W, and Opdam's analysis of permutations of the irreducible representations of W arising from the Knizhnik-Zamolodchikov connection.

Computing reflection length in an affine Coxeter group

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.

Invariant theory for coincidental complex reflection groups

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real r…

Hurwitz numbers for reflection groups I: Generatingfunctionology

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…

Polynomiality of factorizations in reflection groups

We study the number of ways of factoring elements in the complex reflection groups G(r,s,n) as products of reflections. We prove a result that compares factorization numbers in G(r,s,n) to those in the symmetric group on n letters, and we use this comparison, along with the ELSV formula, to deduce a polynomial structure for factorizations in G(r,s,n).

Statistics on Graphs, Exponential Formula and Combinatorial Physics

The concern of this paper is a famous combinatorial formula known under the name "exponential formula". It occurs quite naturally in many contexts (physics, mathematics, computer science). Roughly speaking, it expresses that the exponential generating function of a whole structure is equal to the exponential of those of connected substructures. Keeping this descriptive statement as a guideline, we develop a general framework to handle many different situations in which the exponential formula can be applied.