带广义Jucys-Murphy权重的Coxeter因子分解与反射群的矩阵树定理

Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function

A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that $h_r(J_1,...,J_n),$ the complete homogeneous symmetric polynomial of degree $r$ in the JM elements, coincides with the $r$th term in the asymptotic expansion of the Weingarten function. We use this connection to determine precisely which conjugacy classes occur in the class basis resolution of $h_r(J_1,...,J_n),$ and to explicitly determine the coefficients of the classes of minimal height when $r < n.$ These …

Partial Jucys-Murphy elements and star factorizations

In this paper, we look at the number of factorizations of a given permutation into star transpositions. In particular, we give a natural explanation of a hidden symmetry, answering a question of I.P. Goulden and D.M. Jackson. We also have a new proof of their explicit formula. Another result is the normalized class expansion of some central elements of the symmetric group algebra introduced by P. Biane. To obtain this results, we use natural analogs of Jucys-Murphy elements in the algebra of partial permutations of V. Ivanov and S. Kerov. We investigate their properties and use a formula of A.…

Hecke代数中的分解 I：长轮换分解与Jucys-Murphy元素

Factorizations in Hecke algebras I: long cycle factorizations and Jucys-Murphy elements

本文通过推广Jucys-Murphy元素的分解概念，为A型Iwahori-Hecke代数开创了排列分解理论，给出了对称群中长轮换分解的q-形变，并揭示了q-二项式、q-Catalan和q-Narayana数。

On complete functions in Jucys-Murphy elements

The problem of computing the class expansion of some symmetric functions evaluated in Jucys-Murphy elements appears in different contexts, for instance in the computation of matrix integrals. Recently, M. Lassalle gave a unified algebraic method to obtain some induction relations on the coefficients in this kind of expansion. In this paper, we give a simple purely combinatorial proof of his result. Besides, using the same type of argument, we obtain new simpler formulas. We also prove an analogous formula in the Hecke algebra of $(S_{2n},H_n)$ and use it to solve a conjecture of S. Matsumoto o…

Monomial bases for the centres of the group algebra and Iwahori--Hecke algebra of S_4

G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys--Murphy elements. While we have shown in earlier work that the centre of the group algebra of S_3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S_4 has infinitely many bases consisting of monomial symmetric polynomials in Jucys--Murphy elements, which we characterize completely. The proof of this result involves establishing closed forms for coefficients of class su…

Transitive powers of Young-Jucys-Murphy elements are central

Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,n}, are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of S_n, and show that they have a polynomiality property. These centrality and polynomiality properti…

A new integral basis for the centre of the Hecke algebra of type A

We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally, for n=3, we show that only one such basis exists for the centre of the Hecke algebra, by proving that there are exactly four bases for the centre of the corresponding symmetric group algebra which consist solely of mono…

通过Jucys-Murphy元素之和进行洗牌

Shuffling via sums of Jucys--Murphy Elements

本文研究了一种通过对称群S_m的Jucys-Murphy元素对应的置换进行洗牌的家族，确定了这些洗牌的转移矩阵的本征值，并研究了特定情况下的混合时间，证明了k-星置换洗牌在总变差截止时具有特定的渐进行为。

从实格拉斯曼流形的 Weingarten 演算到单调 Hurwitz 数与 Jucys-Murphy 元的形变

From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements

本文通过实格拉斯曼流形上的积分引入 b-Weingarten 演算，定义 bt-单调 Hurwitz 数，并提出 Jucys-Murphy 元的 b-形变算子及其与 Jack 对称函数相关的若干猜想。

On the fusion procedure for the symmetric group

We give a new version of the fusion procedure for the symmetric group which originated in the work of Jucys and was developed by Cherednik. We derive it from the Jucys-Murphy formulas for the diagonal matrix units for the symmetric group.

Proof of Stanley's conjecture about irreducible character values of the symmetric group

R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.

Partial sum of matrix entries of representations of the symmetric group and its asymptotics

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we expand a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose these objects into a main term and a reminder, proving a central limit theorem for both main terms and a law of …

On representation theory of partition algebras for complex reflection groups

This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tan…