Vertex operators and the class algebras of symmetric groups

A. Lascoux, J. -Y. Thibon

备注：: 23 pages, LaTex. Minor typos corrected

英文摘要：

We exhibit a vertex operator which implements multiplication by power-sums of Jucys-Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of ${\cal W}_{1+\infty}$, to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific e…

2021-10-01math.CO

A classification of centrally-symmetric and cyclic 12-vertex triangulations of $S^2 \times S^2$

In this paper our main result states that there exist exactly three combinatorially distinct centrally-symmetric 12-vertex-triangulations of the product of two 2-spheres with a cyclic symmetry. We also compute the automorphism groups of the triangulations. These instances suggest that there is a triangulation of $S^2 \times S^2$ with 11 vertices -- the minimum number of vertices required.

1998-12-03math.CO

Transpose on vertex symmetric digraphs

We discuss transpose (sometimes called universal exchange or all-to-all) on vertex symmetric networks. We provide a method to compare the efficiency of transpose schemes on two different networks with a cost function based on the number processors and wires needed to complete a given algorithm in a given time.

2014-07-03cs.DSmath.CO

Vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups

A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.

2018-10-21math.COmath.GR

Vertex-transitive graphs with local action the symmetric group on ordered pairs

We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(Γ)$, the local action $G_x^{Γ(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.

2020-10-05math.COmath.GR

On the Hilbert series of vertex cover algebras of Cohen-Macaulay bipartite graphs

We study the Hilbert function and the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is a Cohen-Macaulay bipartite graph.

2009-12-13math.ACmath.CO

更多相关论文收起更多论文

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

2006-06-28math.ACmath.CO

Colored Fermionic Vertex Models and Symmetric Functions

In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$. We establish various combinatorial results for these vertex models and symmetric functions, which include the following. (1) We apply the fusion procedure to the fundamental $R$-matrix for $U_q \big( \widehat{\mathfrak{sl}} (1 | n) \big)$ to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation. (2) We define families of symmetric…

2021-05-10math-phmath.COmath.MP

An $E_8$-approach to the moonshine vertex operator algebra

In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$ associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex o…

2010-09-24math.COmath.GRmath.QA