顶点代数与圈积类代数
Vertex algebras and the class algebras of wreath products
摘要：
引入了任意有限群G的圈积G_n的Jucys-Murphy元，它们在研究G_n的类代数与顶点代数之间的联系中起着重要作用.用群论的方法构造了W_{1+\infty}代数（的一个变体）不可约地作用在G_n的类代数的直和R_G上.建立了JM元的卷积算子与作用在R_G上的Heisenberg代数算子之间的各种关系。作为应用，我们得到了G_n的类代数的两组不同的代数生成元，并建立了G_n的正规共轭类的乘积与Jucys-Murphy元的幂和等的稳定性结果.我们引入了一个稳定代数，它对所有n都编码G_n的类代数结构，其结构常数证明为非负整数.在对称群的情形下（即G是平凡的），我们统一地恢复和加强了Lascoux-Thibon，Kerov-Olshanski和Farahat-Higman等的各种结果.
英文摘要：
The Jucys-Murphy elements for wreath products G_n associated to any finite group G are introduced and they play an important role in our study on the connections between class algebras of G_n for all n and vertex algebras. We construct an action of (a variant of) the W_{1+\infty} algebra acting irreducibly on the direct sum R_G of the class algebras of G_n for all n in a group theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on R_G. As applications, we obtain two distinct sets of algebra generators for the class algebra of G_n and establish various stability results concerning products of normalized conjugacy classes of G_n and the power sums of Jucys-Murphy elements etc. We introduce a stable algebra which encodes the class algebra structures of G_n for all n, whose structure constants are shown to be non-negative integers. In the symmetric group case (i.e. G is trivial), we recover and strengthen in a uniform approach various results of Lascoux-Thibon, Kerov-Olshanski, and Farahat-Higman, etc.
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