Vertex Algebras and the Class Algebras of Wreath Products
Vertex Algebras and the Class Algebras of Wreath Products
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顶点代数和花环积的类代数
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发表时间：
2002
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通讯作者：
Weiqiang Wang
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作者：
Weiqiang Wang
引入了与任何有限群 Γ 相关的花环积 Γn = Γ Sn 的 Jucys-Murphy 元素，它们在我们研究所有 n 的 Γn 类代数与顶点代数之间的联系中发挥着重要作用。我们以群论的方式构造了 W1 + ∞ 代数（的变体）的作用，该作用不可约地作用于所有 n 的 Γn 类代数的直和 RΓ。我们建立了使用 JM 元素的卷积算子和作用于 RΓ 的海森堡代数算子之间的各种关系。作为应用，我们获得了 Γn 类代数的两组不同的代数生成器，并建立了有关 Гn 归一化共轭类乘积和 Jucys-Murphy 元素的幂和等的各种稳定性结果。我们引入了一个稳定代数，它编码所有 n 的 Гn 类代数结构，其结构常数显示为非负整数。在对称群情况下（即当 Г 为平凡时），我们以统一的方法恢复和强化了 Lascoux 和 Thibon、Kerov 和 Olshanski、Farahat 和 Higman 等的各种结果。2000 年数学学科分类 17B69, 20C05。
The Jucys–Murphy elements for wreath products Γn = Γ Sn associated to any finite group Γ are introduced and they play an important role in our study of the connections between class algebras of Γn for all n and vertex algebras. We construct an action of (a variant of) the W1 + ∞‐algebra acting irreducibly on the direct sum RΓ of the class algebras of Γn for all n in a group‐theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on RΓ. As applications, we obtain two distinct sets of algebra generators for the class algebra of Γn and establish various stability results concerning products of normalized conjugacy classes of Γn and the power sums of Jucys‐‐Murphy elements, etc. We introduce a stable algebra which encodes the class algebra structures of Γn for all n, whose structure constants are shown to be non‐negative integers. In the symmetric group case (that is, when Γ is trivial), we recover and strengthen in a uniform approach various results of Lascoux and Thibon, Kerov and Olshanski, and Farahat and Higman, etc. 2000 Mathematics Subject Classification 17B69, 20C05.