On the algebraic decomposition of a centralizer algebra of the hyperoctahedral group
On the algebraic decomposition of a centralizer algebra of the hyperoctahedral group
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超八面体群的中心化代数的代数分解
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R. Orellana
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作者：
R. Orellana
本文给出了超八面体群符号置换表示张量积幂分解的一个组合规则。然后，我们使用这个规则来描述的Bratteli图的中心化代数的这个组在第k张量空间。我们表明，这个代数的基础可以完全描述在集分区，我们给一组发电机和关系。在[J]中Jones给出了中心化代数EndSn（V）的一个刻划，其中V = C，Sn（对称群）通过置换作用在V上，并且它对角地作用在V上.这个代数是由Martin [Ma]独立引入的，并命名为划分代数。研究划分代数的主要动机是推广统计力学中的Temperley-Lieb代数和Potts模型。Halverson和Ram [HR]对划分代数作了一个综述。本文的主要目的是研究分拆代数的一类子代数。本文研究了超八面体群的中心化子代数。这个群是2阶循环群和对称群的圈积，我们将其表示为Gn：= Z/2 ZwrSn。设V = C，则Gn通过有符号置换作用于这个向量空间。代数EndGn（V）的半单分解可以通过利用双中心化子理论将张量积V k分解为简单的Gn-模得到。本文证明了Gn-模V k的不可约分解。由于Gn的表示是由有序Young图对索引的，我们得到了EndGn（V）的不可约表示的有序Young图对的索引集.这个分解规则也产生了一个纯粹的组合规则的杨图对的限制和归纳的简单模块的中心化代数。这个分支规则允许我们写出这些中心化代数的Bratteli图。1991年数学学科分类。小学16 G99，05 E05;中学20 F55，16 S99。
In this paper we give a combinatorial rule for the decomposition of tensor powers of the signed permutation representation of the hyperoctahedral group. We then use this rule to describe the Bratteli diagram of a centralizer algebra of this group over k-th tensor space. We show that a basis for this algebra can be described completely in terms of set partitions and we give a set of generators and relations. Introduction In [J] Jones has given a description of the centralizer algebra EndSn(V ) where V = C and Sn (symmetric group) acts by permutations on V and it acts diagonally on V . This algebra was independently introduced by Martin [Ma] and named the Partition algebra. The main motivation for studying the partition algebra is in generalizing the Temperley-Lieb algebras and the Potts model in statistical mechanics. A survey on the Partition algebra has been written by Halverson and Ram [HR]. The main objective of this paper is to study a family of subalgebras of the Partition algebra. In this paper we look at the corresponding centralizer algebras of the hyperoctahedral group. This group is the wreath product of the cyclic group of order 2 and the symmetric group, which we will denote by Gn := Z/2ZwrSn. Let V = C, then Gn acts on this vector space via signed permutations. The semisimple decomposition of the algebra EndGn(V ) can be obtained by decomposing the tensor product V ⊗k in terms of simple Gn-modules by the double centralizer theory. In this paper we prove the decomposition of the Gn-module V ⊗k in terms of irreducibles. Since the representations of Gn are indexed by ordered pairs of Young diagrams, we obtained an indexing set for the irreducible representations of EndGn(V ) in terms of ordered pairs of Young diagrams. This decomposition rule also yields a purely combinatorial rule in terms of pairs of Young diagrams for the restriction and induction of simple modules of this centralizer algebras. This branching rule allows us to write the Bratteli diagram of these centralizer algebras. 1991 Mathematics Subject Classification. Primary 16G99, 05E05; Secondary 20F55, 16S99.