On table algebras and applications to finite group theory
On table algebras and applications to finite group theory
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DOI：
10.1016/0021-8693(91)90195-e
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发表时间：
1991-04
影响因子：
0.9
通讯作者：
Z. Arad;H. I. Blau
中科院分区：
文献类型：
--
作者：
Z. Arad;H. I. Blau
本文提出了研究有限群的不可约特征或共轭类乘积分解的抽象设置。我们定义了“表代数”的概念，发展了它的一些基本属性，并在这种情况下证明了 Arad 和 Fisman [AFl, AF2]、Arad、Cillag、Herzog 和 Stavi [ACH, AH] 以及 Mann [M] 结果的适当概括。因此，不可约特征和共轭类的独立但相似的情况以统一的方式一起处理。有限群的特征表和类代数的代数和组合推广有着悠久而复杂的历史。关联方案已被广泛研究（有关一些历史参考文献，请参见 [BI，第 1791 页），Brauer 关于伪群的工作 [Br2] 也值得一提。在我们最初定义和探索表代数之后，我们发现它们实际上与一种特殊类型的“C 代数”以精确的方式相关。现在是 C 代数，它概括了交换关联方案。 4.5 年前，Kawada [K] 的一篇论文中介绍了这些内容，作为 Hoheisel [H] 工作的抽象。下面的定理 2.10 表明表代数等价于具有非负结构常数的 C 代数。（C 代数的定义参见 [BI, p. SS]。这也可以从定理 2.10 的证明中推断出来。）137
This paper presents an abstract setting for the study of the decompositions of products of either irreducible characters or conjugacy classes of a finite group. We define the concept of “table algebra,” develop some of its basic properties, and prove in this context suitable generalizations of results of Arad and Fisman [AFl, AF2], Arad, Chillag, Herzog, and Stavi [ACH, AH], and Mann [M]. The separate but analogous cases of irreducible characters and of conjugacy classes are thus treated together in a uniform way.There is a long and complex history of algebraic and combinatoric generalizations of character tables and class algebras of finite groups. Association schemes have been extensively studied (see [BI, p. 1791 for some historical references), and Brauer’s work on pseudo groups [Br2], also bears mentioning. After we initially defined and explored table algebras, we discovered that they are, in fact, related in a precise way to a special type of “C-algebra.” Now C-algebras, which generalize commutative association schemes. were introduced over 4.5 years ago in a paper of Kawada [K], as an abstraction of work of Hoheisel [H]. Theorem 2.10 below shows that table algebras are equivalent to C-algebras with nonnegative structure constants.(See [BI, p. SS] for the definition of a C-algebra. which may also be inferred from the proof of Theorem 2.10.) 137