论文标题
分离融合算法的应用到表代数和关联方案
Applications of the Isolating Fusion Algorithm to Table Algebras and Association Schemes
论文作者
论文摘要
令$ \ mathbf {b} $为$ r $二维代数$ a $ a $在字段或统一的通勤环上的基础。 $ \ mathbf {b} $的半决率是$ \ mathbf {b} $的分区，其特征功能构成了$ a $ $ a $的subgebra的基础，并且融合是尊重$ $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a的基础。 在本文中，我们提供了一种用于计算$ \ mathbf {b} $的最小半扩散（或融合）的算法，该算法隔离了$ \ mathbf {b} $的基本元素的规定差异列表，而当存在这样的半差异（或融合）时。我们将此算法应用于三个问题：（1）计算给定顺序小型关联方案的融合晶格； （2）生成与瞬态自动形态群体的关联方案的明确实现； （3）产生了苏赫里亚协会方案的非胡克里亚融合的例子，其邻接矩阵具有非循环特征值。后者是一个开放问题的感兴趣，询问与横向汽车组的关联方案是否可以具有非循环特征值。
Let $\mathbf{B}$ be a basis for an $r$-dimensional algebra $A$ over a field or commutative ring with unity. The semifusions of $\mathbf{B}$ are the partitions of $\mathbf{B}$ whose characteristic functions form the basis of a subalgebra of $A$, and fusions are semifusions that respect a given involution on $A$. In this paper, we give an algorithm for computing a minimal semifusion (or fusion) of $\mathbf{B}$ that isolates a prescribed list of disjoint sums of basis elements of $\mathbf{B}$, when such a semifusion (or fusion) exists. We apply this algorithm to three problems: (1) computing the fusion lattices for small association schemes of a given order; (2) producing explicit realizations of association schemes with transitive automorphism groups; and (3) producing examples of non-Schurian fusions of Schurian association schemes whose adjacency matrices have noncyclotomic eigenvalues. The latter is of interest to the open question asking whether association schemes with transitive automorphsm groups can have noncyclotomic character values.
