Research on representations of association schemes

本课题的研究成果如下：1。证明了每一个素数阶的关联方案都是可交换的。并且，如果该方案的最小分裂域是一个阿贝尔数域，则该方案与已知方案在代数上同构。将有限群的Clifford理论推广到关联方案。特别地，我们发现了一种从给定的不可约表示构造新的不可约表示的方法。证明了如果一个素数平方阶的方案有一个非平凡的瘦闭子集，则该方案是可交换的。证明了如果一个素数平方阶的格式有一个固有的强正规闭子集，则该格式是可交换的。我们定义了幂零格式，并研究了它们的基本性质。我们考虑了关联方案的模块化标准模块。因此，我们可以通过不变量来区分一些代数同构方案。我们考虑了关联方案的字符积，并给出了该乘积再次为字符的充分条件。我们给出了原始关联方案阶数的一个不等式。从某种意义上说，我们的边界比众所周知的“绝对猎犬条件”要好。

The results of this project is as follows:1. We showed that every association scheme of prime order is commutative. Moreover, if the minimal splitting field of the scheme is an abelian number field, then the scheme is algebraically isomorphic to a well-known one.2. We generalized Clifford theory of finite groups to association schemes. Especially, we found a method to construct new irreducible representations from a given one.3. We showed that, if a scheme of prime square order has a non-trivial thin closed subset, then the scheme is commutative.4. We showed that, if a scheme of prime square order has a proper strongly normal closed subset, then the scheme is commutative.5. We defined nilpotent schemes and investigated their basic properties.6. We considered modular standard modules of association schemes. Consequently, we could distinguish some algebraically isomorphic schemes by their invariants.7. We considered products of characters of association schemes and gave a sufficient condition of the product to be again a character.8. We gave an inequality for the order of a primitive association schemes. In a sense, our bound is better than the well-known "absolute hound condition.