Representation Theory of Non-Commutative Association Schemes of Order

Abstract

In this note, we investigate association schemes of order 6. We prove that non-normal closed subsets of such schemes have order 2 and that normal closed subsets of non-commutative schemes have order 2 or 3. After that, we investigate more closely schemes of order 6 which possess non-normal closed subsets and non-commutative schemes of order 6 which possess a symmetric normal closed subset of order 3. (Non-commutative schemes of order 6 which possess a non-symmetric normal closed subset of order 3 or a normal closed subset of order 2 will be investigated in a forthcoming article.) In both cases, we explicitly give all irreducible (complex) representations of such schemes. Among other structural consequences we obtain that association schemes of order 6 are Coxeter schemes if they have two di erent non-normal closed subsets and that they are semidirect products if they possess a normal and a non-normal closed subset or a thin non-normal closed subset.