Compact cellular algebras and permutation groups Original Research Article

Having in mind the generalization of Birkhoffʹs theorem on doubly stochastic matrices we define compact cellular algebras and compact permutation groups. Arising in this connection weakly compact graphs extend compact graphs introduced by G. Tinhofer. It is proved that compact algebras are exactly the centralizer algebras of compact groups. The technique developed enables us to get nontrivial examples of compact algebras and groups as well as completely identify compact Frobenius groups and the adjacency algebras of Johnsonʹs and Hammingʹs schemes. In particular, Petersenʹs graph proves to be not compact, which answers a question by C. Godsil. Simple polynomial-time isomorphism tests for the classes of compact cellular algebras and weakly compact graphs are presented.