Abstract:
It is well-known that the permutation character of a group action carries valuable information about the action. We investigate the permutation characters of wreath products and show that knowledge about their decomposition gives rise to a characterisation of designs in an association scheme. These designs have a nice geometric interpretation using regular polytopes. Furthermore, we use our characterisation to obtain generalisations of the Livingstone-Wagner theorem.
We also apply similar ideas to perfect matchings. Even though the underlying association scheme does not come from a group, it comes from a Gelfand pair. Its zonal spherical functions can be used to mimick the representation theoretic computations from the group case, allowing us to derive similar results.