Continuing yesterday’s story about relative positions, let be a finite group and let and be finite -sets. Yesterday we showed that -orbits on can be thought of as “atomic relative positions” of “-figures” and “-figures” in some geometry with symmetry group , and further that if and are transitive -sets then these can be identified with double cosets .

Representation theory provides another interpretation of -orbits on as follows. First, if is any permutation representation, then the -fixed points have a natural basis given by summing over -orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations . Because is self-dual, we have

and hence has a natural basis given by summing over -orbits of the action on .

**Definition:** The -morphism associated to a -orbit of via the above isomorphisms is the **Hecke operator** associated to the -orbit (relative position, double coset).

Below the fold we’ll write down some details about how this works and see how we can use the idea that -morphisms between permutations have a basis given by Hecke operators to work out, quickly and cleanly, how some permutation representations decompose into irreducibles. At the end we’ll state another puzzle.