Erdős-Ko-Rado for perfect matchings

A perfect matching of a complete graph K2n is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if F is family of intersecting perfect matchings of K2n, then |F|≤(2(n−1)−1)!! and if equality holds, then F=Fij where Fij is the family of all perfect matchings of K2n that contain some fixed edge ij. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family F is non-Hamiltonian, that is, m∪m′≇C2n for any m,m′∈F, then |F|≤(2(n−1)−1)!!. Our results make ample use of a symmetric commutative association scheme arising from the Gelfand pair (S2n,S2≀Sn). We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.