Near-central permutation factorization and Strahovʼs generalized Murnaghan–Nakayama rule

AbstractThe (p,q,n)-dipole problem is a map enumeration problem, arising in perturbative Yang–Mills theory, in which the parameters p and q, at each vertex, specify the number of edges separating of two distinguished edges. Combinatorially, it is notable for being a permutation factorization problem which does not lie in the centre of C[Sn], rendering the problem inaccessible through the character-theoretic methods often employed to study such problems. This paper gives a solution to this problem on all orientable surfaces when q=n−1, which is a combinatorially significant special case: it is a near-central problem. We give an encoding of the (p,n−1,n)-dipole problem as a product of standard basis elements in the centralizer Z1(n) of the group algebra C[Sn] with respect to the subgroup Sn−1. The generalized characters arising in the solution to the (p,n−1,n)-dipole problem are zonal spherical functions of the Gelʼfand pair (Sn×Sn−1,diag(Sn−1)) and are evaluated explicitly. This solution is used to prove that, for a given surface, the numbers of (p,n−1,n)-dipoles and (n+1−p,n−1,n)-dipoles are equal, a fact for which we have no combinatorial explanation. These techniques also give a solution to a near-central analogue of the problem of decomposing a full cycle into two factors of specified cycle type