Near-central Permutation Factorization and Strahov's Generalized Murnaghan-Nakayama Rule

Abstract

The $(p,q,n)$-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 $\mathbb{C}[\mathfrak{S}_n]$, 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 \emph{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 $Z_1(n)$ of the group algebra $\mathbb{C}[\mathfrak{S}_n]$ with respect to the subgroup $\mathfrak{S}_{n-1}$. The generalized characters arising in the solution to the $(p,n-1,n)$-dipole problem are zonal spherical functions of the Gel'fand pair $(\mathfrak{S}_n\times \mathfrak{S}_{n-1}, \mathrm{diag}(\mathfrak{S}_{n-1}))$ 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.