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

TL;DR

This paper solves a near-central permutation factorization problem related to map enumeration in Yang-Mills theory, using zonal spherical functions and centralizer algebra techniques, revealing symmetry properties of dipole counts on surfaces.

Contribution

It introduces a novel approach to a near-central permutation problem by encoding it via the centralizer algebra and explicit evaluation of generalized characters.

Findings

Equivalence of dipole counts for certain parameters on surfaces.

Explicit evaluation of zonal spherical functions in the problem.

Solution to a near-central analogue of cycle decomposition.

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.