Near-central permutation factorization and Strahov%26apos;s generalized Murnaghan-Nakayama rule

作者：Jackson David M^*; Sloss Craig A

来源：Journal of Combinatorial Theory - Series A, 2012, 119(8): 1856-1874.

DOI：10.1016/j.jcta.2012.06.006

• Centralizers of the symmetric group algebra
• Permutation factorization
• Dipoles in orientable surfaces
• Generalized characters

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 Cinverted right perpendicularG(n)inverted left perpendicular, 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 Z(1) (n) of the group algebra Cinverted right perpendicularG(n)inverted left perpendicular with respect to the subgroup G(n-1). The generalized characters arising in the solution to the (p, n - 1, n)-dipole problem are zonal spherical functions of the Gel%26apos;fand pair (G(n) x Gn-1, diag(G(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.

• 出版日期2012-11