Near-central permutation factorization and Strahov's generalized Murnaghan-Nakayama rule

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Jackson, David M.¹  Sloss, Craig A.¹ 

[1] Univ Waterloo, Dept Combinator & Optimizat, Waterloo, ON N2L 3G1, Canada

关键词: Centralizers of the symmetric group algebra;    Permutation factorization;    Dipoles in orientable surfaces;    Generalized characters;   

DOI  :  10.1016/j.jcta.2012.06.006

来源: Elsevier

【 摘 要 】

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'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. (C) 2012 Elsevier Inc. All rights reserved.

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