2013/11/30 by Ekaterina A. Vassilieva, Vassilieva, Ekaterina A.
Computer Science · Mathematics · #05E05 #05E10 #05E15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1312.0120
openalex publication_date 2013/11/30 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
This article is devoted to the computation of Jack connection coefficients, a\ngeneralization of the connection coefficients of two classical commutative\nsubalgebras of the group algebra of the symmetric group: the class algebra and\nthe double coset algebra. The connection coefficients of these two algebraic\nstructures are of significant interest in the study of Schur and zonal\npolynomials as well as the irreducible characters of the symmetric group and\nthe zonal spherical functions. Furthermore they play an important role in\ncombinatorics as they give the number of factorizations of a permutation into a\nproduct of permutations with given cyclic properties. Usually studied\nseparately, these two families of coefficients share strong similar properties.\nFirst (partially) introduced by Goulden and Jackson in 1996, Jack connection\ncoefficients provide a natural unified approach closely related to the theory\nof Jack polynomials, a family of bases in the ring of symmetric functions\nindexed by a parameter \α that generalizes both Schur (case \α = 1) and\nzonal polynomials (case \α = 2). Jack connection coefficients are also\ndirectly linked to Jack characters, a general view of the characters of the\nsymmetric group and the zonal spherical functions. Goulden and Jackson\nconjectured that these coefficients are polynomials in \α with nice\ncombinatorial properties, the so-called Matchings-Jack conjecture. In this\npaper, we use the theory of Jack symmetric functions and the Laplace Beltrami\noperator to show the polynomial properties of Jack connection coefficients in\nsome important cases. We also provide explicit formulations including notably a\ngeneralization of a classical formula of D 'enes for the number of minimal\nfactorizations of a permutation into transpositions.\n