2006/05/01 by Eugene Strahov, Strahov, Eugene
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CO #math.MP
paper · pdf · doi:10.48550/arxiv.math/0605029
arxiv created 2006/05/01 · openalex publication_date 2006/05/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)× S(n),\Diag S(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an ``unbalanced'' Gelfand pair (S(n)× S(n-1),\Diag S(n-1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n-1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).