TY - JOUR

T1 - Generalized characters of the symmetric group

AU - Strahov, Eugene

PY - 2007/6/20

Y1 - 2007/6/20

N2 - 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).

AB - 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).

KW - Characters

KW - Gelfand pairs

KW - Symmetric functions

KW - Symmetric group

UR - http://www.scopus.com/inward/record.url?scp=34047262708&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2006.09.017

DO - 10.1016/j.aim.2006.09.017

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AN - SCOPUS:34047262708

SN - 0001-8708

VL - 212

SP - 109

EP - 142

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -