Generalized characters of the symmetric group

Abstract

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 ) .