## 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).

Original language | American English |
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Pages (from-to) | 109-142 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 212 |

Issue number | 1 |

DOIs | |

State | Published - 20 Jun 2007 |

Externally published | Yes |

## Keywords

- Characters
- Gelfand pairs
- Symmetric functions
- Symmetric group