## GENERALIZED CHARACTERS OF THE SYMMETRIC GROUP (2006)

### BibTeX

@MISC{Strahov06generalizedcharacters,

author = {Eugene Strahov},

title = {GENERALIZED CHARACTERS OF THE SYMMETRIC GROUP},

year = {2006}

}

### OpenURL

### Abstract

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). 1.

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Citation Context ...(n), the irreducible characters can be computed using either the Frobenius formula, or the determinantal formula, or the Murnaghan-Nakayama rule (see, for example, Macdonald [Mac], Sagan [S], Stanley =-=[St]-=-). Let Λ denote the algebra of symmetric functions, which is a graded algebra, isomorphic to the algebra of polynomials in the power sums p1,p2,.... If we define pρ = pρ1pρ2 ... for each partition ρ =... |

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Citation Context ...tric group, S(n), the irreducible characters can be computed using either the Frobenius formula, or the determinantal formula, or the Murnaghan-Nakayama rule (see, for example, Macdonald [Mac], Sagan =-=[S]-=-, Stanley [St]). Let Λ denote the algebra of symmetric functions, which is a graded algebra, isomorphic to the algebra of polynomials in the power sums p1,p2,.... If we define pρ = pρ1pρ2 ... for each... |

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Citation Context ...nother Gelfand pair, namely (S(n),S(n − k) × S(k)), one obtains the Hahn polynomials. There are generalizations of these results, see Mizukawa [Miz], Akazawa and Mizukawa [AkMiz], Mizukawa and Tanaka =-=[MizTan]-=-. 1.3.3. Our derivation of the Murnaghan-Nakayama rule for the generalized characters (see Theorem 3.2.1) starts from a projection formula, see Section 4.1. This formula was known previously (see Trav... |

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Citation Context ...characteristic map, see [Mac] VII, §2. However, there exist “unbalanced” Gelfand pairs, (S(n) × S(n − 1),diag S(n − 1)). (For a proof that (S(n) × S(n − 1),diag S(n − 1)) is a Gelfand pair see Travis =-=[Trav]-=-, Brender [Bren], and Section 2 below). The algebra C(S(n) ×S(n −1),diag S(n −1)) can be identified with the algebra of complex valued functions defined on the group S(n) invariant with respect to the... |

1 |
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(Show Context)
Citation Context ...ap, see [Mac] VII, §2. However, there exist “unbalanced” Gelfand pairs, (S(n) × S(n − 1),diag S(n − 1)). (For a proof that (S(n) × S(n − 1),diag S(n − 1)) is a Gelfand pair see Travis [Trav], Brender =-=[Bren]-=-, and Section 2 below). The algebra C(S(n) ×S(n −1),diag S(n −1)) can be identified with the algebra of complex valued functions defined on the group S(n) invariant with respect to the conjugations by... |

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1 | On representations of infinite symmetric - Steklov - 1997 |