The monomial symmetric functions and the Frobenius map

Abstract

There is a well known isometry between the center Z( S n ) of the group algebra of the symmetric group S n and the space of homogeneous symmetric functions H n of degree n. This isometry is defined via the Frobenius map F: Z( S n ) → H n , where F(ƒ)= 1 n! Σ σ ∈ S n ƒ(σ)ψ κ(σ) . Let M λ = F −1( m λ ) be the preimage of the monomial symmetric function m λ under F. We give an interpretation of M λ in terms of certain combinatorial structures called λ-domino tabloids. Using this interpretation, a number of properties of M λ can be derived. The combinatorial interpretation of the preimage of the so called forgotten basis of Doubilet and Rota can also be obtained by similar techniques.