An expression for the Frobenius map in the fundamental basis
Angela Hicks, Lehigh University
Abstract: The Schur functions are a well known basis for the symmetric polynomials; many well studied families of symmetric polynomials are positive when expressed in the Schur basis. A key tool in the representation theory of the symmetric group is the classical Frobenius map, which encodes symmetric group modules as symmetric polynomials and irreducible representations as Schur functions. When a polynomial is a positive sum of Schur functions, it thus is the image, under this map, of a representation of the symmetric group.
We discuss a new formula for the Frobenius map on permutation representations in terms of Gessel's fundamental basis, and how it can be used to show polynomials are Schur positive. We also discuss its role in connecting the classical Frobenius map to a similar map of Krob and Thibbon on the representations of the 0-Hecke algebra.
Tea and refreshments available at 3:00 p.m. in the Assmus Conference Room (CU 212).
