Manchester Geometry Seminar 2000/2001

Symmetric products Symⁿ(X) of a space X are described as subsets of the space of measures on X and characterised in terms of identities related to formulas introduced by Frobenius in studying the higher characters of finite groups. The description in the case n=1 is the Gelfand transform. The case where X is a vector space gives a new approach to the invariant theory of the symmetric group and produces explicit formulae for the first syzygies which are the relations satisfied by symmetric polynomials of vector variables.