Rings of continuous functions, symmetric products, and Frobenius algebras

A constructive proof is given for the classical theorem of Gel'fand and Kolmogorov (1939) characterising the image of the evaluation map from a compact Hausdorff space $X$ into the linear space $C(X)^*$ dual to the ring $C(X)$ of continuous functions on $X$. Our approach to the proof enabled us to obtain a more general result characterising the image of the evaluation map from the symmetric products $\operatorname{Sym}^n(X)$ into $C(X)^*$. A similar result holds if $X=\mathbb C^m$ and leads to explicit equations for symmetric products of affine algebraic varieties as algebraic subvarieties in the linear space dual to the polynomial ring. This leads to a better understanding of the algebra of multisymmetric polynomials.
The proof of all these results is based on a formula used by Frobenius in 1896 in defining higher characters of finite groups. This formula had no further applications for a long time; however, it has appeared in several independent contexts during the last fifteen years. It was used by A. Wiles and R. L. Taylor in studying representations and by H.-J. Hoehnke and K. W. Johnson and later by J. McKay in studying finite groups. It plays an important role in our work concerning multivalued groups. Several properties of this remarkable formula are described. It is also used to prove a theorem on the structure constants of Frobenius algebras, which have recently attracted attention due to constructions taken from topological field theory and singularity theory. This theorem develops a result of Hoehnke published in 1958. As a corollary, a direct self-contained proof is obtained for the fact that the 1-, 2-, and 3-characters of the regular representation determine a finite group up to isomorphism. This result was first published by Hoehnke and Johnson in 1992.