The Gelfand map and symmetric products

A short proof of the Buchstaber–Rees theorem

“…A linear map f : A → B is called a (Frobenius) n-homomorphism if f(1) = n and F k = 0 for all k n + 1 [1,8]. Proof.…”

“…Therefore, we can identify the n-homomorphisms as defined by Buchstaber & Rees [1,8] with the linear maps such that their characteristic functions are polynomials of degree n.…”

“…The resulting power series can be identified with our characteristic function. That series was used only in the proof of their theorem 2.9 about the sum of n-and m-homomorphisms, while the central theorem 2.8 concerning the relation of the Frobenius n-homomorphisms with the algebra homomorphisms of the symmetric powers was obtained in Buchstaber & Rees [1] by a long combinatorial argument.…”

“…Establishing it was the main difficulty of the proof in Buchstaber & Rees [1], where it was deduced by complicated combinatorial arguments. In our approach this fact comes about almost without effort.…”

“…This journal is © 2011 The Royal Society A theory of 'n-homomorphisms' (or 'Frobenius n-homomorphisms') of algebras was developed in a series of papers of V. M. Buchstaber and E. G. Rees (see particularly [1][2][3] and references therein). We recall the definition of an n-homomorphism in §3.…”