The talk is devoted to our recent results with Elmer G. Rees (Edinburgh University).
Given a commutative algebra $A$ over the complex numbers $\mathbb C$, we define certain algebraic subsets $\Phi_n$ of the space of all linear maps $\mathop{Hom}(A,\mathbb C)$. The space $\Phi_1$ is the set of ring homomorphisms. The elements of $\Phi_n$ are characterised by algebraic equations similar to those introduced by Frobenius in his definition of $n$-characters of finite groups. We prove that in the case when $A=\mathbb C(X)$ is the ring of complex valued continuous functions on a compact Hausdorff space $X$, the variety $\Phi_n$ is canonically homeomorphic to the symmetric power $\mathop{Sym}^n(X)=X\times\dots\times X/S_n$. The case $n=1$ is the Gelfand correspondence. Another important application of the developed theory is the case $X=\mathbb C^m$ and $A=\mathbb C[u_1,\dots,u_m]$, the ring of polynomial functions on~$X$. We prove that $\Phi_n$ is the symmetric power $\mathop{Sym}^n(\mathbb C)$. The case $n=1$ here is classical as well.
The given description of $\Phi_n$ immediately produces an embedding into the finite dimensional subspace of linear functionals vanishing on monomials of degree~$>n$. The algebraic equation defining $\Phi_n$ in this space is then exactly the first syzygies on the ring of multisymmetric polynomials. These syzygies have been studied for more than a centure and only particular examples were obtained through difficult calculations. Our approach gives all the syzygies in an explicit form. The problem of describing the symmetric powers of $\mathbb C^m$ (and also algebraic subvarieties in $\mathbb C^m$) as algebraic varieties arises in different aspects of classical invariant theory, algebraic geometry and theory of Abelian functions. Besides these questions, we will also discuss modern applications in the theory of integrable systems.
