To the IUM main page

# Victor M. Buchstaber

### Steklov Mathematical Institute, Moscow

## Symmetric powers as algebraic variety

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.