Algebraic theory of symmetric products


Victor M. Buchstaber

Moscow State University

Northeastern University

Thursday, March 1, 2001


Talk at 4:30 p.m. in 509 Lake Hall

Tea at 4:00 p.m. in 544 Nightingale Hall


Abstract:   My talk is a presentation of our recent results with Elmer Rees (Edinburgh University). We define certain algebraic subsets Fn of the space of all linear maps Hom(A,C) where A denotes a commutative algebra; the space F1 is the set of ring homomorphisms. When A is an algebra of functions on a compact Hausdorff space and A = C(X) is the ring of complex valued continuous functions on X the variety Fn coincides with Symn(X).

The case n=1 is classical (the Gelfand transform). Another interesting case is when X = Cm and the algebra is the ring A = C[u1,...,um] of polynomial functions on X. We prove that Fn is the symmetric product Symn (Cm) and the embedding Fn < Hom(A,C) is described by particular equations. The particular description of Fn that we give immediately yields an embedding in the finite dimensional subspace of functions that vanish on monomials of degree > n. The resulting equations are the first syzygies on the ring of multi-symmetric polynomials. Examples of these syzygies have been studied for over a century and obtained through difficult calculations. The approach presented here gives them in explicit form.

The spaces Fn can be discussed in some generality : Fn < Hom(A,B) where A is any associative algebra and B any commutative algebra. The elements of Fn are characterised by formulae similar to those first introduced by Frobenius to define n-characters of finite groups and they also been used more recently in the study of relations in matrix algebras.


Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

There is free parking available for people coming to the colloquium at Northeastern's visitor parking. The entrance is from Columbus Avenue, right next to the parking garage.

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