On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities

TL;DR

This paper establishes polynomial bounds on the multiplicities and the number of Specht modules in the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets, with applications to algebraic complexity and Betti number bounds.

Contribution

It provides the first polynomial bounds on the multiplicities and number of Specht modules in the cohomology of symmetric semi-algebraic sets, advancing understanding of their algebraic structure.

Findings

Polynomial bound on the number of Specht modules with positive multiplicity.

Polynomial bound on the multiplicities of these modules.

Applications to lower bounds on polynomial degrees and Betti number estimates.

Abstract

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant $d$. We prove that if a Specht module, $\mathbb{S}^\lambda$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $\lambda$ is bounded by $O(d)$. This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the above mentioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov and Zell.