On the isotypic decomposition of cohomology modules of symmetric semialgebraic sets: polynomial bounds on multiplicities
Abstract
We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semialgebraic 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 semialgebraic 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 DOI:
 10.48550/arXiv.1503.00138
 arXiv:
 arXiv:1503.00138
 Bibcode:
 2015arXiv150300138B
 Keywords:

 Mathematics  Algebraic Geometry;
 Computer Science  Computational Complexity;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 Primary 14P10;
 14P25;
 Secondary 68W30
 EPrint:
 42 pages, 1 figure. Reorganized for readability. Typos corrected