On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities

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