EQUIVARIANT COHOMOLOGY AND REPRESENTATIONS OF THE SYMMETRIC GROUP

Equivariant Cohomology and Representations of the Symmetric Group

f : Cn(R ) → U(n)/T n from the configuration space of n ordered distinct points of R to the flag manifold of U(n) which is compatible with the natural action of the symmetric group Σn on both spaces. I also noted in [1] that the action of Σn on the rational cohomology of either space coincides with the regular representation but that the homomorphism f induced by f cannot possibly be an isomorp...

Bases for certain cohomology representations of the symmetric group

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we ...

Delocalized Equivariant Cohomology of Symmetric Products

For any closed complex manifold X, we calculate the Poincaré and Hodge polynomials of the delocalized equivariant cohomology H(X, Sn) with a grading specified by physicists. As a consequence, we recover a special case of a formula for the elliptic genera of symmetric products in DijkgraafMoore-Verlinde-Verlinde [8]. For a projective surface X, our results matches with the corresponding formulas...

Cohomology and Group Representations

Freeness of Equivariant Cohomology and Mutants of Compactified Representations

We survey generalisations of the Chang–Skjelbred Lemma for integral coefficients. Moreover, we construct examples of compact manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday’s. The “mutants” we construct are obtained from compactified representations and involve Hopf bundles in a crucial way.