Equivariant cohomology and representations of the symmetric group. (English) Zbl 1057.20007

Atiyah, M.

Chin. Ann. Math., Ser. B 22, No. 1, 23-30 (2001).

Summary: In a recent paper the author constructed a continuous map from the configuration space of \(n\) distinct ordered points in 3-space to the flag manifold of the unitary group \(U(n)\), which is compatible with the action of the symmetric group. This map is also compatible with appropriate actions of the rotation group \(SO(3)\). In this paper the author studies the induced homomorphism in \(SO(3)\)-equivariant cohomology and shows that this contains much interesting information involving representations of the symmetric group.

MSC:

20C30	Representations of finite symmetric groups
20J05	Homological methods in group theory
55R80	Discriminantal varieties and configuration spaces in algebraic topology
55N91	Equivariant homology and cohomology in algebraic topology
57M60	Group actions on manifolds and cell complexes in low dimensions
14L40	Other algebraic groups (geometric aspects)

Keywords:

configuration spaces; flag manifolds; unitary groups; rotation groups; equivariant cohomology; representations of symmetric groups