Algebra/Topology seminar

Speaker: Arthur Garnier

Title: Equivariant cellular structures on spheres and flag manifolds

Abstract: This is a joint work with R. Chirivì and M. Spreafico.
Flag manifolds are homogeneous space attached to Lie groups which play a central role in geometry and Lie theory. The Weyl group W of a complex algebraic group G acts freely on its flag manifold G/B, where B is a Borel subgroup of G. It is well-known the rational cohomology of G/B carries the regular representation of W, but our aim is to determine the integral cochain complex of G/B, as a perfect complex in the derived category of ZW-modules. Naturally, we look for an explicit W-equivariant cell decomposition of G/B.
In this talk we will give a solution to this problem in the case of S_3 acting on SL_3(R)/B(R), which turns out to be a very special case. Indeed, this action is induced by that of the binary octahedral group on the sphere S^3, so we work in that setting instead, and deduce a decomposition of the flag manifold as a corollary. We also deal with the binary icosahedral group for completeness: finite groups acting freely and isometrically on spheres were classified by Milnor and Wolf, and those two cases were the last ones for which an equivariant cellular decomposition was not known before. We use orbit polyopes techniques developed previously by Chirivi-Galves-Neto-Spreafico.
If time permits, we shall also give another cellular structure on this flag manifold which has more cells but is semi-algebraic. This decomposition gives a description of the mod 2 cohomology algebra of this manifold as the mod 2 coinvariant algebra of S_3. I will conclude by some perspectives on the problem about flag manifolds in other types.