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Fixed points of coisotropic subgroups of Γk on decomposition spaces
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study the equivariant homotopy type of the poset of orthogonal decompositions of a finite-dimensional complex vector space. Suppose that n is a power of a prime p, and that D is an elementary abelian p-subgroup of U(n) acting on complex n-space by the regular representation. We prove that the fixed point space of D acting on the decomposition poset of complex n-space contains as a retract the unreduced suspension of the Tits building for GL(k), which a wedge of (k-1)-dimensional spheres. Let Gamma be the projective elementary abelian subgroup of U(n) that contains the center of U(n) and acts irreducibly on complex n-space. We prove that the fixed point space of Gamma acting on the space of proper orthogonal decompositions of complex n-space is homeomorphic to a symplectic Tits building, which is also a wedge of (k-1)-dimensional spheres. As a consequence of these results, we find that the fixed point space of any coisotropic subgroup of Gamma contains, as a retract, a wedge of (k-1)-dimensional spheres. We make a conjecture about the full homotopy type of the fixed point space of D, based on a more general branching conjecture, and we show that the conjecture is consistent with our results.

National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-154155OAI: oai:DiVA.org:su-154155DiVA, id: diva2:1191295
Available from: 2018-03-17 Created: 2018-03-17 Last updated: 2018-03-21

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arXiv:1701.06070

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