2017/01/21 by Gregory Arone, Arone, Gregory, Kathryn Lesh +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 55M20
paper · pdf · doi:10.48550/arxiv.1701.06070
openalex publication_date 2017/01/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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.