Homotopy properties of the complex of frames of a unitary space

Abstract

Let $V$ be a finite-dimensional vector space equipped with a nondegenerate Hermitian form over a field $K$. Let $G\left(V\right)$ be the graph with vertex set the one-dimensional nondegenerate subspaces of $V$ and adjacency relation given by orthogonality. We give a complete description of when $G\left(V\right)$ is connected in terms of the dimension of $V$ and the size of the ground field $K$. Furthermore, we prove that if $dim\left(V\right)>4$, then the clique complex $F\left(V\right)$ of $G\left(V\right)$ is simply connected. For finite fields $K$, we also compute the eigenvalues of the adjacency matrix of $G\left(V\right)$. Then, by Garland's method, we conclude that ${\stackrel{\sim }{H}}_m(F\left(V\right);k)=0$ for all $0⩽m⩽dim\left(V\right)-3$, where $k$ is a field of characteristic 0, provided that $dim{\left(V\right)}^2⩽\left|K\right|$. Under these assumptions, we deduce that the barycentric subdivision of $F\left(V\right)$ deformation retracts to the order complex of the certain rank selection of $F\left(V\right)$ that is Cohen–Macaulay over $k$. Finally, we apply our results to the Quillen poset of elementary abelian $p$-subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of $V$ and the poset of orthogonal decompositions of $V$.