Affiliation:
1. Philipps‐Universität Marburg Fachbereich Mathematik und Informatik Marburg Germany
2. Departamento de Matemática‐IMAS FCEyN Universidad de Buenos Aires Buenos Aires Argentina
3. Philipps‐Universität Marburg, Fachbereich Mathematik und Informatik Marburg Germany
Abstract
AbstractLet be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field . Let be the graph with vertex set the one‐dimensional nondegenerate subspaces of and adjacency relation given by orthogonality. We give a complete description of when is connected in terms of the dimension of and the size of the ground field . Furthermore, we prove that if , then the clique complex of is simply connected. For finite fields , we also compute the eigenvalues of the adjacency matrix of . Then, by Garland's method, we conclude that for all , where is a field of characteristic 0, provided that . Under these assumptions, we deduce that the barycentric subdivision of deformation retracts to the order complex of the certain rank selection of that is Cohen–Macaulay over . Finally, we apply our results to the Quillen poset of elementary abelian ‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of and the poset of orthogonal decompositions of .