On the frame complex of symplectic spaces

Abstract

For a symplectic space V of dimension 2n over Fq, we compute the eigenvalues of its orthogonality graph. This is the simple graph with vertices the 2-dimensional non-degenerate subspaces of V and edges between orthogonal vertices. As a consequence of Garland's method, we obtain vanishing results on the homology groups of the frame complex of V, which is the clique complex of this graph. We conclude that if n<q+3 then the poset of frames of size ≠0,n−1, which is homotopy equivalent to the frame complex, is Cohen-Macaulay over a field of characteristic 0. However, we also show that this poset is not Cohen-Macaulay if the dimension is big enough.