On the frame complex of symplectic spaces

Abstract

For a symplectic space $V$ of dimension $2n$ over ${\mathbb{F}}_q$, 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 $\ne 0,n-1$, 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.