Incidence matrices, permutation characters, and the minimal genus of a permutation group
Let \(V\) be a non-degenerate symplectic, orthogonal, or unitary space of dimension \(n\) over \(\text{GF}(q)\). The authors consider the incidence matrix \(H\) of totally singular 1-spaces of \(V\) against \(k\)-spaces of fixed isometry type (totally singular or non-degenerate). The main theorem is that if \(2\leq k<n\) then \(M:=HH^t\) is non-singular except in a particular case when \(M=0\), or in certain cases when \(k\) is large with respect to \(n\). As a corollary, the following is shown for an almost simple classical group \(G\) with natural module \(V\) of dimension \(n\) (with \(G\) not having a graph automorphism in the linear case): if \(2\leq k<n-1\), \(P\) is the stabiliser of a singular 1-space of \(V\), and \(K\) is the stabiliser of a totally singular or non-degenerate \(k\)-space of \(V\), then the permutation module \(1^G_P\) is a submodule of \(1^G_K\) except in 2 known cases. The excluded cases are when \(K\) is the stabiliser of a maximal totally singular subspace of \(V\) or of a 2-space containing no singular points, or the orthogonal complement of such. There is a further corollary, stating that if \(S\) is a simple classical group, then the minimal genus among all almost simple groups with socle \(S\) is realised by a non-\(k\)-space action or by one of 4 possible actions on subspaces of the natural module.
scientific article; zbMATH DE number 1760669