Lattices generated by orbits of subspaces under finite singular unitary group and its characteristic polynomials
Let \(\mathbb{F}_{q^2}\) be a finite field with \(q^2\) elements and let \(\mathbb{F}_{q^2}^{n+l}\) be the \((n+l)\)-dimensional vector space over \(\mathbb{F}_{q^2}\). The singular unitary group \(U_{n+l,n}(\mathbb{F}_{q^2})\) of degree \(n+l\) over \(\mathbb{F}_{q^2}\) consists of all \((n+l)\times(n+l)\) matrices \(T\) such that \(TI_l\overline T^t=I_l\), where \(I_l=\left(\begin{smallmatrix} I^{(n)}&0\\ 0&0^{(l)}\end{smallmatrix}\right)\). A subspace \(P\) of \(\mathbb{F}_{q^2}^{n+l}\) is mapped into the subspace \(PT\) by \(T\in U_{n+l,n}\). The author also considers singular symplectic and singular orthogonal groups. Let \(\mathcal M\) be an orbit of subspaces and \(\mathcal L\) the set of subspaces that are intersections of subspaces of \(\mathcal M\). Ordering \(\mathcal L\) by ordinary or reverse inclusion produces two lattices. The author studies inclusion relations between lattices.
scientific article; zbMATH DE number 1956804