Finite Grassmannian geometries (Q2725246)

The author studies geometries in a vector space \(V \cong {\mathbb F}_{q}^{n}\) with odd \(q\) which is provided with a non-degenerate bilinear form \(\xi\). The set \(\{U \leq V \mid \dim U = k\}\) is denoted by \({\mathfrak L}_k(V)\). The Grassmann space \({\mathbb G}_{k,m}(V)\) is the incidence structure \(({\mathfrak L}_k(V),{\mathfrak L}_m(V),\leq)\). If \(H \in {\mathfrak L}_{k-1}(V)\) and \(B \in {\mathfrak L}_{k+1}(V)\), then \({\mathbf p}(H,B)= \{U \in {\mathfrak L}_k(V) \mid H < U < B\}\) is called a \(k\)-pencil, \({\mathcal P}_k(V)\) is the collection of all \(k\)-pencils, and \(({\mathfrak L}_k(V),{\mathcal P}_k(V),\in)\) is named a space of pencils. \({\mathbb G}_{k,m}(V,\xi)\) and \({\mathcal P}_k(V,\xi)\) are the restrictions to isotropic elements. NEWLINENEWLINENEWLINEThe relevant parameters of these geometries are determined. NEWLINENEWLINENEWLINEReviewer's remark: The author claims to apply a new method to reduce a quadratic form over \(V\) to canonical form. The 3-page proof uses only standard arguments, however.