The Cramer varieties \(\mathrm{Cr}(r,r+s,s)\) (Q2439281)

The author studies the so-called Cramer varieties \(\mathrm{Cr}(r,t,s)\) via the theory of algebraic groups. More precisely, the variety \(\mathrm{Cr}(r,t,s)\) is first shown to be quasi-homogeneous for the action of \(G=\mathrm{GL}_r \times \mathrm{GL}_t \times \mathrm{GL}_s\). Next, given a point \(v\) of the open orbit of \(\mathrm{Cr}(r,t,s)\) and \(H \subset G\) the stabilizer of \(v\), we denote by \(T_H\) the torus of \(G\) obtained by intersecting a maximal torus with the normalizer of \(H\). Then the main result of the article states that the weight of the canonical differential of \(G/H\) is the determinant of \(T_H\). Besides, the author gives a relation between the Cramer variety \(\mathrm{Cr}(2,4,2)\) and the orthogonal Grassmannian \(\mathrm{OGr}(5,10)\): the variety \(\mathrm{OGr}(5,10)\) is isomorphic to a hyperplane section of \(\mathrm{Cr}(2,4,2)\).