A combinatorial characterization of the Lagrangian Grassmannian \(\mathrm{LG}(3,6)(\mathbb{K})\) (Q2806136)

The Lagrangian Grassmannian \(\mathrm{LG}(3,6)(\mathbb{K})\) is the collection of all points of \(\mathbb P^{19} (\mathbb K)\) on the plane Grassmannian of \(\mathbb P^{5} (\mathbb K)\) which are totally isotropic with respect to some alternating bilinear form. It is known that \(\mathrm{LG}(3,6)(\mathbb{K})\) is contained in a 13-dimensional subspace \(\mathbb P^{13} (\mathbb K)\) of \(\mathbb P^{19} (\mathbb K)\). The authors present a combinatorial characterization of the point-line geometry associated with \(\mathrm{LG}(3,6)(\mathbb{K})\), where lines are those lines of \(\mathbb P^{13} (\mathbb K)\) which are contained in \(\mathrm{LG}(3,6)(\mathbb{K})\). The characterization is given in terms of the point graph of this point-line geometry and of a collection of 4-spaces of \(\mathbb P^{13} (\mathbb K)\) which are required to intersect \(\mathrm{LG}(3,6)(\mathbb{K})\) in parabolic quadrics such that suitable axioms are satisfied.NEWLINENEWLINEThe paper is part of on ongoing project, aiming at a combinatorial characterization of the geometries related to the Freudenthal-Tits magic square. The Lagrangian Grassmannian is the first entry in the third row of this square.