In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is ⁠1/2⁠n(n + 1) (where the dimension of V is 2n). It may be identified with the homogeneous space U(n)/O(n), where U(n) is the unitary group and O(n) the orthogonal group. Following Vladimir Arnold it is denoted by Λ(n). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space V of dimension 2n. It may be identified with the homogeneous space of complex dimension ⁠1/2⁠n(n + 1) Sp(n)/U(n), where Sp(n) is the compact symplectic group. Source: Wikipedia (en)