Some Special Ideals

grassmann_pluecker_ideal([ring::MPolyRing,] subspace_dimension::Int, ambient_dimension::Int)

Given a ring, an ambient dimension and a subspace dimension return the ideal in the given ring generated by the Plücker relations. If the ring is not specified return the ideal in a multivariate polynomial ring over the rationals.

The Grassmann-Plücker ideal is the homogeneous ideal generated by the relations defined by the Plücker Embedding of the Grassmannian. That is given Gr$(k, n)$ the Moduli space of all $k$-dimensional subspaces of an $n$-dimensional vector space, the relations are given by all $d \times d$ minors of a $d \times n$ matrix. For the algorithm see Bernd Sturmfels (1993).

Examples

julia> grassmann_pluecker_ideal(2, 4)
ideal(x[1]*x[6] - x[2]*x[5] + x[3]*x[4])

julia> R, x = polynomial_ring(residue_ring(ZZ, 7), "x" => (1:2, 1:3), ordering=:degrevlex)
(Multivariate polynomial ring in 6 variables over ZZ/(7), zzModMPolyRingElem[x[1, 1] x[1, 2] x[1, 3]; x[2, 1] x[2, 2] x[2, 3]])

julia> grassmann_pluecker_ideal(R, 2, 4)
ideal(x[1, 2]*x[2, 2] + 6*x[2, 1]*x[1, 3] + x[1, 1]*x[2, 3])