Bracket ring
In mathematics, the bracket ring is the subring of the ring of polynomials k generated by the d-by-d minors of a generic d-by-n matrix.
The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding.
For given d ≤ n we define as formal variables the brackets with the λ taken from, subject to = − and similarly for other transpositions. The set Λ of size generates a polynomial ring K over a field K. There is a homomorphism Φ from K to the polynomial ring K in nd indeterminates given by mapping
to the determinant of the d by d matrix consisting of the columns of the xi,j indexed by the λ. The bracket ring B is the image of Φ. The kernel I of Φ encodes the relations or syzygies that exist between the minors of a generic n by d matrix. The projective variety defined by the ideal I is the d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensional space.
To compute with brackets it is necessary to determine when an expression lies in the ideal I. This is achieved by a straightening law due to Young.
