In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij).
generates a polynomial ring K[Λ(n,d)] over a field K. There is a homomorphism Φ(n,d) from K[Λ(n,d)] to the polynomial ring K[xi,j] in nd indeterminates given by mapping [λ1 λ2 ... λd] to the determinant of the d by d matrix consisting of the columns of the xi,j indexed by the λ.
The kernel I(n,d) 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 (n−d)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensional space.
[2] To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d).