For given d ≤ n we define as formal variables the brackets [λ1 λ2 ... λd] with the λ taken from {1,...,n}, subject to [λ1 λ2 ... λd] = − [λ2 λ1 ... λd] and similarly for other transpositions. The set Λ(n,d) of size generates a polynomial ring K[Λ(n,d)] over a fieldK. 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 bracket ringB(n,d) is the image of Φ. The kernelI(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 idealI 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). This is achieved by a straightening law due to Young (1928).[3]