In mathematics, 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).
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 [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
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).