Bracket ring

In mathematics, the bracket ring is the subring of the ring of polynomials "k"["x"11...,"x""dn"] generated by the "d"-by-"d" minors of a generic "d"-by-"n" matrix ("x""ij").
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 ... λ"d"] with the λ taken from {1...,"n"}, subject to [λ1 λ2 ... λ"d"] = − [λ2 λ1 ... λ"d"] and similarly for other transpositions. The set Λ("n","d") of size formula_1 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"["x""i","j"] in "nd" indeterminates given by mapping 
[λ1 λ2 ... λ"d"] to the determinant of the "d" by "d" matrix consisting of the columns of the "x""i","j" indexed by the λ. The "bracket ring" "B"("n","d") is the image of Φ. 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.
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).