Bracket ring

In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x₁₁,...,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.[1]

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 ${\displaystyle {\binom {n}{d}}}$ 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.[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]