Bracket ring explained

In mathematics invariant theory, 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, subject to [λ₁ λ₂ ... λ_''d''] = − [λ₂ λ₁ ... λ_''d''] and similarly for other transpositions. The set Λ(n,d) of size

\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.^[1]

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).^[2]

See also

• Bracket algebra

Notes and References

1. Sturmfels (2008) pp.78–79
2. Sturmfels (2008) p.80