Bracket ring

In

(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

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]

See also

• Bracket algebra

References

• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new",
  Zbl 0196.05802
• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant Theory, Old and New, Boston, MA:
  Zbl 0258.14011
• Zbl 1154.13003
• Zbl 0727.13005, archived from the original
  on 1997-11-15

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