Bracket ring

In mathematics, 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 $\binom{n}{d}$ 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

1. Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl 0944.52006
2. Sturmfels (2008) pp.78–79
3. Sturmfels (2008) p.80

• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525, Zbl 0196.05802
• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102, Zbl 0258.14011
• Sturmfels, Bernd (2008), Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation (2nd ed.), Springer-Verlag, ISBN 3211774165, Zbl 1154.13003
• Sturmfels, Bernd; White, Neil (1990), "Stanley decompositions of the bracket ring", Mathematica Scandinavica 67 (2): 183–189, ISSN 0025-5521, MR 1096453, Zbl 0727.13005