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.

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 [math]\displaystyle{ \binom{n}{d} }[/math] 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).

See also

• Bracket algebra

References

• 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 
• 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 
• Sturmfels, Bernd (2008), Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation (2nd ed.), Springer-Verlag, ISBN 3211774165 
• Sturmfels, Bernd; White, Neil (1990), "Stanley decompositions of the bracket ring", Mathematica Scandinavica 67 (2): 183–189, ISSN 0025-5521, http://www.math.ufl.edu/~white/stanley1.ps