Mathematics – Algebraic Geometry
Scientific paper
1999-08-03
J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366
Mathematics
Algebraic Geometry
18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of Gian-Carlo Rota
Scientific paper
The maximal minors of a p by (m + p) matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m + p)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new `Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties (n=0). We also show that the row-consecutive p by p-minors of a generic matrix form a sagbi basis and we give a quadratic Gr\"obner basis for their algebraic relations.
Sottile Frank
Sturmfels Bernd
No associations
LandOfFree
A sagbi basis for the quantum Grassmannian does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A sagbi basis for the quantum Grassmannian, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A sagbi basis for the quantum Grassmannian will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-374066