1999/08/03 by Frank Sottile, Bernd Sturmfels
Mathematics · #math.AG #math.AC #math.CO #msc:13P10 #msc:13F50 #msc:14M12 #msc:14M15 #msc:14M17
published as J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366 · 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of Gian-Carlo Rota
arxiv created 1999/08/03 · arxiv updated 2009/11/30
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 `Gröbner 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 Plücker relations has a quadratic Gröbner 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 Gröbner basis for their algebraic relations.