A sagbi basis for the quantum Grassmannian Academic Article uri icon

abstract

  • The maximal minors of a p(m+p)-matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The sub-algebra 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 "Grbner 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, and Koszul, and the ideal of quantum Plcker relations has a quadratic Grbner 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 (pp)-minors of a generic matrix form a sagbi basis and we give a quadratic Grbner basis for their algebraic relations. 2001 Elsevier Science B.V.

published proceedings

  • Journal of Pure and Applied Algebra

author list (cited authors)

  • Sottile, F., & Sturmfels, B.

citation count

  • 15

complete list of authors

  • Sottile, Frank||Sturmfels, Bernd

publication date

  • January 2001