|Title||A sagbi basis for the quantum Grassmannian|
|Author(s)|| Frank Sottile, Bernd Sturmfels|
|Type||Article in Journal|
|Abstract||The maximal minors of a p times(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 "Grobner 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 Plucker relations has a quadratic Grobner 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 that the row-consecutives p times p-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.
|Keywords||straightening law, poset, quantum cohomology, Schubert calculus, Grassmannian, Gröbner basis, sagbi basis|
|Journal||Journal of Pure and Applied Algebra|