A sagbi basis for the quantum Grassmannian

Abstract

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.