A SAGBI BASIS FOR THE QUANTUM GRASSMANNIAN

FRANK SOTTILE BERND STURMFELS

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 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 × p-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.