A sagbi basis for the quantum Grassmannian

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 “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, 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.