A sagbi basis for the quantum Grassmannian
Frank Sottile and Bernd Sturmfels
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 "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 $p\times p$-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.