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TitleA sagbi basis for the quantum Grassmannian
Author(s) Frank Sottile, Bernd Sturmfels
TypeArticle in Journal
AbstractThe 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.
Keywordsstraightening law, poset, quantum cohomology, Schubert calculus, Grassmannian, Gröbner basis, sagbi basis
JournalJournal of Pure and Applied Algebra
Translation No
Refereed No