Details:
Title  A sagbi basis for the quantum Grassmannian  Author(s)  Frank Sottile, Bernd Sturmfels  Type  Article in Journal  Abstract  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 pplanes 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 RaviRosenthalWang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, CohenMacaulay, 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 wellknown for the classical Schubert varieties (n = 0). We also show that that the rowconsecutives p times pminors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.  Keywords  straightening law, poset, quantum cohomology, Schubert calculus, Grassmannian, Gröbner basis, sagbi basis  Length  16 
Language  English  Journal  Journal of Pure and Applied Algebra  Volume  158  Pages  347366  Year  2001  Translation 
No  Refereed 
No 
