A sagbi basis for the quantum Grassmannian

✍ Scribed by Frank Sottile; Bernd Sturmfels

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 201 KB
• Volume: 158
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00053-0

✦ Synopsis

The maximal minors of a p × (m + p)-matrix of univariate polynomials of degree n with indeterminate coe cients are themselves polynomials of degree np. The sub-algebra generated by their coe cients is the coordinate ring of the quantum Grassmannian, a singular compactiÿcation 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 at deformation from the quantum Grassmannian to a toric variety gives a new "Gr obner 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 Pl ucker relations has a quadratic Gr obner 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 Gr obner basis for their algebraic relations.