A sagbi basis for the quantum Grassmannian
| dc.creator | Sottile, Frank | |
| dc.creator | Sturmfels, Bernd | |
| dc.date | 1999-08-03 | |
| dc.date.accessioned | 2026-07-07T05:30:11Z | |
| dc.date.available | 2026-07-07T05:30:11Z | |
| dc.description | The maximal minors of a p by (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 by p-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations. | |
| dc.description | 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of Gian-Carlo Rota | |
| dc.identifier | https://arxiv.org/abs/math/9908016 | |
| dc.identifier | http://arxiv.org/abs/math/9908016 | |
| dc.identifier | J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78913 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | Combinatorics | |
| dc.subject | 13P10, 13F50, 14M12, 14M15, 14M17 | |
| dc.title | A sagbi basis for the quantum Grassmannian | |
| dc.type | text |