Sagbi combinatorics of maximal minors and a Sagbi algorithm

SAGBI and SAGBI-Gröbner Bases over Principal Ideal Domains

A Sagbi Basis For

Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by Vn the indecomposable n dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V2 ⊕ V2 ⊕ V3]p .

Combinatorics of Maximal Minors

We continue the study of the Newton polytope Pm,n of the product of all maximal minors of an m x n-matrix of indeterminates. The vertices of Pm,n are encoded by coherent matching fields A = (A z) , where z runs over all m-element subsets of columns, and each Az is a bijection z —> [m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid...

A Sagbi Basis for the Quantum Grassmannian

The maximal minors of a p× (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 generat...

SAGBI Bases Under Composition

Our interest in the subject of this paper is inspired by Hong (1998), where Hoon Hong addresses the problem of the behavior of Gröbner bases under composition of polynomials. More precisely, let Θ be a set of polynomials, as many as the variables in our polynomial ring. The question then is under which conditions on these polynomials it is true that for an arbitrary Gröbner basis G (with respec...