A matching field $\Lambda$ for the Grassmannian Gr($k$, $n$), is a simple combinatorial object. It may be thought of as a choice of initial term for each maximal minor of a generic $k \times n$ matrix of variables. For example, take $k = 2$ and $n = 4$. Let $X = (x_{i,j})$ be a generic $2 \times 4$ matrix of variables. Suppose that a matching field $\Lambda$ has tuples $\{12, 31, 14, 32, 24, 34\}$. This means that $\Lambda$ distinguishes the term $x_{1,1} x_{2,2}$ from the maximal minors on columns $1$ and $2$ of $X$: $x_{1,1} x_{2,2} - x_{1,2} x_{2,1}$. Similarly for the terms $x_{1,3} x_{2,1}$, $x_{1,1} x_{2,4}$, and so on.
If the terms of all maximal minors distinguished by a matching field are their respective initial terms with respect to a fixed weight matrix, then we say that the matching field is coherent. Each such weight matrix induces a weight vector on the Pluecker coordinates of the Grassmannian. If the initial ideal of the Pluecker ideal of the Grassmannian with respect to this weight vector is a toric ideal, i.e. a prime binomial ideal, then we say that the matching field gives rise to a toric degeneration of the Grassmannian. By a result of Sturmfels (1996), a matching field gives rise to a toric degeneration if and only if the maximal minors of $X$ form a subalgebra basis (or SAGBI basis, respectively Khovanskii basis) with respect to the order (respectively valuation) induced by the weight matrix.
This concept naturally generalises to partial flag varieties under the Pluecker embedding.
The MatchingFields package provides functions to construct many of the well-studied examples of matching fields. Given a matching field $L$, the package provides functions to: check whether $L$ is coherent; find a weight matrix that induces it; and test whether $L$ gives rise to a toric degeneration. The package also produces polytopes associated to matching fields and can compute their Newton-Okounkov bodies.
|
|
|
|
This documentation describes version 1.3 of MatchingFields, released February 2, 2026.
If you have used this package in your research, please cite it as follows:
|
The object MatchingFields is a package, defined in MatchingFields.m2.
The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MatchingFields.m2:1325:0.