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 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) with respect to the order induced by the weight matrix.
This concept naturally generalises to partial flag varieties under the Pluecker embedding.
The MatchingFields package gives basic functions, to construct many of the well-studied examples of matching fields. Given a matching field $L$, it is straight forward to check whether $L$ is coherent, what is a weight matrix that induces it, and whether is gives rise to a toric degeneration. The package also produces polytopes associated to matching fields and Newton-Okounkov bodies.
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This documentation describes version 1.2 of MatchingFields.
The source code from which this documentation is derived is in the file MatchingFields.m2.
The object MatchingFields is a package.