matchingFieldPolytope -- The polytope of a matching field

Each matching field defines a projective toric variety whose defining ideal is given by the kernel of the monomial map. See matchingFieldRingMap. The coordinate ring of this toric variety is the Ehrhart ring of the matching field polytope. Note that for flag matching fields, the toric variety is embedded into a high-dimensional projective space via the Segre embedding whose domain is a product of Grassmannians.

Given a matching field $\Lambda$ for the Grassmannian Gr$(k,n)$, the matching field polytope $P(\Lambda)$ is simply the convex hull of the exponent vectors of the image of Pluecker variables under the monomial map of $\Lambda$. The polytope naturally lives in the space $\RR^{k \times n}$.

The columns of the above matrix are the vertices of the matching field polytope $P(\Lambda)$. Each column should be thought of as a $2 \times 4$ matrix whose entries are listed row by row.

A matching field $\Lambda$ for a partial flag variety Fl$(k_1, \dots, k_s; n)$ is a union of matching fields $\Lambda = \bigcup \Lambda_i$ for some Grassmannians. The matching field polytope for a partial flag variety is the Minkowski sum $P(\Lambda) = \sum P(\Lambda_i)$ of Grassmannian matching field polytopes in $\Lambda$. For this sum to make sense, each Grassmannian matching field polytope must be put into the same space, which is taken to be $\RR^{k_{\max} \times n}$ where $k_{\max} = \max\{k_i\}$ is the largest $k$ such that there is a Grassmannian matching field for Gr$(k,n)$ contained in $\Lambda$. If $v \in \RR^{k_i \times n}$ is a vertex for a Grassmannian matching field polytope, then we embed $v$ into $\RR^{k_{\max} \times n}$ by joining a suitably sized matrix of zeros to $v$ from below.

Embedding a Grassmannian matching field polytope into a higher dimensional space as described is done by specifying the optional value ExtraZeroRows.

The above example constructs the diagonal matching field polytope for the partial flag variety Fl$(1, 2; 4)$ as a Minkowski sum. The quick way to do this is as follows.