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matroidSubdivision -- The matroid subdivision induced by the Pluecker weight of a coherent matching field

Synopsis

Description

The hypersimplex $\Delta(k, n) \subseteq \RR^{n}$ is the convex hull of the characteristic vectors of all $k$-subsets of $\{1, \dots, n\}$, and we label each vertex with with its corresponding subset. A regular subdivision of the vertices of $\Delta(k, n)$ is said to be matroidal if, for each maximal cell of the subdivision, the subsets labelling its vertices form the set of bases of a matroid. The well-known result is: a point lies in the Dressian Dr($k$, $n$), the tropical prevariety of all $3$-term Pluecker relation in Gr($k$, $n$), if and only if it induces a matroidal subdivision of the hypersimplex.

i1 : L = grMatchingField(2, 4, {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}})

o1 = Grassmannian Matching Field for Gr(2, 4)

o1 : GrMatchingField
 
i2 : netList matroidSubdivision L -- an octahedron sliced into 2 pieces

     +------+------+------+------+------+
o2 = |{1, 4}|{2, 4}|{1, 3}|{2, 3}|{1, 2}|
     +------+------+------+------+------+
     |{1, 4}|{2, 4}|{3, 4}|{1, 3}|{2, 3}|
     +------+------+------+------+------+

Whenever the function matroidSubdivision is supplied with a Grassmannian matching field, the cached weight that induces the matching field is used for the computation of the matroid subdivision. Note that, if the function is supplied directly with the \textit{plueckerWeight}, then the coordinates are ordered so that the corresponding sets are listed in reverse lexicographic order.

Ways to use matroidSubdivision :

For the programmer

The object matroidSubdivision is a method function.

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