Description
Given a toric vector bundle $\mathcal E$ in Klyachko's description on a toric variety $X = TV(\Sigma)$, it is encoded by increasing filtrations $E^{\rho}(j)$ for each ray $\rho \in \Sigma(1)$. To these filtrations we can associated the set $L(\mathcal E)$ of intersections $\cap_{\rho} E^{\rho} (j_{\rho})$, where $(j_{\rho})_\rho$ runs over all tuples in $\mathbb Z^{\Sigma(1)}$. This set $L(\mathcal E)$ is ordered by inclusion and there is a unique matroid $M(\mathcal E)$ associated to it, see [RJS, Proposition 3.1].
groundSet computes the ground set (i.e. building blocks) of this matroid.
i1 : E = tangentBundle(projectiveSpaceFan 2)
o1 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
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i2 : groundSet E
o2 = {| 1 |, | 1 |, | 0 |}
| 1 | | 0 | | 1 |
o2 : List
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With the ground set, one can compute the parliament of polytopes using
parliament or compute the set of compatible bases using
compatibleBases.