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# groundSet -- computes the ground set of a matroid associated to a toric vector bundle

## Synopsis

• Usage:
g = groundSet E
• Inputs:
• Outputs:
• g, a list, containing elements of ground set as one column matrices
• Consequences:
• The result of groundSet will be stored as a cacheValue in E.cache#groundSet. It will be used by other methods.

## 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 i2 : groundSet E o2 = {| 1 |, | 1 |, | 0 |} | 1 | | 0 | | 1 | o2 : List
With the ground set, one can compute the parliament of polytopes using parliament or compute the set of compatible bases using compatibleBases.

## Caveat

This method works for any toric reflexive sheaf on any toric variety.