weightMatrixCone -- The cone of weight matrices that induce the matching field

Given a coherent matching field $\Lambda$, either for the Grassmannian or partial flag variety, the set of weight matrices that induce $\Lambda$ naturally form a polyhedral cone. The function weightMatrixCone constructs this cone by writing down a collection of inequalities. To illustrate this assume that $(1,2)$ is a tuple of $\Lambda$. A weight matrix $M = (m_{i,j})$ induces a matching field with the tuple $(1,2)$ if and only if $m_{1,1} + m_{2,2} < m_{1,2} + m_{2,1}$. Continuing in this way for all other tuples of $\Lambda$ produces the cone of weight matrices. Note, the inequalities, like the one above, are strict. So, in general, only the interior points of the cone give rise to generic weight matrices that induce the matching field.

In the above example, we can see that adding a vector from the lineality space can be interpreted as adding a constant to each element in a specific row or column of the weight matrix.

For matching fields that are not originally defined by a weight matrix, the cone of weight matrices allows us to test if the matching field is coherent. The matching field is coherent if and only if the cone is full dimensional. This is the strategy implemented by the function isCoherent.

In the example above, the cone naturally lives in $\RR^6$ so it is not full dimensional. Therefore, the matching field is not coherent.