flMatchingField -- Construct a matching field for a partial flag variety

This function is the basic constructor for matching fields for partial flag varieties, often called flag matching fields. The function outputs an instance of type FlMatchingField, which represents the flag matching field and stores all data related and computed about it.

There are three basic ways to define a flag matching field. The first way is to supply a weight matrix that induces the flag matching field. This produces a flag matching field for the full flag variety.

In the above example, we construct the flag matching field for the full flag variety induced by the given weight matrix. The tuples for the flag matching field are listed by their size. Similarly to Grassmannian matching fields: GrMatchingField, the function isToricDegeneration checks the equality of the matchingFieldIdeal and the initial ideal of the plueckerIdeal with respect to the weight of the matching field.

The second way to define a flag matching field is to supply a weight matrix and specify the size of the sets or, in other words, specify the dimensions of the vector spaces in the flags.

The third way to define a flag matching field is by listing out its tuples.

As shown in the example above, the first argument "kList" specifies the size of the sets. The third argument is a list of tuples of the matching field. When a flag matching field is constructed in this way, it is not guaranteed to be coherent, i.e., it may not be induced by a weight matrix. Similarly to Grassmannian matching fields, the function isCoherent checks whether the matching field is coherent and the function getWeightMatrix returns a weight matrix that induces the matching field, if it exists. If the matching field is not coherent, then these methods produce an error.

A note of caution. Two different weight matrices may induce the same matching field. So, the function getWeightMatrix may return a unexpected weight matrix. However, if a matching field is defined by a weight matrix, then that weight matrix will be returned.