plueckerIdeal -- The Pluecker ideal of a matching field

The Pluecker ideal is the defining ideal of a partial flag variety embedded in a product of Grassmannians, where each Grassmannian is embedded, by the Pluecker embedding, into a suitable projective space. In the case of the Grassmannian Gr($k$, $n$), it is concretely given by kernel of the ring map $K[P_I : I \subseteq [n],\ |I| = k] \rightarrow K[x_{i,j} : i \in [k], \ j \in [n]]$ where $P_I$ is mapped to the $k \times k$ maximal minor of the matrix $(x_{i,j})$ whose columns are indexed by the set $I$. It is well-known that this ideal has a Groebner basis consisting of homogeneous quadrics.

The function plueckerIdeal takes a matching field, either for the Grassmannian or a partial flag variety and outputs the Pluecker ideal for that Grassmannian or partial flag variety. The ambient polynomial ring that contains this ideal is constructed to have the term order induced by the matching field.

In the above example, the weights for the ambient ring are not the same as the Pluecker weights of the matching field. This is because of the minimum-maximum convention problem. For compatibility with packages such as Tropical, we use the minimum convention in MatchingFields so the smallest weight with respect to the weight matrix that induces the matching field is the initial term of a Pluecker form. However, the monomial ordering given by Weights uses the maximum convention, so the ambient ring has weights that are based on the negative of the induced Pluecker Weight.

Note that the given matching field must be coherent. If the matching field is not defined in terms of a weight matrix, then the function will attempt to compute a weight matrix for the matching field. If the matching field is not coherent then the function will produce an error.

To construct the pluecker ideal for a non-coherent matching field, set the option MonomialOrder to "none". The resulting ideal is constructed in a polynomial ring with the GRevLex order.