m = plueckerMap LThe ring map for the Pluecker embedding of the Grassmannian sends each Pluecker variable $P_J$, where $J$ is a $k$-subset of $[n]$, to its corresponding maximal minor in a generic $k \times n$ matrix of variables $X = (x_{i,j})$.
The domain and codomain of this ring map are naturally equipped with term orders derived from a weight matrix, which induces the matching field. So, for this function, we require that the matching fields be coherent. If a weight matrix is not supplied, then one is automatically computed. If the matching field is not coherent, then an error is thrown.
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For the above polynomial ring, the monomial order is given by a weight ordering. Note that the weights are based on $-1 \times W$ where $W$ is the weight matrix that induces $L$, displayed using the function getWeightMatrix. The purpose of $-1$ is to transition between the minimum convention of matching fields and the maximum convention of initial terms in Macaulay2.
The ring map for the Pluecker embedding of a partial flag variety is completely analogous.
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The monomial order on the ring of Pluecker variables, shown above, is also based on $-1 \times W$. More concretely, the weight vector of a Pluecker variable $P_J$ is the weight of the initial term of the image of the Pluecker variable $m(P_J) = \det(X_J)$ under the map.
The monomial map associated to the matching field, see matchingFieldRingMap is the map that sends each Pluecker variable $P_J \mapsto \rm{in}(\det(X_J))$ to the lead term of the maximal minor $\det(X_J)$.
The object plueckerMap is a method function with options.