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MatchingFields > pleuckerIdeal

pleuckerIdeal -- The Pleucker ideal of a matching field

Synopsis

Description

The Pleucker ideal is the defining ideal of a partial flag variety embedded in a product of Grassmannians, where each Grassmannian is embedded, by the Pleucker 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 pleuckerIdeal takes a matching field, either for the Grassmannian or a partial flag variety and outputs the Pleucker 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.

i1 : L = grMatchingField(2, 4, {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}})

o1 = Grassmannian Matching Field for Gr(2, 4)

o1 : GrMatchingField
i2 : I = pleuckerIdeal L

o2 = ideal(p   p    - p   p    + p   p   )
            2,3 1,4    1,3 2,4    1,2 3,4

o2 : Ideal of QQ[p   ..p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   1,4   2,4   3,4
i3 : (monoid ring I).Options.MonomialOrder

o3 = {MonomialSize => 32           }
     {Weights => {0, 1, 1, 2, 2, 2}}
     {GRevLex => {1, 1, 1, 1, 1, 1}}
     {Position => Up               }

o3 : VerticalList
i4 : getWeightPleucker L

o4 = {-1, -2, -2, -3, -3, -3}

o4 : List

In the above example, the weights for the ambient ring are not the same as the Pleucker 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 Pleucker 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 Pleucker 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.

i5 : L = grMatchingField(2, 4, {{1,2}, {1,3}, {4,1}, {2,3}, {2,4}, {3,4}})

o5 = Grassmannian Matching Field for Gr(2, 4)

o5 : GrMatchingField
i6 : isCoherent L

o6 = false

Ways to use pleuckerIdeal :

For the programmer

The object pleuckerIdeal is a method function.

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