Macaulay2 » Documentation
Packages » MatchingFields » plueckerMap
next | previous | forward | backward | up | index | toc

# plueckerMap -- The ring map of the Pluecker embedding

- Usage:
    `m = plueckerMap L`
- Inputs:
  - L, an instance of the type GrMatchingField or an instance of the type FlMatchingField,
- Optional inputs:
  - MonomialOrder => a string, default value "default", either "default" or "none" (supply "none" only if $L$ is not coherent)
- Outputs:
  - m, a ring map, the ring map of the Pluecker embedding

## Description

The 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.

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

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

    o1 : GrMatchingField

    i2 : isCoherent L

    o2 = true

    i3 : getWeightMatrix L

    o3 = | 0 0  0  0  |
         | 0 -3 -1 -2 |

                  2       4
    o3 : Matrix ZZ  <-- ZZ

    i4 : plueckerMap L

    o4 = map (QQ[x   ..x   ], QQ[p   ..p   , p   , p   , p   , p   ], {x   x    - x   x   , x   x    - x   x   , - x   x    + x   x   , x   x    - x   x   , - x   x    + x   x   , x   x    - x   x   })
                  1,1   2,4       1,2   1,3   2,3   1,4   2,4   3,4     1,1 2,2    1,2 2,1   1,1 2,3    1,3 2,1     1,3 2,2    1,2 2,3   1,1 2,4    1,4 2,1     1,4 2,2    1,2 2,4   1,3 2,4    1,4 2,3

    o4 : RingMap QQ[x   ..x   ] <-- QQ[p   ..p   , p   , p   , p   , p   ]
                     1,1   2,4          1,2   1,3   2,3   1,4   2,4   3,4

    i5 : describe target plueckerMap L

    o5 = QQ[x   ..x   , Degrees => {8:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32       }]
             1,1   2,4                                                  {Weights => {5:0, 3, 1..2}}
                                                                        {GRevLex => {8:1}         }
                                                                        {Position => Up           }

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.

    i6 : L = diagonalMatchingField({1,2}, 4)

    o6 = Flag Matching Field for Fl(1, 2; 4)

    o6 : FlMatchingField

    i7 : getWeightMatrix L

    o7 = | 0 0 0 0 |
         | 4 3 2 1 |

                  2       4
    o7 : Matrix ZZ  <-- ZZ

    i8 : m = plueckerMap L

    o8 = map (QQ[x   ..x   ], QQ[p ..p , p   ..p   , p   , p   , p   , p   ], {x   , x   , x   , x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   })
                  1,1   2,4       1   4   1,2   1,3   2,3   1,4   2,4   3,4     1,1   1,2   1,3   1,4   1,1 2,2    1,2 2,1   1,1 2,3    1,3 2,1   1,2 2,3    1,3 2,2   1,1 2,4    1,4 2,1   1,2 2,4    1,4 2,2   1,3 2,4    1,4 2,3

    o8 : RingMap QQ[x   ..x   ] <-- QQ[p ..p , p   ..p   , p   , p   , p   , p   ]
                     1,1   2,4          1   4   1,2   1,3   2,3   1,4   2,4   3,4

    i9 : describe source m

    o9 = QQ[p ..p , p   ..p   , p   , p   , p   , p   , Degrees => {10:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32        }]
             1   4   1,2   1,3   2,3   1,4   2,4   3,4                                                   {Weights => {5:0, 2:1, 3:2}}
                                                                                                         {GRevLex => {10:1}         }
                                                                                                         {Position => Up            }

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 $\det(X_J)$.

### Menu

## Ways to use plueckerMap:

- `plueckerMap(FlMatchingField)`
- `plueckerMap(GrMatchingField)`

## For the programmer

The object plueckerMap is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatchingFields.m2:2016:0.