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getWeightPluecker -- weight of the Pluecker variables induced by the weight matrix

Synopsis

Description

Suppose that a coherent matching field is induced by a $k \times n$ weight matrix $M$. The Pluecker forms are minors of a generic matrix of variables. For example, for the Grassmannian the Pluecker forms are the maximal minors. The weight matrix $M$ is generic, which is equivalent to the property: the initial form of each Pluecker form with respect to $M$ is a monomial. The weight of the initial term of each Pluecker form is the induced weight on the ring in the Pluecker variables, which is given by the function getWeightPluecker. By convention, the Pluecker variables are listed such that their subsets are in RevLex order, which is the order given by the function subsets.

An equivalent formulation is: the Pluecker weight vector is the tuple of tropical determinants of $M$, also known as the image of $M$ under the tropical Stiefel map (or its natural generalisation to partial flag varieties).

i1 : L = diagonalMatchingField(2, 4)

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

o1 : GrMatchingField
 
i2 : getWeightMatrix L

o2 = | 0 0 0 0 |
     | 4 3 2 1 |

              2       4
o2 : Matrix ZZ  <-- ZZ
 
i3 : getWeightPluecker L

o3 = {3, 2, 2, 1, 1, 1}

o3 : List

Note that the polynomial rings associated to a matching field have weight vectors based on the weight matrix given by getWeightMatrix and weight vector given by getWeightPluecker. The package MatchingFields uses a minimum convention but the initial terms of polynomials uses the maximum convention so the weight vectors may look a little different.

i4 : m = matchingFieldRingMap 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   })
              1,1   2,4       1,2   1,3   2,3   1,4   2,4   3,4     1,1 2,2   1,1 2,3   1,2 2,3   1,1 2,4   1,2 2,4   1,3 2,4

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 source m

o5 = QQ[p   ..p   , p   , p   , p   , p   , Degrees => {6:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32          }]
         1,2   1,3   2,3   1,4   2,4   3,4                                                  {Weights => {0..1, 1..2, 2:2}}
                                                                                            {GRevLex => {6:1}            }
                                                                                            {Position => Up              }
 
i6 : describe target m

o6 = QQ[x   ..x   , Degrees => {8:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32    }]
         1,1   2,4                                                  {Weights => {5:0, 1..3}}
                                                                    {GRevLex => {8:1}      }
                                                                    {Position => Up        }

See also

Ways to use getWeightPluecker :

For the programmer

The object getWeightPluecker is a method function.

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