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MatchingFields > getWeightPleucker

getWeightPleucker -- weight of the Pleucker 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 Pleucker forms are minors of a generic matrix of variables. For example, for the Grassmannian the Pleucker forms are the maximal minors. The weight matrix $M$ is generic, which is equivalent to the property: the initial form of each Pleucker form with respect to $M$ is a monomial. The weight of the initial term of each Pleucker form is the induced weight on the ring in the Pleucker variables, which is given by the function getWeightPleucker. By convention, the Pleucker 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 Pleucker 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 : getWeightPleucker 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 getWeightPleucker. 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 getWeightPleucker :

For the programmer

The object getWeightPleucker is a method function.

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