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getWeightMatrix -- weight matrix that induces the matching field

Synopsis

Description

If the supplied matching field is coherent, then this function returns a weight matrix that induces the matching field. If the matching field was originally defined by a weight matrix then that weight matrix is returned. Otherwise, a weight matrix is computed. The weight matrix is computed by computing the weight matrix cone, which can be returned with the function weightMatrixCone. If the supplied matching field is not coherent, then the function gives an error.

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

o3 = Grassmannian Matching Field for Gr(2, 4)

o3 : GrMatchingField
 
i4 : isCoherent L

o4 = true
 
i5 : getWeightMatrix L

o5 = | 0 0  0  0  |
     | 0 -3 -1 -2 |

              2       4
o5 : Matrix ZZ  <-- ZZ

The weight on the ring containing the Pluecker forms, i.e., minors of a generic matrix, is based on the weight matrix returned by getWeightMatrix. Note that the package MatchingFields uses the minimum convention but polynomial ring weight vectors use the maximum convention so some conversion is required.

i6 : plueckerMap L

o6 = 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

o6 : 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
 
i7 : R = target plueckerMap L

o7 = R

o7 : PolynomialRing
 
i8 : describe R

o8 = 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           }

See also

Ways to use getWeightMatrix :

For the programmer

The object getWeightMatrix is a method function.

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