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matchingFieldRingMap -- monomial map of the matching field

Synopsis

Description

Each tuple $J = (j_1, j_2, \dots, j_k)$ of a matching field defines a monomial given by $m(J) = c x_{1, j_1} x_{2, j_2} \dots x_{k, j_k}$ where the coefficient $c \in \{+1, -1\}$ is the sign of the permutation that permutes $J$ into ascending order. Equivalently, $c = (-1)^d$ where $d = |\{(a, b) \in [k]^2 : a < b, j_a > j_b \}|$ is the number of descents of $J$. The monomial $m(J)$ is the lead term of the corresponding Pluecker form with respect to the weight order given by the matching field.

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

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

o1 : GrMatchingField
 
i2 : getTuples L

o2 = {{2, 1}, {3, 1}, {3, 2}, {1, 4}, {2, 4}, {3, 4}}

o2 : List
 
i3 : matchingFieldRingMap L

o3 = 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      1,2 2,1    1,3 2,1    1,3 2,2   1,1 2,4   1,2 2,4   1,3 2,4

o3 : 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
 
i4 : plueckerForms = matrix plueckerMap L

o4 = | -x_(1,2)x_(2,1)+x_(1,1)x_(2,2) -x_(1,3)x_(2,1)+x_(1,1)x_(2,3)
     ------------------------------------------------------------------------
     -x_(1,3)x_(2,2)+x_(1,2)x_(2,3) x_(1,1)x_(2,4)-x_(1,4)x_(2,1)
     ------------------------------------------------------------------------
     x_(1,2)x_(2,4)-x_(1,4)x_(2,2) x_(1,3)x_(2,4)-x_(1,4)x_(2,3) |

                            1                     6
o4 : Matrix (QQ[x   ..x   ])  <-- (QQ[x   ..x   ])
                 1,1   2,4             1,1   2,4
 
i5 : leadTerm plueckerForms

o5 = | -x_(1,2)x_(2,1) -x_(1,3)x_(2,1) -x_(1,3)x_(2,2) x_(1,1)x_(2,4)
     ------------------------------------------------------------------------
     x_(1,2)x_(2,4) x_(1,3)x_(2,4) |

                            1                     6
o5 : Matrix (QQ[x   ..x   ])  <-- (QQ[x   ..x   ])
                 1,1   2,4             1,1   2,4
 
i6 : leadTerm plueckerForms == matrix matchingFieldRingMap L

o6 = true

Note that the polynomial rings have weight-based term orders that depend on a weight matrix that induces the matching field. So if the matching field supplied is not coherent then function gives an error. To check that a matching field is coherent use the function isCoherent.

See also

Ways to use matchingFieldRingMap :

For the programmer

The object matchingFieldRingMap is a method function with options.

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