matchingFieldRingMap -- monomial map of the matching field

Usage:

m = matchingFieldRingMap L
Inputs:
- L, an instance of the type MatchingField, 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 when $L$ is not coherent)
Outputs:
- m, a ring map, monomial ring map whose kernel is the matching field ideal

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. If the matching field supplied is not coherent then the rings will have the default GRevLex monomial order. If no weight matrix was supplied in the construction of the matching field, then one will be computed, see getWeightMatrix. To check that a matching field is coherent use the function isCoherent.

Ways to use matchingFieldRingMap:

matchingFieldRingMap(MatchingField)

For the programmer

The object matchingFieldRingMap is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MatchingFields.m2:2585:0.

matchingFieldRingMap -- monomial map of the matching field

Description

See also

Menu

Ways to use matchingFieldRingMap:

For the programmer