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matchingFieldIdeal -- The toric ideal of a matching field

Synopsis

Description

A matching field $\Lambda$ for the Grassmannian Gr$(k,n)$ associates to each subset $J = \{j_1 < \dots < j_k\}$ an ordering of that subset $\Lambda(J) = (j_{\sigma(1)}, \dots, j_{\sigma(k)})$ for some permutation $\sigma \in S_k$. The monomial map associated to a matching field $\Lambda$ is defined as the map that sends each Pluecker coordinate $p_J$ to the monomial sgn$(\sigma)x_{1, \Lambda(J)_1} x_{2, \Lambda(J)_2} \cdots x_{k, \Lambda(J)_k}$ where sgn$(\sigma) \in \{+1, -1\}$ is the sign of the permutation. The matching field ideal is the kernel of this monomial map.

i1 : L = diagonalMatchingField(2, 4)

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

o1 : GrMatchingField
 
i2 : m = matchingFieldRingMap L

o2 = 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

o2 : 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
 
i3 : I = matchingFieldIdeal L

o3 = ideal(p   p    - p   p   )
            2,3 1,4    1,3 2,4

o3 : Ideal of QQ[p   ..p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   1,4   2,4   3,4
 
i4 : ker m === I

o4 = true

The analogous setup holds for flag matching fields. A flag matching field can be thought of as a union of Grassmannian matching fields. The inclusion of the Grassmannian matching field naturally extends to an inclusion of the corresponding ideals. The flag matching field ideals are also generated by 'incident relations' that involve Pluecker coordinates from distinct pairs of Grassmannians within the flag variety.

i5 : L = diagonalMatchingField({1,2}, 4)

o5 = Flag Matching Field for Fl(1, 2; 4)

o5 : FlMatchingField
 
i6 : I = matchingFieldIdeal L

o6 = ideal (p p    - p p   , p p    - p p   , p p    - p p   , p p    -
             3 1,4    1 3,4   2 1,4    1 2,4   2 1,3    1 2,3   3 2,4  
     ------------------------------------------------------------------------
     p p   , p   p    - p   p   )
      2 3,4   2,3 1,4    1,3 2,4

o6 : Ideal of QQ[p ..p , p   ..p   , p   , p   , p   , p   ]
                  1   4   1,2   1,3   2,3   1,4   2,4   3,4

The functions matchingFieldIdeal and plueckerIdeal both construct ideals that belong to the same polynomial ring. Similarly, the ring maps constructed by the function plueckerMap and matchingFieldRingMap have the same target ring.

i7 : I' = plueckerIdeal L

o7 = ideal (p   p    - p   p    + p   p   , - p   p    + p   p    - p   p   ,
             2,3 1,4    1,3 2,4    1,2 3,4     2,3 1,4    1,3 2,4    1,2 3,4 
     ------------------------------------------------------------------------
     p   p    - p   p    + p   p   , - p   p    + p   p    - p   p   , -
      2,3 1,4    1,3 2,4    1,2 3,4     2,3 1,4    1,3 2,4    1,2 3,4   
     ------------------------------------------------------------------------
     p p    + p p    + p p   , - p p    + p p    + p p   , - p p    + p p   
      2 1,3    1 2,3    3 1,2     2 1,4    1 2,4    4 1,2     3 1,4    1 3,4
     ------------------------------------------------------------------------
     + p p   , - p p    + p p    + p p   )
        4 1,3     3 2,4    2 3,4    4 2,3

o7 : Ideal of QQ[p ..p , p   ..p   , p   , p   , p   , p   ]
                  1   4   1,2   1,3   2,3   1,4   2,4   3,4
 
i8 : ring I === ring I'

o8 = true
 
i9 : source plueckerMap L

o9 = QQ[p ..p , p   ..p   , p   , p   , p   , p   ]
         1   4   1,2   1,3   2,3   1,4   2,4   3,4

o9 : PolynomialRing
 
i10 : source plueckerMap L === source matchingFieldRingMap L

o10 = true
 
i11 : target plueckerMap L

o11 = QQ[x   ..x   ]
          1,1   2,4

o11 : PolynomialRing
 
i12 : target plueckerMap L === target matchingFieldRingMap L

o12 = true

If the matching field is not coherent, then the matching field ideal can be constructed by setting the option MonomialOrder to "none". Doing this sets the monomial order of the target polynomial ring above to GRevLex.

i13 : L = grMatchingField(2, 5, {{2,1}, {3,2}, {4,3}, {1,4}, {2,4}, {1,3}, {1,5}, {5,2}, {3,5}, {4,5}})

o13 = Grassmannian Matching Field for Gr(2, 5)

o13 : GrMatchingField
 
i14 : isCoherent L

o14 = false
 
i15 : I = matchingFieldIdeal(L, MonomialOrder => "none")

o15 = ideal(- p   p    - p   p   )
               3,4 1,5    1,3 4,5

o15 : Ideal of QQ[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
                   1,2   1,3   2,3   1,4   2,4   3,4   1,5   2,5   3,5   4,5
 
i16 : (options ring I).MonomialOrder

o16 = {MonomialSize => 32                       }
      {GRevLex => {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}
      {Position => Up                           }

o16 : VerticalList

The option Strategy determines how the matching field ideal is computed. The default uses the package FourTiTwo. This strategy works by passing in the matrix of the toric ideal to toricGroebner with the correct weight vector. In the case of Grassmannian matching fields, the columns of the matrix of the toric ideal are exactly the vertices of the matching field polytope. For a flag matching field $\Lambda$, the matrix of the toric ideal is the top-justified juxtaposition of such matrices for the Grassmannian matching fields contained in $\Lambda$. On the other hand, the strategy "M2" simply uses the in-built function to compute the kernel of the map matchingFieldRingMap.

Caveat

For some versions of the package FourTiTwo, the strategy "4ti2" may not correctly take into account the weights. See the caveat in the documentation of the function toricGroebner. If there are any problems, it may be more reliable to use the option "M2".

Ways to use matchingFieldIdeal :

For the programmer

The object matchingFieldIdeal is a method function with options.

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