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diagonalMatchingField -- the diagonal matching field

Description

The diagonal matching field is defined to be the matching field whose tuples are all in ascending order. It is a coherent matching field so it is induced by a weight matrix.

The weight matrix used to construct the diagonal matching field us given by $M = (m_{i,j})$ with $m_{i,j} = (i-1)(n-j+1)$.

i1 : L = diagonalMatchingField(3, 6)

o1 = Grassmannian Matching Field for Gr(3, 6)

o1 : GrMatchingField
 
i2 : getWeightMatrix L

o2 = | 0  0  0 0 0 0 |
     | 6  5  4 3 2 1 |
     | 12 10 8 6 4 2 |

              3       6
o2 : Matrix ZZ  <-- ZZ

The function diagonalMatchingField can be used in three different ways. If it is supplied two integers $(k,n)$ then it produces the diagonal matching field for the Grassmannian, as shown in the above example. If it is supplied a single integer $n$, then it produces the diagonal matching field for the full flag variety. The matching fields of the full flag variety have tuples of size $1, 2, \dots, n-1$. The function can be made to produce diagonal matching fields for partial flag varieties by supplying it a list $tupleSizeList$ and integer $n$. The sizes of the tuples are the entries of $tupleSizeList$.

i3 : L = diagonalMatchingField 4;
 
i4 : netList getTuples L

     +-+-+-+
o4 = |1| | |
     +-+-+-+
     |2| | |
     +-+-+-+
     |3| | |
     +-+-+-+
     |4| | |
     +-+-+-+
     |1|2| |
     +-+-+-+
     |1|3| |
     +-+-+-+
     |2|3| |
     +-+-+-+
     |1|4| |
     +-+-+-+
     |2|4| |
     +-+-+-+
     |3|4| |
     +-+-+-+
     |1|2|3|
     +-+-+-+
     |1|2|4|
     +-+-+-+
     |1|3|4|
     +-+-+-+
     |2|3|4|
     +-+-+-+
 
i5 : L = diagonalMatchingField({1, 2}, 5);
 
i6 : netList getTuples L

     +-+-+
o6 = |1| |
     +-+-+
     |2| |
     +-+-+
     |3| |
     +-+-+
     |4| |
     +-+-+
     |5| |
     +-+-+
     |1|2|
     +-+-+
     |1|3|
     +-+-+
     |2|3|
     +-+-+
     |1|4|
     +-+-+
     |2|4|
     +-+-+
     |3|4|
     +-+-+
     |1|5|
     +-+-+
     |2|5|
     +-+-+
     |3|5|
     +-+-+
     |4|5|
     +-+-+

Diagonal matching fields always give rise to toric degenerations of Grassmannians and flag varieties. In the literature, this toric degeneration is also known as Gelfand-Tsetlin degeneration. The matching field polytopes for the diagonal matching field, which can be constructed with the function matchingFieldPolytope, are unimodularly equivalent to Gelfand-Tsetlin polytopes.

See also

Menu

Ways to use diagonalMatchingField:

  • diagonalMatchingField(List,ZZ)
  • diagonalMatchingField(ZZ)
  • diagonalMatchingField(ZZ,ZZ)

For the programmer

The object diagonalMatchingField is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MatchingFields.m2:2541:0.