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diagonalMatchingField -- the diagonal matching field

Synopsis

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 $kList$ and integer $n$. The sizes of the tuples are the entries of $kList$.

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

Ways to use diagonalMatchingField :

For the programmer

The object diagonalMatchingField is a method function.

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