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# getTuples -- The tuples of a matching field

- Usage:
    `tuples = getTuples L`
- Inputs:
  - L, an instance of the type GrMatchingField or an instance of the type FlMatchingField,
- Outputs:
  - tuples, a list, A list of subsets of $1, \dots, n$; the tuples of the matching field

## Description

A matching field $\Lambda$ for the Grassmannian Gr$(k, n)$ is a collection tuples $\Lambda(J)$ for each $k$-subset $J \subseteq [n]$. The entries of the tuple form a permutation of $J$, so in some literature $\Lambda(J)$ is taken to be the element of the symmetric group $\sigma \in S_k$ such that $\Lambda(J) = (j_{\sigma(1)}, j_{\sigma(2), \dots, j_{\sigma(k)}})$ where $J = \{j_1 < j_2 < \dots < j_k\}$.

    i1 : L = diagonalMatchingField(2, 4)

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

    o1 : GrMatchingField

    i2 : getTuples L

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

    o2 : List

The tuples are stored such that their underlying sets are in RevLex order, which is the order produced by the method subsets.

For flag matching fields, the tuples are stored as a list of list of tuples for each Grassmannian matching field contained within.

    i3 : L = diagonalMatchingField({1,2}, 4)

    o3 = Flag Matching Field for Fl(1, 2; 4)

    o3 : FlMatchingField

    i4 : netList getTuples L

         +------+------+------+------+------+------+
    o4 = |{1}   |{2}   |{3}   |{4}   |      |      |
         +------+------+------+------+------+------+
         |{1, 2}|{1, 3}|{2, 3}|{1, 4}|{2, 4}|{3, 4}|
         +------+------+------+------+------+------+

## See also

- grMatchingField -- Construct a matching field for the Grassmannian variety
- flMatchingField -- Construct a matching field for a partial flag variety
- diagonalMatchingField -- the diagonal matching field

### Menu

## Ways to use getTuples:

- `getTuples(FlMatchingField)`
- `getTuples(GrMatchingField)`
- getTuples(TopeField) -- tuples of a tope field

## For the programmer

The object getTuples is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatchingFields.m2:2198:0.