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getTuples -- The tuples of a matching field

Synopsis

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

Ways to use getTuples :

For the programmer

The object getTuples is a method function.

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