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flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank

Function: flats
Usage:

flats(n,A)
Inputs:
- n, an integer, rank
- A, a hyperplane arrangement, hyperplane arrangement
Outputs:
- a list, a list of flats of rank $n$

Description

If $A$ is a CentralArrangement, the flats are computed using the flats method from the Matroids package. Otherwise, $A$ is computed using the Orlik--Solomon algebra.

i1 : A = typeA(3)

o1 = {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
       1    2   1    3   1    4   2    3   2    4   3    4

o1 : Hyperplane Arrangement

i2 : flats(2,A)

o2 = {{3, 4, 5}, {1, 2, 5}, {0, 5}, {0, 2, 4}, {1, 4}, {2, 3}, {0, 1, 3}}

o2 : List

If the rank is omitted, the flats of each rank are listed.

i3 : flats A

o3 = {{{}}, {{5}, {4}, {3}, {2}, {1}, {0}}, {{3, 4, 5}, {1, 2, 5}, {0, 5},
     ------------------------------------------------------------------------
     {0, 2, 4}, {1, 4}, {2, 3}, {0, 1, 3}}, {{0, 1, 2, 3, 4, 5}}}

o3 : List

See also

circuits(CentralArrangement) -- list the circuits of an arrangement
flats(Matroid) -- flats of a matroid

Ways to use this method:

flats(Arrangement)
flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
flats(ZZ,CentralArrangement)

The source of this document is in HyperplaneArrangements.m2:2440:0.