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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject PermutationAction

from application group

derived objects for permutation representations

derived from:
Action

### Orbits

Dealing with orbits under permutation groups.

SWITCH_TABLE

A switch table is a tool for finding lex-maximal or -minimal in orbits under the action of a permutation group. Its main ingredient is an upper-left triangular matrix with group elements as entries. See https://arxiv.org/abs/1709.04746 The output contains the support at every level, i.e. a number and a Set<Int>, the number is the size of the support and and the Set<Int> are the indices of those entries that can be permuted to the index of the current level while keeping previous level indices fixed. I.e. entry [i,j] will keep the first i entries of a vector fixed, while moving the j-th entry to position i. Note that we start counting at 0!

Type:
SwitchTable
Example:

 > $P = new PermutationAction(GENERATORS=>[[1,2,0,4,5,3],[2,1,0,5,4,3]]); > print$P->SWITCH_TABLE;
Supports: (size, content)
Level 0: 3 {0 1 2}
Level 1: 2 {1 2}
Entries:
[0,0]: 0 1 2 3 4 5
[0,1]: 1 0 2 4 3 5
[0,2]: 1 2 0 4 5 3
[1,1]: 0 1 2 3 4 5
[1,2]: 0 2 1 3 5 4

### no category

BASE

A base for STRONG_GENERATORS.

Type:
Array<Int>

N_STRONG_GENERATORS

The number of STRONG_GENERATORS.

Type:
Int

STRONG_GENERATORS

Strong generating set with respect to BASE.

Type:
Array<Array<Int>>

TRANSVERSALS

Transversals along the stabilizer chain.

Type:
Array<Array<Int>>

TRANSVERSAL_SIZES

The number of group elements per transversal.

Type:
Array<Int>

### Orbits

Dealing with orbits under permutation groups.

lex_maximal

Assume the group acts on a vector by permuting its entries. Then this method gives the lex-maximal vector from the orbit of the input vector under the group action. See https://arxiv.org/abs/1709.04746

lex_minimal

Similar to lex_maximal.

• documentation/latest/group/permutationaction.txt