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7 Signed Permutations

We provide signed permutations, that is permutations that can additionally change the sign of their result.

Assume \(n \in \mathbb{N}\), then a signed permutation on \(n\) points is a permutation \(\pi\) on \(\{ 1 \ldots n \}\) together with signs \(sgn : \{ 1 .. n \} \rightarrow \{-1,1\}\). A signed permutation on \(n\) points acts on the set \(\{ -n \ldots 1, 1 \ldots n \}\) by \( \omega ^ { (\pi, sgn) } = sgn(\omega)\cdot sgn(|\omega|^\pi) \cdot (|\omega|^\pi) \).

We provide two representations of signed permutations, one as a list of images IsSignedPermListRep (7.2-8) and one formed as pair of a permutation and a sign map IsSignedPermRep (7.2-7). Our benchmarks indicate that a list of images is the better representation, and hence this is the default.

To get started with signed permutations consider the following example

gap> s := SignedPerm([2,-1]);
<signed permutation in list rep>
gap> 1 ^ s;
2
gap> 2 ^ s;
-1
gap> OnPoints(2, s);
-1

One can form groups out of signed permutations

gap> r := SignedPerm([-1,3,-2,4]);; t := SignedPerm([3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32
gap> Orbit(G, 1, OnPoints);
[ 1, -1, 3, -3, -2, 4, 2, -4 ]
gap> Stabilizer(G, 1, OnPoints);
<group of size 4 with 9 generators>

Note that currently the package does not make an effort to exploit the special structure of signed permutation groups as permutation groups.

7.1 Different Representations

To create signed permutations in the different representations, we provide a constructor.

gap> r := NewSignedPerm(IsSignedPermRep, [-1,3,-2,4]);;
gap> t := SignedPerm(IsSignedPermRep, [3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32
gap> r := NewSignedPerm(IsSignedPermListRep, [-1,3,-2,4]);;
gap> t := SignedPerm(IsSignedPermListRep, [3,1,4,2]);;
gap> G := Group(r,t);
<group with 2 generators>
gap> Size(G);
32

7.2 Low-Level Descriptions

7.2-1 IsSignedPerm

Returns: true or false

Category of signed permutations

7.2-2 ListSignedPerm

Convert a signed permutation into a list of images, equivalent to List([1..LargestMovedPoint(s)], x -> x^s);

7.2-3 ListSignedPerm

Convert a signed permutation to a list of images of length len. Arguments perm, len

7.2-4 SignedPerm

Given a list of signed images create a signed permutation object in IsSignedPermListRep (7.2-8).

7.2-5 NewSignedPerm

7.2-6 NewSignedPerm

7.2-7 IsSignedPermRep

Returns: true or false

Representation of signed permutations as a permutation and a vector of signs.

7.2-8 IsSignedPermListRep

Returns: true or false

Representation of signed permutations as a list of signed images

7.2-9 OnPosPoints

Only act as a permutation on \(\{ 1\ldots n\}\)

7.2-10 LargestMovedPoint

The largest point that is moved by the signed permutation, where moving includes changing the sign.

7.2-11 RandomSignedPermList

Create a random list of images that can be used to create a signed permutation.

7.2-12 RandomSignedPerm

Create a random signed permutation

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