3-bit Walsh permutation

Walsh permutation

There are Sloane'sA002884(3) = 4 * 6 * 7 = 168 invertible binary 3×3 matrices.

They form the general linear group GL(3,2). (As all non-zero determinants are 1 in the binary field, it is also the special linear group.)

It is isomorphic to the projective special linear group PSL(2,7), the symmetry group of the Fano plane.

Each of these maps corresponds to a permutation of seven elements, which can be seen as a collineation of the Fano plane.

Representations

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overview

These images show the connection between the 3×3 matrices and the permutations:
Each row of the 3×3 matrix can be interpreted as a number in  . The corresponding Walsh function (row of a Walsh matrix) is shown to its right.
The columns of the resulting 3×8 matrix can be interpreted as a permutation of  .
(The small gray cube shows the result of applying the permutation. Reading it as a sequence corresponds to reading the permutation matrix by rows.)

 
124
 
471
 
456
 
316
 
341

Conjugacy classes

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The group has  A006951(3) = 6 conjugacy classes. They almost correspond to the cycle type,
but there are two different conjugacy classes with 7-cycles. (For the distinction between them, see here.)

The permutations in 2+2 are self-inverse, and their fixed points correspond to Walsh functions. They can be found here.

The 3×3 matrices in the same conjugacy class are similar.

Cycle shapes

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In a symmetric representation like the Fano plane, there are 33 different cycle shapes (or 24 if the direction is ignored).
A complete list can be found here. They are denoted by city names, followed by a qualifier of the direction, where needed.

hexagon Shanghai


  and   have the same cycle shape, if there is a   from the symmetric subgroup, so that  . (The 3×3 matrix of   is a permutation matrix.)

This is a refinement of matrix similarity, where   is allowed to be any element of the group. (Cycle shapes are a refinement of conjugacy classes.)

Powers and cycle graph

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The cycle graph of this group has 28 triangles, 21 squares and 8 heptagons.

Each of the following rows is an example of a cycle. (Each one is closed by the neutral element, which is not shown.)
It shows consecutive powers of the first element from left to right. (Also of the last element from right to left.)
Elements in symmetric positions are inverse to each other.

Triangles

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Each of the 28 triangles contains two inverse permutations of cycle type 3+3.

Squares

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This shows why there are two permutations of cycle type 2+4 for each one with 2+2.

Toronto, Rome
Toronto Rome Toronto

Heptagons

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Cairo, Alexandria, Tripoli
Cairo b Alexandria b Tripoli a Tripoli b Alexandria a Cairo a