3-bit Walsh permutation



There are (3) = 4 * 6 * 7 = 168 binary 3×3 matrices.

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

It is isomorphic to the , the symmetry group of the.

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

Representations
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 $$\{1..7\}$$. The corresponding (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 $$(0..7)$$. (The small gray cube shows the result of applying the permutation. Reading it as a sequence corresponds to reading the permutation matrix by rows.)

These images show why the author chose the term Walsh permutation. A permutation is Walsh iff applying it to the columns of a creates a matrix that can also be described as a Walsh matrix with permuted rows.

Here the binary 3×3 matrices are used as transformation matrices. A list of these files can be found here.

These are transforms with inverses of binary matrices, which contain some negative 1s. A list of these files can be found here.

These images show the connection between transforms and permutations: The vertices of the periodic gray cubes have values from 0 to 7, and the transformation moves each vertex of the colored cube to some vertex in some gray cube. This file shows essentially the same as this one (shown below), but the colored small vertices make it clearer how it corresponds to this. (This is the result of applying the permutation. It corresponds to the gray cube in the overview files above.)

The colors are in the same vertices of the gray cube as in the transform images above. This is the result of applying the permutation. The number values of the colors are like those in the gray cubes in the overview files (in the first box above). Complementary vertices are connected by brown edges. The patterns are numbered 1..7.

Conjugacy classes
The group has (3) = 6. They almost correspond to the , 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.

Cycle shapes
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.

Inverse 3×3 matrices differ in their determinant. Those in the inner triangle have 1, and those in the outer have −1.

Inverse 3×3 matrices differ in their number of ones. Those in the inner hexagon have 5, and those in the outer have 6.

$$A$$ and $$B$$ have the same cycle shape, if there is a $$P$$ from the symmetric subgroup, so that $$B = P^{-1} A P$$. (The 3×3 matrix of $$P$$ is a permutation matrix.)

This is a refinement of, where $$P$$ is allowed to be any element of the group. (Cycle shapes are a refinement of es.)

Powers and cycle graph
The 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 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
Each of the 28 triangles contains two inverse permutations of cycle type 3+3.

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