Category:Hexadecachoric group; lattice edges with equations

The vertices of the lattice denote permutations. It is instructive to assign a permutation ${\displaystyle p}$ to each edge between ${\displaystyle a}$ and ${\displaystyle b}$ that characterizes how they are related.

There are two ways to do that:

• ${\displaystyle a(p)=b}$
• ${\displaystyle p(a)=b}$

In both cases these edge permutations are from the same two self-inverse conjugacy classes:

• The 24 * 32 = 768 tesseract edges correspond to the four permutations like  .
• The 16 * 36 = 576 permutohedron edges correspond to permutations like  . Either to all twelve, or to the three with rank 1.

In the image on the right they are shown as dark and light red.

The POV-Ray source of these images is black_edges_perm.pov using black_edges.inc.

a(p) = b[edit]

For each edge between ${\displaystyle a}$ and ${\displaystyle b}$ there is a permutation ${\displaystyle p}$ for which ${\displaystyle a(p)=b}$ and ${\displaystyle b(p)=a}$.
There are 16 such permutations, namely all those in the two red conjugacy classes.

Tesseract edges[edit]

Each of the 4 permutation in the dark red conjugacy class corresponds to 24 * 8 = 192 tesseract edges.
Scrolling through the imagestack on the left shows how the edge directions in a tesseract are described by the finite permutation that belongs to the respective permutohedron vertex.

The template Imagestack requires additional javascript-code. It doesn't work if javascript is switched off.

Permutohedron edges[edit]

Each of the 12 permutations in the light red conjugacy class corresponds to 6 * 8 = 48 permutohedron edges.
The left imagestack shows the parallel edges. The one on the right shows the crossing ones.
Each ${\displaystyle p}$ is a pair (m, n). Those with the same n have the same permutohedron edge direction (seen in the image above). The edges are parallel if m is 0, otherwise crossing.

The template Imagestack requires additional javascript-code. It doesn't work if javascript is switched off. parallel

The template Imagestack requires additional javascript-code. It doesn't work if javascript is switched off. crossing

p(a) = b[edit]

tesseract

permutohedron

For each edge between ${\displaystyle a}$ and ${\displaystyle b}$ there is a permutation ${\displaystyle p}$ for which ${\displaystyle p(a)=b}$ and ${\displaystyle p(b)=a}$.
There are 7 such permutations, namely those with rank 1.
Each of them corresponds to 192 of the 1344 edges in the lattice.
Their assignment to the edges so simple, that no images are required:

Tesseract edges:

•   corresponds to those parallel to the edge between and .
•   corresponds to those parallel to the edge between and .
•   corresponds to those parallel to the edge between and .
•   corresponds to those parallel to the edge between and .

Permutohedron edges:

•   corresponds to the blue edges.
•   corresponds to the green edges.
•   corresponds to the red edges.

(Not to be confused with the blue, green and red edges in the permutohedron. These are the edge colors shown in the lattice diagram and its simplification on the right. They are the same as in the Cayley graph.)