Category:Hexadecachoric group; lattice
This lattice is for this group, what the permutohedron is for the symmetric group. (Compare Weak order of permutations.)
TV = tesseract vertices = 16, TE = tesseract edges = 32,
PV = permutohedron vertices = 24, PE = permutohedron edges = 36.
It has TV*PV = 16*24 = 384 vertices
and TV*PE + TE*PV = 16*36 + 32*24 = 1344 edges.
Each vertex has tesseract degree + permutohedron degree = 4 + 3 = 7 edges, so this is a 7-regular graph.
The neutral permutation 
and the permutation 
are the bottom and top of this bounded lattice.
Each vertex has a rank between 0 (bottom) and 10 (top), so this is a graded poset. (Compare subcategory lattice rank.)
Definition[edit]
The lattice can be seen as a generalization of the weak order of permutations.
There is an edge from (m1, n1) to (m2, n2) iff two conditions are true:
- n1 is ≤ n2 in the weak order of permutations (seen in the permutohedron on the right).
- The horizontal equivalent of m1 is bitwise ≤ the horizontal equivalent of m2.
m corresponds to the sign distribution in the matrix rows. So the horizontal equivalent is the same for matrix columns.
Relationship to Cartesian product[edit]
This graph is isomorphic to the Cartesian product of the 4-cube and the 4-permutohedron. It has the same vertices and the same number of edges. Most of the edges are the same. They change only one of the two coordinates, and the change of the other one corresponds to an edge in the tesseract or the permutohedron. But there are 36*8 = 288 edges that change both coordinates. (See the crossing edges here.)
_to_(11,_8),_PT.png)
| Python script to generate the edges |
|---|
tess_edges = [(0, 1), (2, 3), (4, 5), (6, 7), (8, 9), (10, 11), (12, 13), (14, 15), (0, 2), (1, 3), (4, 6), (5, 7), (8, 10), (9, 11), (12, 14), (13, 15), (0, 4), (1, 5), (2, 6), (3, 7), (8, 12), (9, 13), (10, 14), (11, 15), (0, 8), (1, 9), (2, 10), (3, 11), (4, 12), (5, 13), (6, 14), (7, 15)]
perm_edges = [
[(0, 1), (6, 7), (16, 22), (17, 23), (3, 5), (9, 11), (12, 18), (14, 20), (2, 4), (8, 10), (13, 19), (15, 21)], # blue
[(0, 2), (7, 13), (10, 16), (21, 23), (1, 3), (6, 12), (11, 17), (20, 22), (4, 5), (8, 14), (9, 15), (18, 19)], # green
[(0, 6), (1, 7), (16, 17), (22, 23), (3, 9), (5, 11), (12, 14), (18, 20), (2, 8), (4, 10), (13, 15), (19, 21)] # red
]
permutations = [
[0, 2, 1, 3, 4, 6, 5, 7, 8, 10, 9, 11, 12, 14, 13, 15], # blue
[0, 1, 4, 5, 2, 3, 6, 7, 8, 9, 12, 13, 10, 11, 14, 15], # green
[0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15] # red
]
lattice_edges = []
for p in range(24):
for t_low, t_high in tess_edges:
lattice_edges.append(
((t_low, p), (t_high, p))
)
for color_index in range(3):
color_permutation = permutations[color_index]
color_edges = perm_edges[color_index]
for p_low, p_high in color_edges:
for t_low in range(16):
t_high = color_permutation[t_low]
lattice_edges.append(
((t_low, p_low), (t_high, p_high))
)
print(sorted(lattice_edges))
|
Subcategories
This category has the following 2 subcategories, out of 2 total.