What is the complete decomposition of $E_8$ roots in terms of the objects $A_0$, $A_1$, $A_2$, $B_0$, and $B_1$

Question

Question:

What is the complete decomposition of $E_8$ roots in terms of the objects $A_0$, $A_1$, $A_2$, $B_0$, and $B_1$

Answer:

Wendy Krieger, an MSE contributor, has kindly provided the answer to this question as given in the "Answer" below (see the "Answer" which I posted - it is Wendy's decomposition, which I lifted from a different thread.)

I find this answer extremely useful because it succinctly gathers in one place various pieces of information which are not in one concise place anywhere else on the web.

Also, this answer may be useful to anyone considering the bounty question which is about to expire:

Is there an internally consistent nearest-neighbor relation in this complete linearization of the 240 roots of $E_8$?

Answer 1

This is a projection of 4_21 beginning with a simplex. It shows in this section all sorts of interesting things that have been discussed here.

 +3    x---o---o---o---o---o---o   8   A0   B1  E0
 +2    o---o---o---o---o---x---o  28   A1   B0  E0
 +1    o---o---x---o---o---o---o  56   A2   B1  E0
  0    x---o---o---o---o---o---x  56   A0   B0  E0
 -1    o---o---o---o---x---o---o  56   A1   B1  E0
 -2    o---x---o---o---o---o---o  28   A2   B0  E0
 -3    o---o---o---o---o---o---x   8   A0   B1  E0

A0 is the root polytope or eutactic star of the A8 system.

A1, A2 have 84 vertices are the birectate 8-simplex.

B0 is a polytope of 112 vertices, being the rectate 8-orthotope.

B1 has 128 vertices, is the half-8cube

E0 has 240 vertices, is the eutactic star of 4_21

Sections show the Coxeter-Dynkin diag of each of the sections.

Answer 2

For completeness sake, here is E7 and E6 on the A7 and A6.

           1_22                            2_21
       E6 eutactic star                E6 tiling ptope

  6  o---o---o---o---o    1     
  3  o---o---x---o---o   20        x---o---o---o---o    6
  0  x---o---o---o---x   30        o---o---o---x---o   15
 -3  o---o---x---o---o   20        x---o---o---o---o    6
 -6  o---o---o---o---o    1     

The sections at +3, -3, make prisms of the 2/2 and /4 ie the bitruncated 6-simplex and the simplex itself. The dual of the ooxoo is interesting, in as far as it is comprised entirely of chords of sqrt(2) and sqrt(3), which are shortchords of polygons. This leads to a facinating range of hyperbolic tilings in six dimensions.

The second projection was used to demonstrate that the 2_21 can be built from a lace-tower of A5 polytopes, being for the benefit of Dr Klitzing.

These polytopes are the real versions of 3(3)3(3)3 and 3(3)4(2)3.

We now turn to some E7

      2_31 (E7 eutactic star)            1_32 (E7 tiling ptope)

  +4   o---o---o---o---o---x   7
  +2   o---o---x---o---o---o  35
   0   x---o---o---o---o---o  42
  -2   o---o---o---x---o---o  35
  -4   x---o---o---o---o---o   7

E7 does not get as much attention as E6 or E8, because complex polytopes do not live here. Must look at the models again.

Hamming-Code information

The geometric version of the hammingcode gives the 4D4^2 reading of the the E8, which leads in turn to some interesting notions about the polytopes of 112 and 128 vertices, being 1/5A (rectified-8-orthotope), and 6A/ half-8-cube.

The hamming-code is a set of 16 coordinates, that have the greatest spacing:

 {00, 0F, 33, 3C, 55, 5A, 66, 69, 96, 99, A5, AA, C3, CC, F0, FF}

This breaks down to four sets

{00 0F F0 FF}, {33 3C C3 CC} {55 5A A5 AA} and {66 69 96 99}

These further become pairs, (0,F), (3,C), (5,A) and (6,9), which represent some four-dimensional lattices: specifically, they are all body-centred 4-cubics, that together make a semi-4-cubic.

If we write a coordinate like 3 out in binary, we get 0011. Replacing 1 by ½, we get 0,0,½,½. With any integers, this makes a four-cubic. Adding ½ to each coordinate gives ½,½,0,0 and together, these make a full body-centred cubic.

The bcc is the most efficient packing in 4d, corresponds to {3,3,4,3}. It has four stations, where it can freely stand in the same symmetry. These give the four pairs above, ie (0,F), (3,C), (5,A), (6,9) as above.

Supposing these four represent 0,1,2,3 respectively, the E8 represents a body-centred polytope, where the coordinates are matching sets, eg 0,F + 0,F gives 00, 0F, F0, FF as 8 half-units coordinates. This is true for each station. The stations are too shallow to take a sphere itself, but their prism is deep enough.

So, the Hamming-code above represents the 'body-centred' version, like 00, 11, 22, 33.

This code effortlessly extends to 12, 16, 20, dimensions, yielding dense packings of spheres, when the rule is changed from x=y to sum=0 mod 4, so

000, 013 022, 031, 103, 112, 121, 130, 202, 211, 220, 233, 301, 310, 323, 332

Each of 0,1,2,3 can be replaced with either of the letter-pairs 0F, 3C, 55, 69.

So 130 stands for 360, 36F, 390, 39F, C60, C6F, C90, C9F.

The packing is twice that of the quartercubic, but the Coxeter-Todd is more efficient at 2.370370 of the same. A quarter-cubic corresponds to putting a sphere of diam sqrt(2) per unit volume. E8 has a packing of 1 Q, while the leech-lattice has a packing of 4096 Q. The 24d hamming-packing gives a packing of 16 Q.

Hamming in E8

For a vertex in 00, the adjacent vertices here make a 8-orthotope, or 16 vertices. F0 adds a tesseract in the first four axies, 0F in the other four, these are also 16 each. No points are close enough in FF to touch 00. All together 48 vertices.

The 'other' branches, which go from 00 to 33,3C,C3,CC the only vertices that touch 00 is a cell of {3,3,4,3}. Since this gives a duoprism (ie x-x prism), we find there are 8*8 = 64 such points, as a 4-orthotope duoprism.

The three other branches are identical in this way, so we end up with

same 1 48 two orthogonal 24-chora. other 3 64 bi-16choron prism. = 4-orthotope duoprism

One gets the pairings

 48+64       =  112     the 1/5A   or root of B8
 64+64       =  128     the 6/A    or half-8cube
 64+64+48    =  176
 64+64+64+48 =  240     the 6/B or 4_21 itself.

Answer 3

one of my favorite polytopes is the 8 dimensional fy = Gosset figure 4_21 aka the hull of the roots of E8.

Today a very simple diagram dawned to me, which depicts very easily the interplay of E8, D8, A8, E7 x A1, D7 x A1, A7 x A1, and A6 x A1 x A1 by means of a simple projection of this very polytope, in fact by means of an according lace city display. - Here it comes:

fy = o3o3o3o *c3o3o3o3x
                                   _+-------- x3o3o3o3o3o3o (oca)
                                _-  _+------- o3o3o3o3o3x3o (inv. roc)
                             _-  _-  _+------ o3o3x3o3o3o3o (broc)
                          _-  _-  _-  _+----- x3o3o3o3o3o3x (suph)
                       _-  _-  _-  _-  _+---- o3o3o3o3x3o3o (inv. broc)
                    _-  _-  _-  _-  _-  _+--- o3x3o3o3o3o3o (roc)
                 _-  _-  _-  _-  _-  _-  _+-- o3o3o3o3o3o3x (dual oca)
              _-  _-  _-  _-  _-  _-  _-

        p              -- o3o3o3o *c3o3o3o (point)

  h   R   r   H        -- o3o3o3o *c3o3o3x (naq)

H   b   S   B   h      -- x3o3o3o *c3o3o3o (laq)

  h   R   r   H        -- o3o3o3o *c3o3o3x (naq)

        p              -- o3o3o3o *c3o3o3o (point)

      \   \   \   \   \
       \   \   \   \   +-- o3o3o *b3o3o3o3x (zee)
        \   \   \   +----- x3o3o *b3o3o3o3o (hesa)
         \   \   +-------- o3o3o *b3o3o3x3o (rez)
          \   +----------- o3o3x *b3o3o3o3o (alt. hesa)
           +-------------- o3o3o *b3o3o3o3x (zee)

where:
p = o3o3o3o3o3o (point)
h = x3o3o3o3o3o (hop)
H = o3o3o3o3o3x (dual hop)
r = o3x3o3o3o3o (ril)
R = o3o3o3o3x3o (inv. ril)
b = o3o3x3o3o3o (bril)
B = o3o3o3x3o3o (inv. bril)
S = x3o3o3o3o3x (staf)

together with the relations to the following subdiagrams / partial deletions in this lace city:

relation to 
brene = o3o3x3o3o3o3o3o
        .        

  .   .   r   .  

.   b   .   .   h

  .   .   r   .  

        .        

with A8 subsymmetry,

relation to
soxeb = x3o3o3o3o3o3o3x
        p        

  h   .   .   H  

.   .   S   .   .

  h   .   .   H  

        p        

again with A8 subsymmetry,

relation to
rek = o3o3o *b3o3o3o3x3o
        .        

  h   .   r   .  

H   .   S   .   h

  .   R   .   H  

        .        

with D8 subsymmetry, and

relation to
hocto = x3o3o *b3o3o3o3o3o
        p        

  .   R   .   H  

.   b   .   B   .

  h   .   r   .  

        p        

again with D8 subsymmetry.

--- rk