Consider the root system of $E_8$, written in its standard "even" coordinate system: i.e., it is the set of all $240$ vectors in $\mathbb{R}^8$ which whose coordinates are either all integers or all integers-plus-a-half, have an even sum, and whose sum of squares is $2$.

For any choice of a set of positive roots, the half-sum of the positive roots is known as the corresponding "Weyl vector". For example, if we take the positive roots to be those whose last nonzero coordinate is positive, the Weyl vector is $(0,1,2,3,4,5,6,23)$. Since the Weyl vector lies in the interior of the Weyl chamber, the Weyl group acts freely on its orbit, and there are $\#W(E_8) = 2^{14} \cdot 3^5 \cdot 5^2 \cdot 7 = 696\,729\,600$ Weyl vectors, exactly one for each choice of positive roots.

My question is essentially whether we can describe this set of $696\,729\,600$ Weyl vectors in a simple way through its coordinates.

One obvious reduction is that the Weyl group of $D_8$, which is a subgroup (of order $8!\times 2^7 = 2^{14}\cdot 3^2\cdot 5\cdot 7 = 5\,160\,960$) of that of $E_8$ acts by permuting the $8$ coordinates in any way and changing the sign of an even number of them (see remarks below). So all that need to be described are the $696\,729\,600 / 5\,160\,960 = 3^3\cdot 5 = 135$ orbits of Weyl vectors modulo this action. It's not difficult to list them explicitly, e.g., a simple computation gives me:

```
(0, 1, 2, 3, 4, 5, 6, 23)
(0, 1, 2, 3, 4, 13, 14, 15)
(0, 1, 2, 3, 8, 9, 10, 19)
(0, 1, 2, 5, 6, 7, 8, 21)
(0, 1, 2, 5, 6, 11, 12, 17)
(0, 1, 2, 7, 8, 9, 14, 15)
(0, 1, 2, 9, 10, 11, 12, 13)
(0, 1, 3, 4, 5, 6, 7, 22)
(0, 1, 3, 4, 5, 12, 13, 16)
(0, 1, 3, 4, 7, 8, 9, 20)
(0, 1, 3, 4, 7, 10, 11, 18)
(0, 1, 3, 6, 7, 10, 13, 16)
(0, 1, 3, 8, 9, 10, 13, 14)
(0, 1, 4, 5, 6, 9, 10, 19)
(0, 1, 4, 5, 6, 11, 14, 15)
(0, 1, 4, 5, 8, 9, 12, 17)
(0, 1, 4, 7, 8, 11, 12, 15)
(0, 1, 5, 6, 7, 8, 11, 18)
(0, 1, 5, 6, 7, 12, 13, 14)
(0, 1, 5, 6, 9, 10, 11, 16)
(0, 1, 6, 7, 8, 9, 10, 17)
(0, 2, 3, 4, 7, 9, 10, 19)
(0, 2, 3, 5, 6, 11, 13, 16)
(0, 2, 3, 5, 7, 10, 12, 17)
(0, 2, 3, 7, 8, 10, 13, 15)
(0, 2, 4, 5, 7, 9, 11, 18)
(0, 2, 4, 6, 7, 11, 13, 15)
(0, 2, 4, 6, 8, 10, 12, 16)
(0, 2, 5, 6, 8, 9, 11, 17)
(0, 3, 4, 5, 7, 11, 12, 16)
(0, 3, 4, 6, 7, 10, 11, 17)
(1/2, 3/2, 5/2, 7/2, 9/2, 11/2, 13/2, -45/2)
(1/2, 3/2, 5/2, 7/2, 9/2, 25/2, 27/2, 31/2)
(1/2, 3/2, 5/2, 7/2, 15/2, 17/2, 19/2, -39/2)
(1/2, 3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 37/2)
(1/2, 3/2, 5/2, 9/2, 11/2, 13/2, 15/2, 43/2)
(1/2, 3/2, 5/2, 9/2, 11/2, 23/2, 25/2, -33/2)
(1/2, 3/2, 5/2, 9/2, 13/2, 15/2, 17/2, 41/2)
(1/2, 3/2, 5/2, 9/2, 13/2, 21/2, 23/2, -35/2)
(1/2, 3/2, 5/2, 11/2, 13/2, 21/2, 25/2, 33/2)
(1/2, 3/2, 5/2, 13/2, 15/2, 19/2, 27/2, -31/2)
(1/2, 3/2, 5/2, 15/2, 17/2, 19/2, 27/2, 29/2)
(1/2, 3/2, 5/2, 17/2, 19/2, 21/2, 25/2, -27/2)
(1/2, 3/2, 7/2, 9/2, 11/2, 23/2, 27/2, 31/2)
(1/2, 3/2, 7/2, 9/2, 13/2, 17/2, 19/2, 39/2)
(1/2, 3/2, 7/2, 9/2, 13/2, 19/2, 21/2, -37/2)
(1/2, 3/2, 7/2, 9/2, 15/2, 19/2, 23/2, 35/2)
(1/2, 3/2, 7/2, 11/2, 13/2, 21/2, 27/2, -31/2)
(1/2, 3/2, 7/2, 11/2, 15/2, 19/2, 25/2, -33/2)
(1/2, 3/2, 7/2, 13/2, 15/2, 21/2, 25/2, 31/2)
(1/2, 3/2, 7/2, 15/2, 17/2, 21/2, 25/2, -29/2)
(1/2, 3/2, 9/2, 11/2, 13/2, 17/2, 21/2, 37/2)
(1/2, 3/2, 9/2, 11/2, 13/2, 23/2, 27/2, 29/2)
(1/2, 3/2, 9/2, 11/2, 15/2, 17/2, 23/2, -35/2)
(1/2, 3/2, 9/2, 11/2, 17/2, 19/2, 23/2, 33/2)
(1/2, 3/2, 9/2, 13/2, 15/2, 23/2, 25/2, -29/2)
(1/2, 3/2, 9/2, 13/2, 17/2, 21/2, 23/2, -31/2)
(1/2, 3/2, 11/2, 13/2, 15/2, 17/2, 21/2, 35/2)
(1/2, 3/2, 11/2, 13/2, 17/2, 19/2, 21/2, -33/2)
(1/2, 5/2, 7/2, 9/2, 13/2, 21/2, 25/2, -33/2)
(1/2, 5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 37/2)
(1/2, 5/2, 7/2, 11/2, 13/2, 19/2, 23/2, -35/2)
(1/2, 5/2, 7/2, 11/2, 13/2, 23/2, 25/2, 31/2)
(1/2, 5/2, 7/2, 11/2, 15/2, 21/2, 23/2, 33/2)
(1/2, 5/2, 7/2, 13/2, 15/2, 21/2, 27/2, -29/2)
(1/2, 5/2, 7/2, 13/2, 17/2, 19/2, 25/2, -31/2)
(1/2, 5/2, 9/2, 11/2, 15/2, 19/2, 21/2, 35/2)
(1/2, 5/2, 9/2, 11/2, 15/2, 21/2, 25/2, -31/2)
(1/2, 5/2, 9/2, 13/2, 15/2, 19/2, 23/2, -33/2)
(1/2, 7/2, 9/2, 11/2, 13/2, 21/2, 23/2, -33/2)
(1, 2, 3, 4, 5, 12, 14, 15)
(1, 2, 3, 4, 6, 7, 8, 21)
(1, 2, 3, 4, 6, 11, 12, -17)
(1, 2, 3, 4, 8, 9, 11, 18)
(1, 2, 3, 5, 6, 8, 9, 20)
(1, 2, 3, 5, 6, 10, 11, -18)
(1, 2, 3, 6, 7, 10, 14, -15)
(1, 2, 3, 6, 7, 11, 12, 16)
(1, 2, 3, 6, 8, 9, 13, -16)
(1, 2, 3, 8, 9, 11, 12, -14)
(1, 2, 4, 5, 6, 12, 13, 15)
(1, 2, 4, 5, 7, 8, 10, 19)
(1, 2, 4, 5, 7, 10, 13, -16)
(1, 2, 4, 5, 8, 10, 11, 17)
(1, 2, 4, 6, 7, 9, 12, -17)
(1, 2, 4, 7, 8, 11, 13, -14)
(1, 2, 4, 7, 9, 10, 12, -15)
(1, 2, 5, 6, 7, 9, 10, 18)
(1, 2, 5, 6, 8, 11, 12, -15)
(1, 2, 5, 7, 8, 10, 11, -16)
(1, 3, 4, 5, 6, 10, 12, -17)
(1, 3, 4, 5, 8, 9, 10, 18)
(1, 3, 4, 6, 8, 10, 13, -15)
(1, 3, 4, 7, 8, 9, 12, -16)
(1, 3, 5, 6, 7, 10, 12, -16)
(3/2, 5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 41/2)
(3/2, 5/2, 7/2, 9/2, 11/2, 21/2, 23/2, -35/2)
(3/2, 5/2, 7/2, 9/2, 11/2, 25/2, 27/2, 29/2)
(3/2, 5/2, 7/2, 9/2, 17/2, 19/2, 21/2, 35/2)
(3/2, 5/2, 7/2, 11/2, 13/2, 15/2, 19/2, 39/2)
(3/2, 5/2, 7/2, 11/2, 15/2, 19/2, 27/2, -31/2)
(3/2, 5/2, 7/2, 13/2, 15/2, 17/2, 25/2, -33/2)
(3/2, 5/2, 7/2, 15/2, 17/2, 23/2, 25/2, -27/2)
(3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 23/2, -29/2)
(3/2, 5/2, 9/2, 11/2, 13/2, 19/2, 25/2, -33/2)
(3/2, 5/2, 9/2, 11/2, 15/2, 17/2, 19/2, 37/2)
(3/2, 5/2, 9/2, 13/2, 17/2, 21/2, 25/2, -29/2)
(3/2, 5/2, 9/2, 15/2, 17/2, 19/2, 23/2, -31/2)
(3/2, 5/2, 11/2, 13/2, 15/2, 21/2, 23/2, -31/2)
(3/2, 7/2, 9/2, 11/2, 17/2, 19/2, 27/2, -29/2)
(3/2, 7/2, 9/2, 13/2, 15/2, 19/2, 25/2, -31/2)
(2, 3, 4, 5, 6, 7, 9, 20)
(2, 3, 4, 5, 8, 9, 14, -15)
(2, 3, 4, 6, 7, 8, 9, 19)
(2, 3, 4, 6, 7, 9, 13, -16)
(2, 3, 4, 7, 9, 11, 12, -14)
(2, 3, 4, 8, 9, 10, 11, -15)
(2, 3, 5, 6, 9, 10, 13, -14)
(2, 3, 5, 7, 8, 10, 12, -15)
(2, 4, 5, 6, 8, 9, 13, -15)
(5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2, 39/2)
(5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 27/2, -31/2)
(5/2, 7/2, 9/2, 13/2, 19/2, 21/2, 25/2, -27/2)
(5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 23/2, -29/2)
(5/2, 7/2, 11/2, 13/2, 17/2, 19/2, 25/2, -29/2)
(5/2, 9/2, 11/2, 13/2, 15/2, 17/2, 27/2, -29/2)
(3, 4, 5, 6, 7, 8, 14, -15)
(3, 4, 5, 6, 10, 11, 12, -13)
(3, 4, 5, 7, 9, 10, 12, -14)
(3, 4, 6, 7, 8, 9, 13, -14)
(7/2, 9/2, 11/2, 13/2, 19/2, 21/2, 23/2, -27/2)
(7/2, 9/2, 11/2, 15/2, 17/2, 19/2, 25/2, -27/2)
(4, 5, 6, 7, 9, 10, 12, -13)
(9/2, 11/2, 13/2, 15/2, 17/2, 21/2, 23/2, -25/2)
(5, 6, 7, 8, 9, 10, 11, -12)
```

(Again, any element of this list is defined only up to permutation of the coordinates and an even number of sign changes: here I've sorted the coordinates in absolute value and written the minus sign, if necessary, on the last coordinate, but the representatives in question might not be the best.)

I can see no clear pattern in this list. Maybe I'm looking at it in all the wrong way.

**Question:** How can we describe this set simply?

(A followup question might be whether we can easily multiply two elements of $W(E_8)$ represented as transformations on such Weyl vectors. But the first step is, of course, to recognize them.)

**Remarks:**

The embedding of $W(D_8)$ into $W(E_8)$ comes from the fact that the root system $D_8$ is a closed subsystem of $E_8$: as such, it can be described by Borel-de Siebenthal theory: take the extended (i.e., affine) Dynkin diagram of $E_8$ and remove node number $1$ in Bourbaki's numbering of the node, this leaves us with the Dynkin diagram of $D_8$. But more concretely, in the chosen coordinate system, the root system of $D_8$ consists of those $112$ roots having shape $(\pm1,\pm1,0,0,0,0,0,0)$ (for some choice of the two signs and some permutation of the coordinates) inside that of $E_8$ which additionally consists of the $128$ roots having shape $(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})$ (for some choice of the signs such that an even number are minus).

The motivation of the problem is to understand $W(E_8)$ better and see how its elements can be represented (also, now that I experimentally found a list of numbers, I'm naturally inclined to try to find patterns in it…). To perhaps better explain why I think this is a natural question, consider the analogous case of $A_n$ in the standard system of $n+1$ coordinates all integer with sum zero: the Weyl vector is $(0,1,2,\ldots,n)$ minus whatever constant is necessary to make it sum to zero (viz., $\frac{n-1}{2}$); its orbit under $W(A_n) \cong \mathfrak{S}_{n+1}$ consists of all permutations of $(0,1,2,\ldots,n)$ (minus constant), these are very easy to recognize, and it is fairly natural to represent an element $w$ of $\mathfrak{S}_{n+1}$ by the corresponding Weyl vector $\rho_w = w\cdot\rho_0$ (where $\rho_0$ is a fixed Weyl vector, say the one I just wrote); in fact, trying to compute $w$ (as product of Coxeter generators) from $\rho_w$ is essentially a sorting algorithm. The case of the other classical root systems is similarly easy; so I thought it natural to try to look at the exceptional root systems, and, of these, $E_8$ seems the most interesting because there is a coordinate system that is really pleasant (because of the relation with $D_8$).