8
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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 $D_8$).

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  • $\begingroup$ One observation is that on those having integer entries - always four numbers are odd. For those containing halves - subtract vector $(1/2,...,1/2)$ and you also obtain four numbers odd. Second thing - I would check these vectors in another $E_8$ lattice - which come from octonion integers. $\endgroup$ – Marek Mitros Mar 28 '18 at 14:47
  • $\begingroup$ @MarekMitros Interesting remark. Your observation follows from the fact that the class of the half Weyl vector $\rho/2$ modulo the root lattice is invariant under the Weyl group (because $\rho/2$ is the half sum of the fundamental weights, so every simple reflection acts on it by subtracting the corresponding simple root, giving the same class). $\endgroup$ – Gro-Tsen Mar 28 '18 at 15:15
  • $\begingroup$ A few comments: It would help to say something about how $W(D_8)$ occurs as a subgroup of $W(E_8)$, since it isn't a parabolic subgroup. The group orders are compatible, but why is there a natural embedding? (Do you need to appeal to Lie theory, or just look at the finite groups?) Also, it's more usual to write the group orders in terms of prime factorization, such as $|W(E_8)| = 2^{14} \; 3^5 \; 5^2 \; 7$. And I wonder if your question has a specific motivation, or what consequences it might have? $\endgroup$ – Jim Humphreys Mar 28 '18 at 18:32
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    $\begingroup$ @JimHumphreys I added some remarks to try to address your questions. $\endgroup$ – Gro-Tsen Mar 29 '18 at 12:32
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How about following approach. Assume that $E_8$ vectors have length $1$. Take one vector $u_1$ from $E_8$ lattice. Let $L$ be hyperplane perpendicular to it. Consider sets of positive roots which contain $u_1$ and $56$ vectors having skalar product $1/2$ with $u_1$; finally $63$ vectors forming positive roots of $E_7$ lying on $L$. Sum of $u_1$ and $56$ mentioned vectors form $29*u_1$ (there are 28 pairs which add up to $u_1$).

Next fix $u_2$ in $E_7$ lattice in $L$, let $K$ denote hyperplane perpendicular to $u_2$ in $L$. There are $60$ vectors in $K$ forming $D_6$ root lattice. Sets of positive roots containing $u_2$ and $32$ skewed vectors - forming skalar product $1/2$ with $u_2$ contain also positive roots of $D_6$ lying on $K$. Sum of $u_2$ and $32$ mentioned vectors is $17*u_2$ (there are 16 pairs which add up to $u_2$ each).

Now assuming that you know how Weyl vectors of $D_6$ lattice are defined. We have following formula for Weyl vector of $E_8$ lattice.

$(29*u_1+17*u_2)/2+w$

where $u_1,u_2$ are any perpendicular vectors in $E_8$ and $w$ is Weyl vector of $D_6$ lattice perpendicular to $u_1$ and $u_2$.

Counting the number we have $\frac{696729600}{240*126}=23040$ which is size of Weyl group for $D_6$.

Selecting two vectors $(1,1,0...0), (1,-1,0...0)$ from $E_8$ mentioned in the question we obtain from formula above partial answer:

$(23,6,1,2,3,4,5,0)$

and permutations and sign changes on last 6 coordinates.

Addition

If we continue the same way we obtain following formula for Weyl vector in $E_8$:

$w=(29*u_1+17*u_2+9*u_3+u_4+5*u_5+u_6+u_7+u_8)/2$

where $u_1, u_2, u_3$ are any triple of perpendicular vectors in $E_8$. Vector $u_4$ is determined up to the sign as the one forming $D_4$ sublattice with the first three vectors. Finally $u_5$ is any vector in perpendicular $D_4$ and last three vectors are determined up to the sign. In order to find the patterns for Weyl vectors we need to consider cases for first three vectors - which types of vectors they are. Original root system consist of type 1 vectors - having two non-zero coordinates and type 2 vectors - having all non-zero coordinates.

Using this procedure we can calculate size of Weyl group as $240*126*60*2* 24*2*2*2$.

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    $\begingroup$ I'm afraid I don't understand your answer (note that there may be a problem with the word "summarize", which in ordinary English means "prepare a summary", "condense", "recapitulate", "review" and which I don't understand as mathematical English). I agree that giving a basis of simple roots for $E_8$ is equivalent to giving two orthogonal roots and then a basis of simple roots for the $D_6$ that is orthogonal to them, but I don't see how this relates to my question. (contd.) $\endgroup$ – Gro-Tsen Apr 3 '18 at 11:17
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    $\begingroup$ (contd.) In particular, I don't see how to obtain a simple description of the coordinates of the Weyl vectors of $E_8$ from your remarks (one problem is that the $D_6$ being considered depends on the two orthogonal roots). I should emphasize that my question is whether a simple elementary procedure can be found to generate a list of representatives such as the one I gave. $\endgroup$ – Gro-Tsen Apr 3 '18 at 11:22
  • $\begingroup$ I changed "summary" to "sum" and "summarized" to "add up" meaning that I want to add vectors. Can you share formulas for Weyl vectors for $D_6$ lattice ? Then I would try to apply it to my formula. I used "sum of positive roots" as Weyl vectors. Now I see that it should be divided by 2. I agree that $D_6$ depends on two selected roots. I will try to give it a thought in the evening. $\endgroup$ – Marek Mitros Apr 3 '18 at 11:56
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    $\begingroup$ The Weyl vectors for $D_6$ in the usual orthonormal coordinate system (where the roots are of the form $(\pm1,\pm1,0,0,0,0)$) are $(0,\pm1,\pm2,\pm3,\pm4,\pm5)$ with arbitrary signs and after an arbitrary permutation of the coordinates. (You can easily check this by computing the fundamental weights.) $\endgroup$ – Gro-Tsen Apr 3 '18 at 12:44

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