15
$\begingroup$

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)
(-1/2, 3/2, 5/2, 7/2, 9/2, 11/2, 13/2, 45/2)
(0, 1, 3, 4, 5, 6, 7, 22)
(1/2, 3/2, 5/2, 9/2, 11/2, 13/2, 15/2, 43/2)
(0, 1, 2, 5, 6, 7, 8, 21)
(1, 2, 3, 4, 6, 7, 8, 21)
(1/2, 3/2, 5/2, 9/2, 13/2, 15/2, 17/2, 41/2)
(3/2, 5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 41/2)
(0, 1, 3, 4, 7, 8, 9, 20)
(1, 2, 3, 5, 6, 8, 9, 20)
(2, 3, 4, 5, 6, 7, 9, 20)
(-1/2, 3/2, 5/2, 7/2, 15/2, 17/2, 19/2, 39/2)
(1/2, 3/2, 7/2, 9/2, 13/2, 17/2, 19/2, 39/2)
(3/2, 5/2, 7/2, 11/2, 13/2, 15/2, 19/2, 39/2)
(5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2, 39/2)
(0, 1, 2, 3, 8, 9, 10, 19)
(0, 2, 3, 4, 7, 9, 10, 19)
(0, 1, 4, 5, 6, 9, 10, 19)
(1, 2, 4, 5, 7, 8, 10, 19)
(2, 3, 4, 6, 7, 8, 9, 19)
(1/2, 3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 37/2)
(-1/2, 3/2, 7/2, 9/2, 13/2, 19/2, 21/2, 37/2)
(1/2, 5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 37/2)
(1/2, 3/2, 9/2, 11/2, 13/2, 17/2, 21/2, 37/2)
(3/2, 5/2, 9/2, 11/2, 15/2, 17/2, 19/2, 37/2)
(0, 1, 3, 4, 7, 10, 11, 18)
(-1, 2, 3, 5, 6, 10, 11, 18)
(1, 2, 3, 4, 8, 9, 11, 18)
(0, 2, 4, 5, 7, 9, 11, 18)
(0, 1, 5, 6, 7, 8, 11, 18)
(1, 3, 4, 5, 8, 9, 10, 18)
(1, 2, 5, 6, 7, 9, 10, 18)
(-1/2, 3/2, 5/2, 9/2, 13/2, 21/2, 23/2, 35/2)
(-3/2, 5/2, 7/2, 9/2, 11/2, 21/2, 23/2, 35/2)
(1/2, 3/2, 7/2, 9/2, 15/2, 19/2, 23/2, 35/2)
(-1/2, 5/2, 7/2, 11/2, 13/2, 19/2, 23/2, 35/2)
(-1/2, 3/2, 9/2, 11/2, 15/2, 17/2, 23/2, 35/2)
(3/2, 5/2, 7/2, 9/2, 17/2, 19/2, 21/2, 35/2)
(1/2, 5/2, 9/2, 11/2, 15/2, 19/2, 21/2, 35/2)
(1/2, 3/2, 11/2, 13/2, 15/2, 17/2, 21/2, 35/2)
(0, 1, 2, 5, 6, 11, 12, 17)
(-1, 2, 3, 4, 6, 11, 12, 17)
(0, 2, 3, 5, 7, 10, 12, 17)
(-1, 3, 4, 5, 6, 10, 12, 17)
(0, 1, 4, 5, 8, 9, 12, 17)
(-1, 2, 4, 6, 7, 9, 12, 17)
(1, 2, 4, 5, 8, 10, 11, 17)
(0, 3, 4, 6, 7, 10, 11, 17)
(0, 2, 5, 6, 8, 9, 11, 17)
(0, 1, 6, 7, 8, 9, 10, 17)
(-1/2, 3/2, 5/2, 9/2, 11/2, 23/2, 25/2, 33/2)
(1/2, 3/2, 5/2, 11/2, 13/2, 21/2, 25/2, 33/2)
(-1/2, 5/2, 7/2, 9/2, 13/2, 21/2, 25/2, 33/2)
(-1/2, 3/2, 7/2, 11/2, 15/2, 19/2, 25/2, 33/2)
(-3/2, 5/2, 9/2, 11/2, 13/2, 19/2, 25/2, 33/2)
(-3/2, 5/2, 7/2, 13/2, 15/2, 17/2, 25/2, 33/2)
(1/2, 5/2, 7/2, 11/2, 15/2, 21/2, 23/2, 33/2)
(-1/2, 7/2, 9/2, 11/2, 13/2, 21/2, 23/2, 33/2)
(1/2, 3/2, 9/2, 11/2, 17/2, 19/2, 23/2, 33/2)
(-1/2, 5/2, 9/2, 13/2, 15/2, 19/2, 23/2, 33/2)
(-1/2, 3/2, 11/2, 13/2, 17/2, 19/2, 21/2, 33/2)
(0, 1, 3, 4, 5, 12, 13, 16)
(0, 2, 3, 5, 6, 11, 13, 16)
(0, 1, 3, 6, 7, 10, 13, 16)
(-1, 2, 4, 5, 7, 10, 13, 16)
(-1, 2, 3, 6, 8, 9, 13, 16)
(-2, 3, 4, 6, 7, 9, 13, 16)
(1, 2, 3, 6, 7, 11, 12, 16)
(0, 3, 4, 5, 7, 11, 12, 16)
(0, 2, 4, 6, 8, 10, 12, 16)
(-1, 3, 5, 6, 7, 10, 12, 16)
(-1, 3, 4, 7, 8, 9, 12, 16)
(0, 1, 5, 6, 9, 10, 11, 16)
(-1, 2, 5, 7, 8, 10, 11, 16)
(1/2, 3/2, 5/2, 7/2, 9/2, 25/2, 27/2, 31/2)
(1/2, 3/2, 7/2, 9/2, 11/2, 23/2, 27/2, 31/2)
(-1/2, 3/2, 7/2, 11/2, 13/2, 21/2, 27/2, 31/2)
(-1/2, 3/2, 5/2, 13/2, 15/2, 19/2, 27/2, 31/2)
(-3/2, 5/2, 7/2, 11/2, 15/2, 19/2, 27/2, 31/2)
(-5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 27/2, 31/2)
(1/2, 5/2, 7/2, 11/2, 13/2, 23/2, 25/2, 31/2)
(1/2, 3/2, 7/2, 13/2, 15/2, 21/2, 25/2, 31/2)
(-1/2, 5/2, 9/2, 11/2, 15/2, 21/2, 25/2, 31/2)
(-1/2, 5/2, 7/2, 13/2, 17/2, 19/2, 25/2, 31/2)
(-3/2, 7/2, 9/2, 13/2, 15/2, 19/2, 25/2, 31/2)
(-1/2, 3/2, 9/2, 13/2, 17/2, 21/2, 23/2, 31/2)
(-3/2, 5/2, 11/2, 13/2, 15/2, 21/2, 23/2, 31/2)
(-3/2, 5/2, 9/2, 15/2, 17/2, 19/2, 23/2, 31/2)
(0, 1, 2, 3, 4, 13, 14, 15)
(1, 2, 3, 4, 5, 12, 14, 15)
(0, 1, 4, 5, 6, 11, 14, 15)
(-1, 2, 3, 6, 7, 10, 14, 15)
(0, 1, 2, 7, 8, 9, 14, 15)
(-2, 3, 4, 5, 8, 9, 14, 15)
(-3, 4, 5, 6, 7, 8, 14, 15)
(1, 2, 4, 5, 6, 12, 13, 15)
(0, 2, 4, 6, 7, 11, 13, 15)
(0, 2, 3, 7, 8, 10, 13, 15)
(-1, 3, 4, 6, 8, 10, 13, 15)
(-2, 4, 5, 6, 8, 9, 13, 15)
(0, 1, 4, 7, 8, 11, 12, 15)
(-1, 2, 5, 6, 8, 11, 12, 15)
(-1, 2, 4, 7, 9, 10, 12, 15)
(-2, 3, 5, 7, 8, 10, 12, 15)
(-2, 3, 4, 8, 9, 10, 11, 15)
(3/2, 5/2, 7/2, 9/2, 11/2, 25/2, 27/2, 29/2)
(1/2, 3/2, 9/2, 11/2, 13/2, 23/2, 27/2, 29/2)
(-1/2, 5/2, 7/2, 13/2, 15/2, 21/2, 27/2, 29/2)
(1/2, 3/2, 5/2, 15/2, 17/2, 19/2, 27/2, 29/2)
(-3/2, 7/2, 9/2, 11/2, 17/2, 19/2, 27/2, 29/2)
(-5/2, 9/2, 11/2, 13/2, 15/2, 17/2, 27/2, 29/2)
(-1/2, 3/2, 9/2, 13/2, 15/2, 23/2, 25/2, 29/2)
(-1/2, 3/2, 7/2, 15/2, 17/2, 21/2, 25/2, 29/2)
(-3/2, 5/2, 9/2, 13/2, 17/2, 21/2, 25/2, 29/2)
(-5/2, 7/2, 11/2, 13/2, 17/2, 19/2, 25/2, 29/2)
(-3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 23/2, 29/2)
(-5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 23/2, 29/2)
(0, 1, 5, 6, 7, 12, 13, 14)
(-1, 2, 4, 7, 8, 11, 13, 14)
(0, 1, 3, 8, 9, 10, 13, 14)
(-2, 3, 5, 6, 9, 10, 13, 14)
(-3, 4, 6, 7, 8, 9, 13, 14)
(-1, 2, 3, 8, 9, 11, 12, 14)
(-2, 3, 4, 7, 9, 11, 12, 14)
(-3, 4, 5, 7, 9, 10, 12, 14)
(-3/2, 5/2, 7/2, 15/2, 17/2, 23/2, 25/2, 27/2)
(-1/2, 3/2, 5/2, 17/2, 19/2, 21/2, 25/2, 27/2)
(-5/2, 7/2, 9/2, 13/2, 19/2, 21/2, 25/2, 27/2)
(-7/2, 9/2, 11/2, 15/2, 17/2, 19/2, 25/2, 27/2)
(-7/2, 9/2, 11/2, 13/2, 19/2, 21/2, 23/2, 27/2)
(0, 1, 2, 9, 10, 11, 12, 13)
(-3, 4, 5, 6, 10, 11, 12, 13)
(-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 first 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 $D_8$).

Edit (2019-06-17): To answer a question by André Henriques in the comments, the Sage code used to produce the list above is here (there are a few comments in the code explaining how it works). The code also produces a representation of the same list as the graph of the decomposition of the Weyl chamber of $W(D_8)$ as $W(E_8)$-cells (with adjacency being given by the $W(E_8)$-reflection hyperplanes, colored according to the nodes of the $E_8$ Dynkin diagram), and it looks like this: Graph of the W(D₈) Weyl chamber divided into W(E₈) cells

Improve this question
$\endgroup$
7
  • $\begingroup$ 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. $\endgroup$
    – user21230
    Commented Mar 28, 2018 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
    Commented Mar 28, 2018 at 15:15
  • 2
    $\begingroup$ @JimHumphreys I added some remarks to try to address your questions. $\endgroup$
    – Gro-Tsen
    Commented Mar 29, 2018 at 12:32
  • 1
    $\begingroup$ @Gro-Tsen. You say "It's not difficult to list them explicitly, e.g., a simple computation gives me: [list]". I would find it pretty daunting if I had to produce that list myself... May I ask you what your method was? $\endgroup$
    – André Henriques
    Commented Jun 16, 2019 at 22:36
  • 1
    $\begingroup$ @AndréHenriques I added a link to the code. There is nothing particularly smart, but, of course, this was done with a computer (maybe this wasn't clear). Basically, the algorithm to generate the cosets $wH$ of $H≤G$ (here $G=W(E_8)$ and $H=W(D_8)$) is to try to left-multiply every known coset by a set of generators $g_i$ of $G$ until nothing new is produced. (Note that I store elements $wH$ as a $W(D_8)$-dominant vector $v=ρw$ together with the element $w$ itself which is needed to compute $g_i w$.) $\endgroup$
    – Gro-Tsen
    Commented Jun 17, 2019 at 12:48

2 Answers 2

Reset to default
2
$\begingroup$

Concerning the first question in the header (and some of your preparatory remarks), it's useful to keep in mind the Planche VII for $E_8$ at the end of Chapters 4-6 of Bourbaki's treatise Groupes et algebres de Lie (English translation, Springer). For example, (VII) gives explicit coordinates for $2\rho$ in terms of both the standard basis of $\mathbb{R}^8$ and the simple roots. There is a lot of useful information in these planches, though I wish they had used some of the blank space to fill in all of the positive roots arranged by height (as T.A. Springer does in his IHES paper for exceptional types). It would also be helpful to include a planche for the "exceptional" type $D_4$.

Concerning your other more specialized question about Weyl group orbits, it might help to give some motivation. Aside from that, it is probably not easy to compute these orbits or interpret the results in this case.

Improve this answer
$\endgroup$
2
1
$\begingroup$

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 $240*126*60*2* 24*2*2*2$.

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ 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
    Commented Apr 3, 2018 at 11:17
  • 2
    $\begingroup$ (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
    Commented Apr 3, 2018 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. $\endgroup$
    – user21230
    Commented Apr 3, 2018 at 11:56
  • 1
    $\begingroup$ 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
    Commented Apr 3, 2018 at 12:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .