Signs in Chevalley's commutator formula

Question

I am trying to understand presentations of twisted groups of Lie type (specifically $^2D_5$) over finite fields using Steinberg's presentations (for instance from Gorenstein, Lyons and Solomon, Number 3, Chapter 2.4). Such presentations (for instance from Theorem 2.4.8 in the above book) typically involve the Chevalley commutator formula, which involves terms often of the form (in the most straightforward case)

$[x_{\tilde{\alpha}}(t),x_{\tilde{\beta}}(u)] = x_{\tilde{\alpha} + \tilde{\beta}}(\epsilon_{\alpha, \beta} t u)$

where $\epsilon_{\alpha,\beta} = \pm 1$ is a parameter dependent only on the roots $\alpha$ and $\beta$. This is the extent of the information I can find about $\epsilon_{\alpha, \beta}$. I suspect that I am not free to choose these completely arbitrarily, but I'm not sure on what restrictions I need to place on these choices; in other words, how to tell whether a given set of choices of values for $\epsilon_{\alpha,\beta}$ would result in a valid presentation.

If anyone has any information or references on how to understand these signs in more detail, I would be very grateful.

Answer 1

These signs have a history of causing problems. For simply-laced groups (type A-D-E), a general approach to these sign ambiguities can be found in Jacob Lurie's undergraduate thesis at http://www.math.harvard.edu/~lurie/papers/thesis.pdf, around Section 3.

But maybe it's best here to just write down some signs that work. For simply-laced groups again, I learned an approach from Gordan Savin's "On unramified representations of covering groups" (Crelle, 2004), which he attributes to the book on loop groups by Pressley and Segal. Here's the idea:

Order the simple roots $\{ \alpha_1, \ldots, \alpha_\ell \}$. Normalize the Killing form so that $\langle \alpha, \alpha \rangle = 2$ for all roots $\alpha$. (They all have the same length in type A-D-E). Let $B$ be the bilinear form on the root lattice satisfying $$B(\alpha_i, \alpha_j) = \begin{cases} 0 & \text{if } i < j \\ 1 & \text{if } i = j \\ \langle \alpha_i, \alpha_j \rangle & \text{if } i > j. \end{cases}$$

One can and should order the simple roots in such a way that the bilinear form above is invariant under the automorphism group of the Dynkin diagram. This allows one to work with twisted groups, I think.

For two roots $\alpha, \beta$, define $$\epsilon(\alpha, \beta) = (-1)^{B(\alpha, \beta)}.$$ These signs work!

Answer 2

There are lots of ways to fix the signs of the structure constants, so I will just provide a bunch of references I use when needed:

• Section 2 of A third look at weight diagrams gives an easy and unified treatment for the ADE-type groups;
• Section 9 of Chevalley Groups Over Commutative Rings I. Elementary Calculations works out groups of all normal types in a bit harder form, but has some tables of signes inside;
• Can one see the signs of structure constants? is mostly about the microweight representations for $\mathsf{E}_6$ and $\mathsf{E}_7$, but contains a lot of reference on the methods of determining the sighs.

Now all of the above is for the groups of normal types and you indicated you are interested in twisted groups such as ${}^2\mathsf{D}_\ell$. For those you can always fix the signs in the corresponding non-twisted groups and use the fact that the elementary generators of the twisted group are the products of the elementary generators over the orbits of roots — see the Steinberg lectures on the Chevalley groups (original mimeographic notes (PDF), TeXified version (PDF)) or Carters's Simple Groups of Lie Type.

Answer 3

Here are some supplementary comments, in community-wiki format:

For either the original Chevalley groups or the twisted variants, the concrete, detailed treatment in Roger Carter's 1972 book here is also a good resource: see Chapters 4-5 (especially 5.2) for the split groups and their complicated Chevalley commutation relations, as well as Chapter 13 for the twisted groups. It's useful to note that there are two kinds of integer constants involved: those occurring with signs in a Chevalley basis of the simple Lie algebra over $\mathbb{C}$ and those coming from the root strings when two linearly independent roots are given. Both are relatively simple for the simply-laced types, especially classical types such as $D_\ell$ where root vectors can be specified in natural ways. But the twisted groups add some complications, and I haven't seen examples of rank as large as 5 treated explicitly. (Though people have by now done a lot of computation.)

Any way it's approached, the basic idea is to start with a Chevalley basis and then work with exponential power series in characteristic 0; after that one reduces modulo a prime, etc. Another version, dealing with more than the adjoint groups, is found in Steinberg's 1967-68 Yale lectures (in the early sections). These notes used to be available online, but since his death the homepage at UCLA has been taken down as noted elsewhere on MO. But there are copies including LaTeX versions circulating, apart from libraries.

One other comment is that some previous questions on MO have focused on the sign choices in a Chevalley basis, and they too provide some theoretical references which may (or may not) be helpful. For practical advice, you might try Frank Luebeck here.