3
$\begingroup$

The rank of the BN-pair of a Chevalley group is the number of simple roots of its Lie algebra, which is the index of the name of its Dynkin diagram ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4,G_2 $).

The finite simple groups of Lie type and not Chevalley, have also a BN-pair.
Question: Is there a reference in which their rank are computed?

Remark: the Steinberg group $^2A_3(2^2) ≃ B_2(3)$, so the rank of its BN-pair is $2$ (and not $3$).

Improve this question
$\endgroup$
9
  • $\begingroup$ This looks too elementary for this site (and the tag 'lie-groups' isn't relevant). Keep in mind that the rank of a BN-pair is defined to be the rank of the Coxeter system $(W,S)$ where $W:=N/H$ with $H:=B \cap N$ (so the rank is $|S|$ and is easily computed for finite groups of Lie type). $\endgroup$
    – Jim Humphreys
    Nov 15, 2016 at 13:56
  • 1
    $\begingroup$ @JimHumphreys I disagree that the question is off-topic here: even if easy (for cognoscenti), I think questions about BN-pairs are something that a PhD student or non-expert mathematician can ask about here; it's certainly beyond the undergraduate curriculum as I know it. $\endgroup$
    – Todd Trimble ♦
    Nov 15, 2016 at 19:23
  • 1
    $\begingroup$ Yes, look in Carter's book "Simple Groups of Lie Type" (I misremembered the title when I wrote my previous comment). In particular, see 14.1; the 2nd page of 14.1 specifically addresses the Suzuki and Ree groups (and look in the bibliography too). That book treats all finite groups of Lie type, or rather various things about group of rational points mod the center (to get actual simple groups), but in so doing he works out the BN-pairs for the algebraic groups underlying those examples. There are ways to understand this using more about algebraic groups, but you don't mention your background. $\endgroup$
    – nfdc23
    Nov 15, 2016 at 22:19
  • 1
    $\begingroup$ There are a lot of relevant sources, such as Carter's 1972 book and my more recent one (LMS Lecture Note Series, Cambridge, 2006), as well as Steinberg's 1967-68 Yale lectures: math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf Or try Chapter 2 of the series of books by Gorenstein-Lyons-Solomon (AMS, vol. 40, no 3, 1998): especially 2.3.2, e.g., the BN-pair ranks of (Weyl groups of) Suzuki and Ree groups are half the Lie rank. But notation does vary a lot in the sources. $\endgroup$
    – Jim Humphreys
    Nov 15, 2016 at 22:34
  • 1
    $\begingroup$ Concerning the remark, the group in question has two completely different BN-pairs. In one of them B is the normalizer of a Sylow 2-subgroup (because the group is $^2A_3(2^2)$). In the other B is the normalizer of a Sylow 3-subgroup. Both BN pairs have rank 2. The subscript $3$ indicates that the rank of the BN pair in the overlying algebraic group is 3. $\endgroup$
    – Richard Lyons
    Dec 21, 2018 at 20:59

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.