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The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and triangular $B$ matrices and come with a "Weyl group" $N/(B\cap N)$, that has to be a Coxeter group.

Conversely, Tits showed in 1974 that a group with a spherical BN-pair of rank at least 3 is of "Lie type". (Ironically, the general group identification needed in the classification for lowest rank 3 was the "quasi-thin" case solved last by Aschbacher).

Now the existence of a rank 1 BN-pair for a group $G$ is equivalent to the existence of a doubly-transitive action of $G$ on a set $X$ (which can be taken to be $G/B$). This implies that also the sporadic simple Mathieu groups have a BN-pair (of rank 1).

Now my question: The construction of the Monster group uses also a rank 1 "BN-construction" that is not proper: The "triality" element in the Weyl group $S_3$ takes the (non-normalizing) role the transpositions ought to have. Sometimes the term "weak BN-pair" is mentioned. However, I could not find a proper definition. Can one briefly explain this concept and how it is related to the usual BN-pairs? Is it as generic or rather ad-hoc? Can one suggest good introductory literature?

EDIT: Found e.g. in appendix F of Aschbachers "Classification of Quasithin groups"

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1) It's unclear to me how far the BN-pair formalism can be helpful in rank 1 situations (especially if weakened further). It seems no one yet finds the sporadic groups approachable, uniformly or otherwise, in the spirit of groups of Lie type. Tits relied heavily on associated geometry, which is visible only for rank at least 3. In rank 2 the narrower notion of "split" BN-pair led by more algebraic methods to a definitive treatment by Fong-Seitz in their Invent. Math. papers (1973-74). 2) "Sometimes the term ... is mentioned": any recollection of where or by whom? –  Jim Humphreys Apr 8 '12 at 19:53
    
I'll have a look at the paper you suggested, thanx! Well, the term already appears in appendix F of Aschbachers highly influencial "Classification of Quasi-Thin Groups" (As I understand that's the topic on which he closed the classification theorem?) But I have NO-CLUE what he's talking about and I can't see the connection e.g. to the introduction "Buildings" by Brian Lehmann. (HELP ;-) ) Also, it's certainly no accidient, the BN-pair of the Monster is named that way and exactly looks like the one of a doubly transitive group? And this seems like THE road to the monster? –  Simon Lentner Apr 8 '12 at 23:06
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Just to clearify notions: You speak about the Tits Buildings and say their theory (which part?) is only that much help, if it's "thick", i.e. each residue has at least three chambers? (In contrast a Coxeter building, which is what the "appartments" are, has only exactly two) –  Simon Lentner Apr 9 '12 at 9:20
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Just noting: The rank 1 situation can be formalized to what is called "split BN-pair of rank one" or "Moufang sets" (which do not include the Mathieu groups). See for example this survey by Tom De Medts and Yoav Segev: cage.UGent.be/~tdemedts/preprints/moufsets.pdf . –  Koen S Apr 9 '12 at 9:32
    
For weak BN-pairs of rank 2 you can take a look at: A. Delgado, B. Stellmacher Weak BN-pairs of rank 2, in A. Delgado, D. Goldschmidt, B. Stellmacher, Groups and graphs: new results and methods, DMV Seminar, 6. Birkhaeuser Verlag, Basel, (1985) 244 pp. and at math.uni-bielefeld.de/groups2012/talks/… and at math.msu.edu/~meier/Preprints/CGP/cgp_abstract.html –  j.p. Apr 12 '12 at 10:07

2 Answers 2

up vote 4 down vote accepted

There has been a lot of work done on various generalizations of the concept of the building, to apply them to sporadic groups. These generalizations are variously known as diagram geometries, chamber systems, etc. Names like G.Stroth, S.Smith, M.Ronan, A. Delgado, D. Goldschmidt, B. Stellmacher, etc. spring to mind. There is an "elementary" book on diagram geometries by A.Pasini (a review of the latter is here.) There is a series of books by A.A.Ivanov (some of them are jointly with S.Shpectorov) developing a theory of this sort to deal with a majority of sporadics.

Indeed, one needs a weakening of the classical buildings to cover sporadics. Instead of starting from a weak BN-pair, one can weaken Tits' axioms from his "Local approach to buildings" to develop a theory dealing with sporadics. E.g. Witt designs for Mathieu groups (already from 1938) are extensions (in certain well-defined way) of the affine plane of order 3 and of the projective plane of order 4. Similar things can be done with $HS$, $Suz$, Fischer's sporadic groups, $He$, $McL$, $Co_3$, $Co_2$, and $BM$. (E.g. --- cannot resist citing myself here: the 3-transposition graph for $Fi_{22}$ can be characterized as the extension of the polar space for $U_6(2)$.) This appears to work when the underlying combinatorics is not too complicated (and the corresponding permutation representation has low rank).

Regarding the $BN$-pairs approach, I must say I don't recall details, having done very little work on these things in past 15 years. In a nutshell, one cannot hope for "real" apartments, etc., so one instead looks at amalgams of parabolic subgroups. Instead of a definition, let me giev you a toy example, $GL_4(2)$ and its Borel subgroup $B$ (taken to be the upper-traingular matrices, say). Then you have "minimal parabolics" $P_i$, i.e. subgroups generated by $B$ and $e_{i+1,i}$, for $i=1,2,3$ (here $e_{ij}$ denotes the matrix with 1 at position $ij$ and on the main diagonals, and 0s elsewhere). Then, you get maximal parabolics, $P_{ij}$, generated by $P_i\cup P_j$. This is what is called a rank 3 amalgam (as you have 3 minimal parabolics).

Your geometry then consists of cosets of $B$, $P_i$'s, $P_{ij}$'s in the whole group and in each other. The amalgam is now the set-theoretic union of $B$, $P_i$'s, $P_{ij}$'s, and you can study its universal completion, i.e. the biggest group where is can be embedded into. By tweaking the groups which can arise as $B$, $P_i$'s, $P_{ij}$'s, one covers more cases than buildings, and tries to stay away from infinite universal completions for ranks at least 3.

PS. IMHO, Aschbacher sometimes tends to ignore prior work, re-inventing the wheel in different terminology.

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GREAT, THANK YOU! I know of course some Witt designs, most importantly the Golay code ;-) I understand your answer as these should be treated as a generalization of Tits Buildings and their automorphisms form sporadic groups (thanx also for your contribution ;-) ) But it's still unclear to me, how this describes the group in a BN-pair manner....is there somethings like apartments (maybe the "lines"?) And how come, the monster still doesn't appear - isn't there any "geometry" behind the choices $N\tilde S_3\times M_24$ and $B\tilde Co_1$ as appartment- and chamber-stabilizers? –  Simon Lentner Apr 15 '12 at 16:41
    
I updated my answer, trying to address this. –  Dima Pasechnik Apr 15 '12 at 17:55
    
Thank you again, I think I start to understand now :-) (and I think that counts for an answer!)...I just still wonder, how the Monster and it's wellknown $B,N$ appear in this context. Your list contains no "classified Geometry" associated to it (there still might be an "arbitrary one"?).....BUT your rank-3-amalgams reminded me VERY MUCH on a simplified construction by Conway I know - can you confirm that? Do the $N_{x,y,z}$ correspond to some $P_{1,2,3}$? –  Simon Lentner Apr 16 '12 at 18:19
    
I gave a list of sporadics for which "combinatorial" characterizations can be shown (i.e. you construct an object for which the group in question turns out to be the automorphism group), as opposed to the amalgams approach, where one assumes right from the start that a group is around (the geometry is still there, but is assumed to be flag-transitive right from the start). Amalgams were used to characterize groups like $J_4$, $M$, $BM$, other large (or not so large) sporadic groups (see e.g. aforementioned books by A.A.Ivanov et al). –  Dima Pasechnik Apr 17 '12 at 2:39

Thanx to Humphreys and Koen for providing, that rank 1,2 is "geometrically unsatisfying" in the sense of Tits buildings and suggesting the notion of split BN-pairs as a tightning. But looking at the suggested source (and others) I'm less convinced this helps answering the question....(?)

Split BN-Pairs of rank 2 (or "Moufang polygons"?) were classified (as you said) by Fong-Seitz and the result is (online e.g. in ">this secondary PDF), that the're basically:

$$A_2,\;,B_2,\;^2A_4,\; ^2A_5, \; G_2,\;^3D_4,\;^2F_4$$

Split BN-Pairs of rank 1 (or Moufang sets) were shortly before classified by Hering, Kantor, Seitz (online as PDF here) as close to (does the occasional double-occurence to above worry me?):

$$A_1,\; ^2B_2, \; ^2A_2,\;^2G_2,\;\text{or sharply 2-transitive}$$

However none of these appears to me a sporadic simple group? It still seems to me, that the prominent $B,N$-construction (e.g. Griess himself) of the monster is rather a weakening. Keeping things provable might be the reason for the complicated definition of a weak BN-Pair in Aschbachers Book (s.a.)- any clues on this? I couldn't even convince myself this IS the notion we look for?

...BUT I found another source: Parker e.g. uses weak BN-pairs of rank 2 (!) to distinguish odd characteristic Lie groups paper e.g. here. And I found (appearently quite different) weak BN-Pairs of odd order p...

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