Symmetric Generation of Not-So-Finite (Real) Reflection Groups by Ben Fairbairn

Symmetric Generation of Not-So-Finite (Real) Reflection Groups Abstract This is joint work with RT Curtis and J M¨ller. The remarkable u techniques of symmetric generation have furnished almost effortless constructions of a variety of interesting groups. In this talk we shall first discuss the existence of involutory symmetric generating sets for real reflection groups meeting certain finiteness conditions. This represents the first examples of infinite groups to succumbe to the techniques of symmetric generation. 1 (Real) Reflection Groups We first recall some basic definitions pertaining to real reflections groups. Since much of the audience should already be familiar with most of this, we only give a surface scratching overview. Furthermore much of it can easily be found in standard references such as [2], [3] or [10] Let V be an n dimensional R vector space endowed with a positive definite symmetric bilinear form (and thus a sensible concept of angle). Recall that a linear map α ∈ End(V ) is a reflection if there exists a vector v ∈ V such that α(v) = −v and the subspace v ⊥ is fixed pointwise. (We remark that stated differently, v ⊥ is the 1-eigenspace of α and v is a -1 eigenvector. The “Real” in the title of this talk is only in parentheses as it would be desirable to generalize this to the situation where v is a ζ eigenvector where ζ is a root of unity, a natural generalization given what comes next. This of course would require using a field other than R such as C. There are certain technical issues blocking this that need to be resolved before I can do this. Please contact me later, if you any ideas on how to do this.) A group G ≤ GL(V ) is a reflection group (or Coxeter Group) if it is generated by reflections. Of course a randomly chosen set of reflections will almost certainly generate an infinite group, making finite reflection groups quite exceptional and therefore quite interesting. These were complete classified by Coxeter in the 1930s (though if Conway and Sloan are to be believed Mitchel “had already solved an essentially harder problem”.) Whilst we will not describe the full classification here (its not hard, but takes us somewhat off coarse) we will describe some of the ideas used to classify them as they will be of interest later. Given any reflection group G generated by the reflections α1 , . . . , αr it may be shown that G is uniquely determined by the presentation x1 , . . . , xr |x2 = . . . = x2 = (xi xj )|αi αj | = 1 1 r with such a presentation we associate a Coxeter diagram (or Coxeter-Dynkin diagram) defined as follows. Let Γ be a graph with one vertex (or node) for each reflection αi . We adjoin the vertices with edges labeled with positive integers according to the following rules. • if |αi αj | = 2 then the nodes i and j will not be adjoined by an edge. • if |αi αj | = 3 then the nodes will be adjoined by an unlabeled edge. 1 • if |αi αj | = mij > 3 then the nodes will be adjoined by an edge labeled with the integer mij . The labels on the edges are called the Coxeter numbers. Note in particular that we always have mii = 1. This allows us to describe many of the properties of our generating set in graph theoretic language. This proves useful in classifying the finite reflection groups as one ends up proving results such as ‘the Coxeter diagram of a finite reflection group is acyclic’ or ‘no node of the Coxeter diagram of a finite reflection group has degree greater than 3’ etc. This makes the classification substantially easier and indeed possible. An example: Let {e1 , . . . , en } be an orthonormal basis for V . The symmetric group Sn can clearly act on V by simply permuting these vectors. How does the transposition (12) act? Note first that the vector e1 − e2 is mapped to the vector −(e1 − e2 ). Furthermore the vectors e1 + e2 , e3 , . . . , en are fixed, so transpositions are reflections. In particular note that the whole of Sn may be generated by the transpositions (12), (23), . . . , (n − 1, n). The Coxeter diagram is thus little more than a path with (confusingly) n − 1 nodes. We call these diagrams the diagrams of type An (when there are n nodes) for historical and archaic reasons. Given a Coxeter diagram of type Φ we write W (Φ) for the corresponding reflection group, called the Weyl group of the diagram (after Herman Weyl). We will also mention one further class of reflection groups that prove useful in classifying the finite reflection groups. An affine reflection is a reflection about a hyperplane that does not necessarily pass through the origin (so is a genuine reflection followed by a translation in some direction not fixing the hyperplane). Note in particular that every reflection is an affine reflection. A group generated by affine reflections is an affine reflection group. These are always infinite, but in a very controlled way: defining the Coxeter diagram in an analogous way (except for allowing edges to be formally labeled ∞ if the product does not have finite order) it turns out that the Coxeter diagram of an affine reflection group has the property that removing any node leaves the Coxeter diagram of a finite reflection group. This will prove to be quite interesting in what follows. We note that in this talk we shall only be considering Coxeter diagrams that are connected. This not only avoids certain technical issues later on (though our main theorem can indeed be adapted to the more general situation where our diagram is not connected) but is also extremely natural given that the reflection group corresponding to a diagram that is not connected is little more than the direct product of the reflection groups corresponding to the connected components of the diagram. 2 Involutory Symmetric Generation Even if you got a bit lost in the last section, we shall now (temporarily) talk about something completely different. We write 2 n for the free group generated by n involutions. We shall write ti for one of its generators so that 2 n = ti |1 ≤ i ≤ n . We shall call these ti s the symmetric generators. Let N ≤ Sn be a permutation group acting transitively on n points. The permutation π ∈ N may be used to define an automorphism 2 of 2 n in the natural way: ˆ tπ := tπ(i) i using this action we are able to define the semi-direct product P := 2 n : N . (Members of the audience may be more familiar with the notation 2 n N , but I shall stick to Atlas notation to avoid confusing myself and thus by extension the entire audience!) We shall call P a progenitor and the group N the control group of the progenitor. Note we shall slightly abuse notation and terminology by referring to both N and its image in G as the control group and similarly with the symmetric generators. The above action is defined in such a way that we are able to write any element of P in the form πw where π ∈ N and w is a word in the ti s. Consequently if H ≤ P is a finitely generated subgroup we can write H = π1 w1 , . . . , πr wr . In particular if H is a normal subgroup of P we can write the factor group as 2 n:N =: G. π1 w1 , . . . , πr wr In this situation we refer to the words πi wi as relations and say that we have factored the progenitor P by the relations π1 w1 , . . . , πr wr . We call such an expression a symmetric presentation of G. How can we determine whether or not G is finite? We use the orbit stabilizer theorem. To this end we define the coset stabilizing subgroup of the word in the symmetric geenrators w to be the subgroup of the control group defined by N (w) := {π ∈ N |N wπ = N w}. This is clearly a subgroup of N and there are |N : N (w) | right cosets of N (w) in the double coset N wN ⊂ G. We can enumerate these cosets using procedures such as the celebrated Todd-Coxeter algorithm. The sum of these numbers then gives the index of N in G. We can thus determine the order of G and in particular prove it is finite. Given a particular progenitor which relations should we factor it by? The following lemma, whilst easy to prove (although the constraints of time preclude us from giving a proof here), turns out to be extremely powerful in determining such relations. Lemma 1 ti , tj ∩ N ≤ CN (StabN (i, j)) This provides motivation for considering symmetric generating sets in the first place: given a progenitor, simply asking which relations are the most natural to factor it by can lead to extremely elementary constructions of extremely non-trivial groups, most notably the sporadic simple groups, though we shall not make any reference to these constructions here. (The interested listener should consult Rob Curtis’ recent book [6].) In addition I will mention another, more practical, application of these ideas that I worked on earlier in my PhD. The Conway group ·0 has order 8315553613086720000. Consequently representing elements of this group is not easy. Using a symmetric presentation of ·0 due to Bray and Curtis [4] we can represent elements of ·0 as a string of at most 64 symbols and typically far fewer. This represents a considerable saving compared to representing an element of 3 ·0 as a permutation of 196560 symbols or as a 24 × 24 matrix (ie as a string of 242 =576 symbols). A couple of years ago myself and Rob Curtis figured out how to interpret the double cosets geometrically in terms of configurations of vectors in the Leech lattice on which ·0 naturally acts called crosses and as a consequence we were able to devise an algorithm for multiplying together elements of ·0 that are symmetrically represented in the manner described above. This will hopefully appear in the journal of symbolic computation in due course [7]. 3 Some Natural Actions of Sn What could be more natural than the action of Sn on n objects? We thus consider the progenitor 2 n : Sn . Applying Lemma 1 we have StabSn (1, 2)“ ∼ ”Sn−2 and so CSn (StabSn (1, 2))“ ∼ ” (12) = = ∼ ” to mean that for all but finitely many values of n we have where we write “ = an isomorphism. (In the few cases of small n where we have different values of n interesting things do happen, but we shall not mention those here.) We thus have that a word in t1 and t2 can only possibly equal the permutation (12). The relation (12) = t1 t2 cannot possibly work since we then also have (12) = t2 t1 and (13) = t1 t3 and so (123) = (t2 t1 )(t1 t3 ) = t2 t3 = (23) and so the image of the control group is not faithful. We are thus naturally led to the relation (12) = t1 t2 t1 . Can we enumerate (id) (id) these cosets? Well, we first see that Sn = Sn and so |Sn : Sn | = 1. Next (t ) (t ) Sn 1 ∼ Sn−1 and so |Sn : Sn 1 | = n. If we try to continue this process, however, = (t ) (t t ) we find that t1 t2 = (12)t1 ∈ Sn t1 ⊂ Sn t1 Sn thus Sn 1 2 = Sn 1 and the process stops here. We thus have |G : Sn | = n + 1. To actually verify that G is what we think it is, we need to actually exhibit elements of the group we are hoping for. We aim for permutations. The smallest number of points we can make the control group, Sn , act on faithfully is n points, but since we already have every permutation on n points to hand, the smallest number of points we can make our target group permute is n + 1. Which permutations in Sn+1 commute with (t ) Sn 1 ? Visibly the only non-trivial permutation that does this is (1, n + 1). It is easily checked that this satisfies the additional relation and visibly has order two, so we have that G ∼ W (An ) and so = 2 n : Sn ∼ = Sn+1 ∼ W (An ). = (12)t1 t2 t1 How can we generalize this? It may seem slightly odd at first, but if you think about it the action of Sn on a set of n objects is exactly the same as the action of Sn on the subsets of size one. It is therefore natural to consider the n action on the subsets of size two. Consider the progenitor 2 ( 2 ) : Sn for n ≥ 4. Again applying Lemma 1 we have StabSn (12, 13)“ ∼ ”Sn−3 and so CSn (StabSn (12, 13))“ ∼ ” (12), (123) . = = motivated by the relation that worked so well in the first case, we consider the relation (23)t12 t13 t12 . This time the coset enumeration is a little more 4 involved requiring a couple of general lemmata, but roughly speaking you can show that representatives for the double cosets Sn gSn are all of the form g = t12 t34 . . . t2k−1,2k for some k ≤ n/2 ie there is one double coset for every subset of {1, . . . , n} of even size. In particular we have that |G : Sn | = 2n−1 . Furthermore, by a little playing with the relation it may be shown that the elements of the form tij (ij) all commute, ie they generate a copy of the elementary abelian group of order 2n−1 and lies entirely outside the control group. We are thus naturally led to consider a representation in which the control group acts as permutation matrices and the symmetric generators act as the matrices   -1   -1     1 t12 =  .   ..   . 1 In particular we have verified the following symmetric presentation 2 ( 2 ) : Sn ∼ = W (Dn ) ∼ 2n−1 : Sn . = (23)t12 t13 t12 Well, if a trick works once... We consider the action of the symmetric group Sn on subsets of size three n and thus define the progenitor 2 ( 3 ) : Sn for n ≥ 6. Again applying Lemma 1 we have that StabSn (123, 124)“ ∼ ” (1, 2) ×Sn−4 and so CSn (StabSn (123, 124))“ ∼ ” (12), (3, 4) . = = motivated by the relations that worked so well in the earlier cases, we consider the relation (34)t123 t124 t123 . This time the coset enumeration needs to be done on a case by case basis (using the double coset enumeration program of Bray [1]). We find that the enumeration only terminating in the cases n = 6, 7 and 8. The process of finding matrices satisfying the relations is the same as before. Again representing our control group with permutation matrices we are compelled to find matrices satisfying the conditions 1. commute with the stabilizer of a symmetric generator 2. have order two 3. satisfy the relation. Condition 1 forces us to consider matrices of the form:     aI3 +bJ3   =      c J3 n             cJ3 a I3 +b J3 5 t123 while conditions 2 and 3 leads to the relations: (aI3 + bJ3 )2 + 3cc J3 = (a I3 + b J3 )2 + 3cc J3 = I3 c(a + a + 3b + 3b ) = c (a + a + 3b + 3b ) = 0 a2 = a 2 = 1 2ab + 3b2 + 3cc = 2a b + 3b 2 + 3cc = 0. (a + 3b)(a + 3b ) = −1. After some staring ummming and errring we eventually led to the matrices.     2  I3 + J3 3   =      03 1 3 J3            t123 I3         =       I3 - 2 J3 3 03×4 1 3 J4×3 t123 I4            , t123 =            I3 - 2 J3 3 03×5 1 3 J5×3 I5            How to identify the groups? Well, note that the above matrices can all be turned into matrices over the field of two elements by multiplying all of them by 3 and reading them mod 2. The matrices now become the following.        =      I3 03 J3 I3 t123                 =      I3 03×4 6 J4×3 I4 t123               =      I3 03×5 J5×3 I5 t123          What’s the group? Well, since we’re considering a matrix group over F2 we seek a bilinear form that may be preserved by these. Note that since we have a large symmetric group floating around, there’s only three possible form any bilinear form preserved by both the control group and the above matrices namely i=n i=n x2 or i i=1 i=j xi xj or i=1 x2 + i i=j xi xj . Well, three is not a big number to check and indeed in each case one of the above is preserved by the above matrices. In each case we therefore have a quadratic bilinear form preserved by the group and in particular, since we know the orders (the enumerations provide upper bounds and the characteristic 0 matrices a lower bound), we can now identify our groups as being the Coxeter groups of type W (En ) n = 6, 7, 8 (accepting that in the F2 representations the center must act trivially). This result was written up and sent to the Journal of Algebra - who promptly rejected it! The rejection was, however, a little non-standard. The referee, recognising that I was very junior, decided to break anonymity and explain why the paper was rejected: the referee, J¨rgen M¨ller, smelt something more u u general lurking in the shadows. This at first seems a little strange. Deleting an arbitrary node from a (finite) Coxeter diagram gives a control group that is far from natural and deleting the neighboring nodes gives the full centralizer of the reflection corresponding to the deleted node and thus enables us to give a symmetric presentation of the reflection group under consideration in terms of a strange action of strange group. Rather pointless and unnatural, surely? 4 What Happened Next... So after a drastic rewrite of the introduction, the paper was then submitted to Communications in algebra, who are still considering its fate [8]. However, in November of 2008, J¨rgen M¨ller emailed back telling me that u u he had actually proved the more general result he was after. In fact in carefully analysing why my earlier results worked he was able to show that it can work even more generally than in the case of finite groups leading to the results of [9]. To explain what was proved we shall first need a little more notation. Let W be a finitely generated Weyl group with distinguished generators {s1 , . . . , sn } and with Coxeter numbers mij ∈ N ∪ {∞} (mij = ∞ if si sj does 7 not have finite order) recalling that mii = 1 for all i. We write F (σ1 , . . . , σn ) for the free group generated by σ1 , . . . , σn so that by definition W ∼ F (σ1 , . . . , σn )/ (σi σj )mij , 1 ≤ i, j ≤ n = Now, let W ∼ F (σ1 , . . . , σn−1 )/ (σi σj )mij , 1 ≤ i, j ≤ n − 1 . = In addition to the above we are going to make the additional finiteness assumption: m := |W : CW (sn )| < ∞. Under this additional assumption we define w1 := e, w2 , . . . , wm ⊂ W to be a set of right coset representatives for CW (sn ) in W and (suggestively) set tk := (sn )wk . This enables us to define a permutation representation π : W → Sm . We can thus define the progenitor P := F (σ1 , . . . , σn−1 , τ1 , . . . , τm )/ (σi σj )mij , τ 2 , (τk )σi τkπ(σi ) ∼ 2 = With this we prove the following. Theorem 2 (BTF and J M¨ ller - 2009) u W ∼ P/ (σi τ1 )m1n such that m1n ≥ 3 = By way of a “sketch proof” we shall only say the following - the thinly veiled disguise (the σs are the ss and the τ s are the ts) is removed using a natural epimorphism in each direction. Full details may be found in [9]. Note that this result gives us the very first examples of symmetric presentations of infinite groups - deleting any node of any affine reflection group leaves the diagram of a finite group, so the finiteness assumption on CW (sn ) must hold and the above theorem applies meaning any node in the diagram of any affine reflection group may be used to provide a generating set. (For those who know about such things, most nodes in every diagram corresponding to a compact hyperbolic reflection group also exhibit this property, making those examples too.) m :W References [1] JN Bray and RT Curtis “Double Coset Enumeration of Symmetrically Generated Groups” J. Group Theory 7 (2004), p.167-185 [2] CT Benson and LC Grove “Finite Reflection Groups” Grad. Texts in Math. no 99, Springer-Verlag (1985) ´e [3] N Bourbaki “El´ments de math´matique Fasc. XXXIV: Groupes et alg`bres e e de Lie, Chapitre IV: Groups de Coxeter et syst´mes de racines” Acualit´s e e Scientifiques et Industrielles 1337, Herman, (1968) 8 [4] JN Bray and RT Curtis “The Leech Lattice, Λ and the Conway Group ·0 Revisited” accepted by the Transactions of the AMS [5] JH Conway, RT Curtis, SP Norton, RA Parker and RA Wilson “An ATLAS of finite groups”, OUP (1985) [6] RT Curtis “Symmetric Generation of Groups with Applications to many of the Sporadic Finite Simple Groups”, Encyclopedia of Mathematics and Its Applications 111, CUP (2007) [7] RT Curtis and BT Fairbairn “Symmetric Representation of the Elements of the Conway Group ·0” accepted subject to amendments, J. Symbolic Comput. [8] BT Fairbairn “Symmetric Presentations of Coxeter Groups” submitted to Comm. in Algebra [9] BT Fairbairn and J M¨ller “Symmetric Generation of Coxter Groups” acu cepted by Archiv der Mathematik [10] JE Humphreys “Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics)” Cambridge studies in advanced mathematics, CUP (1997) 9