26. Groups generated by reflections; root systems.

The groups considered in this Note appeared in connection with various questions of Geometry, Analysis and the Theory of Lie groups, sometimes in the form of permutation groups, sometimes in the from of groups of displacements in Euclidean or hyperbolic geometry, and these various points of view have only been co-ordinated in recent times.

Historically, the beginnings of the theory are a good deal older than the introduction of the concept of a group: it takes its source indeed in the studies on the “regularity” or the “symmetries” of geometrical figures, and notably in the determination of regular polygons and polyhedra (going back no doubt to the Pythagoreans), which constitute the crowning achievement of the Elements of Euclid, and one of the most admirable creations of the Greek genius. Later, notably with the Arab authors of the high Middle Ages, then with Kepler, the beginnings of a mathematical theory of regular “tilings” of the plane or the sphere by polygons congruent in pairs (but not necessarily regular) appear, no doubt linked to the origin of the different types of ornament thought up by the ancient and Arab civilisations (that can be considered with good cause as an authentic part of the mathematics developed by these civilisations [291]).

Towards 1830-1840, studies in crystallography (Hessel, Bravais, Möbius) lead to considering a problem that is exactly that of the determination of the finite groups of displacements in Euclidean space of 3 dimensions, although the authors quoted above do not use yet the language of the theory of groups; this latter hardly comes into use until around 1860, and it is in the context of the classification of groups that Jordan, in 1869 [174 b], determines the discrete subgroups of displacements in $\mathbf{R}^3$ which preserve orientation (and more generally, all the closed subgroups of the groups of displacements preserving orientation).

Until the last years of the XIXth century, this stream of ideas is developed in many directions, of which the most marked are the following:

1. Conforming to a tendency that appears very early on in the theory of finite groups, “presentations” of finite groups of displacements by generators and relations of a simple type are sought. It is thus that Hamilton, already in 1856 [145 b], proves that the finite groups of rotations in Euclidean space

R$^3$ are generated by two generators $S, T$ linked by the relations $S^p = T^q = (ST)^3 = 1$ for appropriate values of $p$ and $q$.

2. Discrete groups of displacements may or may not contain reflections. Already in 1852, Möbius determines substantially the finite groups of displacements in spherical geometry generated by reflections (which is equivalent to the same problem for finite groups of Euclidean displacements in $\mathbf{R}^3$); he finds that, with the exception of the cyclic groups, such a group has as fundamental domain a spherical triangle having angles of the form $\pi/p, \pi/q, \pi/r$, where $p, q, r$ are three integers > 1 such that $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$ ([223], v. II, pp. 349-360 and pp. 561-708). He also notices that these groups contain all the finite groups of displacements as subgroups.

3. This last stream of ideas finds a new amplification when following work of Riemann and Schwarz on hypergeometric functions and conformal representations, the study of "tilings" of the complex plane or of the half-plane by means of figures bounded by arcs of a circle is started; Klein and Poincaré make it the foundation of the theory of "automorphic functions" and recognise in it (for the case of arcs of a circle orthogonal to a fixed straight line) a problem equivalent to that of the search for the discrete subgroups of the group of displacements of the non-Euclidean hyperbolic plane (identified with the "Poincaré half-plane") [118].

4. The notions of regular polyhedron and of the tiling of $\mathbf{R}^3$ by such polyhedra are extended to all the Euclidean spaces $\mathbf{R}^n$ by Schlafli, in work that goes back to the neighbourhood of 1850, but was only published much later and remained ignored for a long time ([273], v. I, pp. 167-387); he determines completely the regular "polytopes" in each $\mathbf{R}^n$, the group of displacements leaving invariant such a polytope, and a fundamental domain of this group, which, as in the case $n = 3$ studied by Möbius, is a "room" whose trace on the sphere $S_{n-1}$ is a spherical simplex. However, he does not tackle the inverse problem of the search for the finite displacement groups generated by reflections in $\mathbf{R}^n$; this problem will only be solved much later, by Goursat [132] for $n = 4$, and, for arbitrary $n$, its solution will have to wait for the work of E. Cartan ([52 a], v. I_2, pp. 1003-1020) and of Coxeter [70 c], to which we will return later on.

Around 1890, with the first work of Killing and of E. Cartan on Lie groups, begins a new stream of ideas that for a long time will develop without links to the preceding work. Killing [180] and Cartan ([52 a], v. I_1, pp. 137-287), in their study of the structure of complex semisimple Lie algebras, will plan straightaway a primordial role for certain linear forms $\omega_\alpha$ on a "Cartan sub algebra" $\mathfrak{h}$ of such a Lie algebra $\mathfrak{g}$; they are the "roots" relative to $\mathfrak{h}$, so called because with Killing they appear as the roots of the characteristic equation $\det(\operatorname{ad}_\mathfrak{g}(x) - T) = 0$, considered as functions of $x \in \mathfrak{h}$. The properties of these "roots" established by Killing and Cartan reduce to affirming that, in the actual geometric language, they form a "reduced system of roots";

they show then that the classification of the complex semisimple Lie algebras is reduced to that of associated "systems of roots", which itself reduces to the determination of certain matrices with integer coefficients (later called "Cartan matrices"). Killing and Cartan also bring out, for every root $\omega_\alpha$, the existence of an involutive permutation $S_\alpha$ of the set of roots; they use in an essential way the transformation $C = S_{\alpha_1} S_{\alpha_2} \ldots S_{\alpha_l}$, the product of permutations associated with $l$ roots making up a fundamental system (a transformation today called a "Coxeter transformation"); they even extend this permutation into a linear transformation of the vector space generated by the fundamental roots $\omega_{\alpha_i} (1 \leq i \leq l)$, and study its eigenvalues ([180], II, p. 20; [52 a], v. I_1, p. 58). But neither Killing, nor at first Cartan, seem to have thought of considering the group $G'$ generated by the $S_\alpha$; and when Cartan, a little later ([52 a], v. I_1, pp. 293-353), determines the Galois group $G$ of the characteristic equation

of a "general element" $x \in \mathfrak{h}$, he studies it at first without bringing in the $S_\alpha$; thirty years later, already under the influence of the work of H. Weyl, he proves ([52 a], v. I_1, pp. 555-568) that $G$ has as a normal subgroup the group $G'$ and determines in all cases the structure of the quotient group $G/G'$ which (for a simple algebra $g$) is or order 1 or 2 except for the type $D_4$ where it is isomorphic to $S_3$; it is also on this occasion that he interprets $G'$ as the group induced by the inner automorphisms of a complex semisimple Lie algebra, leaving fixed a Cartan subalgebra.

The work of H. Weyl, to which we have just alluded, is that which inaugurated the geometric interpretation of the group $G'$ (since called the "Weyl group" of $g$); in the same way as Killing and Cartan had done it for the transformation $C$, he had the idea of considering the $S_\alpha$ as reflections in the vector space of linear forms on $\mathfrak{h}$. It is also in the memoir of H. Weyl [331 b] that the fundamental domain of the "affine Weyl group" is seen to appear (without besides the link with the "Weyl group" $G'$ being clearly indicated); Weyl uses it in order to prove that the fundamental group of a compact semisimple group is finite, a crucial point in his proof of the complete reducibility of the linear representations of a complex semisimple Lie algebra. Soon after, E. Cartan brings to fruition the synthesis of the global points of view of H. Weyl, of his own theory of real or complex semisimple Lie algebras, and of the theory of symmetric Riemann spaces that he was building at this time. In the memoir ([52 a], v. I_2, pp. 793-840), he completes the determination of the fundamental polytopes of the Weyl group and of the affine Weyl group, and introduces the weight systems and radical weight systems; he extends this discussion to symmetric spaces, and meets thus notably the first examples of non reduced root systems ([52 a], v. I_2, pp. 1003-1010). Finally the article ([52 a], v. I_2, pp. 867-989) gives the first proof of the fact that every finite group generated by reflections in $\mathbf{R}^n$ and irreducible has a fundamental domain having as trace on $S_{n-1}$ a spherical simplex; it is also in this work that he proves the uniqueness of the largest root (under an arbitrary lexicographic ordering on the system of roots) by geometrical considerations.

A little later, van der Waerden [317 b], relying on the memoir of H. Weyl, shows that the classification of the complex semisimple Lie algebras is equivalent to that of the reduced root systems, which he carries out by elementary geometric considerations (whereas, with Killing and Cartan, this classification is a result of complicated calculations with determinants). At about the same time, Coxeter determines explicitly all the finite irreducible groups of Euclidean displacements which are generated by reflections [70 c]; he completes thus the results of the memoir ([52 a], v. I_2, pp. 793-840) of E. Cartan, who had only determined the "crystallographic" groups (i.e. associated to a root system, or also susceptible of being embedded in an infinite discrete group of displacements). The following year [70 d], Coxeter shows that the finite groups generated by reflections are the only finite groups (up to isomorphism) admitting a presentation by involutive generators $R_i$ subject to relations of the form $(R_i R_j)^{m_{ij}} = 1$ ($m_{ij}$ integers), whence the name of "Coxeter groups" given since to groups (finite or not) admitting such a presentation.

The first link between the two streams of research that we have described above seems to have been established by Coxeter [70 e], then by Witt [337 c]. They notice that the infinite irreducible groups of Euclidean displacements generated by reflections correspond bijectively (up to isomorphism) to complex simple Lie algebras. Witt gives a new determination of discrete groups of this type, and extends also the theorem of Coxeter [70 d] recalled above by characterising as well the Coxeter groups isomorphic to infinite discrete groups of Euclidean displacements. This result, and the fact that the analogous groups in hyperbolic geometry are also Coxeter groups,$^1$ have led to the open tackling of the study of these latter, first by putting the emphasis on a geometric realisation ([71], [308 a]), then, following J. Tits [308 b and c] in a purely algebraic framework.

Starting with the work of Witt, the theory of semisimple Lie groups and that of discrete groups generated by reflections will not cease from reacting extremely fruitfully the one on the other. Already from 1941, Stiefel [296] remarks that the Weyl groups are exactly the finite groups generated by reflections, and that leave a network invariant. Chevalley [62 e] and Harish-Chandra [149 a] give in 1948-51 proofs a priori of the bijective correspondence between "crystallographic" groups and complex semisimple Lie algebras; it was not possible until then to verify separately this correspondence for each type of simple Lie algebra.

Around 1950, it is noticed on the other hand that the polynomials invariant under the Weyl group play an important role in the theory of linear representations of infinite dimension [149 a] and in the topology of Lie groups.

$^1$ These groups, studied in depth in the case of dimension 2, have not been considered in dimension $\geq 3$ except incidentally until these last years.

On his side, Coxeter [70 g], taking up again the study of the transformation $C$, product of the fundamental reflections of a finite group $W$ generated by reflections, notices (by means of a separate examination of each type) that the algebra of polynomials invariant under $W$ is generated by algebraically independent elements, whose degrees are linked in a simple way to the eigenvalues of $C$. Proofs a priori of these results were subsequently given by Chevalley [62 f] for the first, and by Coleman [68] and Steinberg [292] for the second.

With the work of A. Borel on linear algebraic groups [30] new developments in the theory of Lie groups start which were to lead to a notable enlargement of this latter. A. Borel brings out the importance of the maximal connected soluble subgroups (since called "Borel subgroups") in a Lie group, and makes of them the principal tool for transposing a large part of the classical theory to algebraic groups over an algebraically closed field (without however obtaining yet a classification of the simple algebraic groups$^2$). The Borel subgroups (in the case of real or complex classical groups) had already turned up some years previously in the work of Gelfand and Neumark on the representations of infinite dimension; and in 1954, F. Bruhat had discovered the remarkable fact that, for the classical simple groups, the decomposition of the group into double cosets over a Borel group is indexed in a canonical way by the Weyl group [40]. This result was subsequently extended to all real and complex semisimple groups by Harish-Chandra [149 b]. On the other hand, in 1955, Chevalley [62 g] had succeeded in associating with every complex semisimple Lie algebra $\mathfrak{g}$ and with every commutative field $k$, a group of matrices with coefficients in $k$, possessing a Bruhat decomposition; and he used this last fact to show that, with a small number of exceptions, the group thus defined was simple (in the sense of the theory of abstract groups). He "explained" thus the coincidence, already noticed since Jordan and Lie, between the simple Lie groups (in the sense of the theory of Lie groups) of types $A, B, C, D$ and the classical simple groups defined in a purely algebraic way over an arbitrary field (a coincidence that until then could not be extended except to the exceptional types $G_2$ and $E_6$ by Dickson [88 c and d]). In particular, by taking a finite field $k$, the construction of Chevalley supplied, for each type of complex simple Lie algebra, a family of finite simple groups, containing a large part of the then known finite simple groups, as well as three new series (corresponding to the types of simple Lie algebras $F_4, E_7$ and $E_8$). Soon after, by various procedures, using modifications of the methods of Chevalley, several authors (Hertzig, Suzuki, Ree, Steinberg and Tits) on the one hand showed that one can obtain in an analogous way the other finite simple groups known at that time, with the exception of the al-

$^2$ An algebraic group of dimension $> 0$ is said to be simple (in the sense of algebraic geometry) if it does not contain any normal algebraic subgroup of dimension $> 0$ other than itself. It is said to be semisimple if it is isogeneous to a product of simple non commutative groups.

ternating groups and the Mathieu groups, and on the other hand constructed other series of new finite simple groups (cf. [54]).

At almost the same time, Chevalley [62 h], still using the technique of Bruhat decompositions, joined to a key result on the normaliser of a Borel subgroup, took up again the study of linear algebraic groups and reached the result that over an algebraically closed field $k$ of arbitrary characteristic, the theory of semisimple linear algebraic groups$^3$ leads essentially to the same types as in the Killing-Cartan classification for $k = \mathbf{C}$. Following on, J. Tits [308 a and b], analysing the methods of Chevalley, reached an axiomatised version (the “BN-pairs”) of the Bruhat decompositions, in a remarkably flexible form, only bringing in the structure of the group; it is this notion that is at present known under the name “Tits system”. All the simple groups (in the various meanings of the word) of which we have been speaking above are canonically supplied with Tits systems, and Tits himself [308 c] has proved that the existence of such a system in an abstract group $G$, linked with a few supplementary properties from pure group theory, allows the proof that $G$ is simple, a theorem which covers the majority of the proofs of simplicity given until then for these groups. In collaboration with A. Borel, he has on the other hand generalised the results of Chevalley of [62 h], by showing the existence of Tits systems in the group of rational points of a semisimple linear algebraic group over an arbitrary field [31].

All the Tits systems occurring in these questions have a finite Weyl group. Another category of examples was discovered by Iwahori and Matsumoto [170]; they have shown that if, in Chevalley’s construction of [62 g], $k$ is a $p$-adic field, then the group obtained has a Tits system whose Weyl group is the affine Weyl group of the complex semisimple Lie algebra with which one started. This result has just been extended by Bruhat and Tits [41] to all semisimple algebraic groups over a local field.

$^3$ The existence of numerous “pathological” simple Lie algebras over a field of characteristic $p > 0$ could have led some to doubt the universal character of the Killing-Cartan classification.