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**Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV, V et VI: Groupes de Coxeter et systèmes de Tits. Groupes engendrés par des réflexions. Systèmes de racines.**
*(French)*
Zbl 0186.33001

Actualités Scientifiques et Industrielles. 1337. Paris: Hermann & Cie. 288 p. F 48,00 (1968).

It is known that a unique root system corresponds to each complex semisimple Lie algebra, connected compact semisimple Lie group or connected semisimple algebraic group defined over an arbitrary field and the groups have structure of Tits systems (i.e. structure of BN-pairs). Further, each semisimple algebraic group over a local field has also a structure of Tits system with an affine Weyl group. Thus the structure of Coxeter groups, Tits systems and the root systems play important roles in the theory of (analytic or algebraic) semisimple groups. This book provides a very good up to date introduction to them which are of great interest in themselves and can be read independent of the theory of semisimple groups or Lie algebras.

Chapter IV. Coxeter groups and Tits systems:

Definitions and fundamental properties of the Coxeter groups and Tits systems are introduced. The Coxeter group is by definition a group which has a presentation by involutive generators, \(S = \{s_i; i\in I\}\) with the relations of the form \((s_is_j)^{m_{ij}} =1\) for any pair \((i,j)\in I\times I\), where \(m_{ij}\) is a positive integer or \(\infty\). (The matrix \(M = (m_{ij})\) is called Coxeter matrix of the group.) The group is also characterized by the so called cancellation axiom. The Tits system is a group \(G\) with two subgroups \(B\) and \(N\) of \(G\) which satisfies some axioms derived from the structure of a Borel subgroup \(B\) and the normalizer \(N\) of a maximal torus of \(B\) in a connected algebraic group. Then \(B/B\cap N\) has a structure of a Coxeter group. Especially, we have a useful criterion of simplicity of the group which is applicable to all simple groups of Lie types.

Chapter V. Groups generated by reflections:

The Coxeter group can be identified with a subgroup of \(\mathrm{GL}(E)\), where \(E=R(S)\) is a vector space over \(R\) with the canonical base \(\{e_i; i\in I\}\). Let \(B_H\) be the bilinear form on \(E\) defined by \(B_H(e_i,e_j) = -\cos(\pi/m_{ij})\). Thus we have a geometric interpretation of the groups. The group is finite if and only if \(B_H\) is positive, non-degenerate (this fact plays an important role in the classification of the finite Coxeter groups), and also we have a structure of the group such that \(B_H\) is positive, degenerate. Further, a finite group in \(\mathrm{GL}(V)\), where \(V\) is a finite dimensional vector space over a field, generated by pseudo-reflections is characterized by the properties of the \(G\)-invariant subalgebra \(R\) of the symmetric algebra \(S\) of \(V\) or the \(R\)-module \(S\) manipulating Poincaré series. The exponents of the finite Coxeter group are defined using the eigenvalues of the Coxeter transformation and the order of the group can be compute by them.

Chapter VI. Root systems:

Definitions and fundamental properties of the root systems and affine Weyl groups are given with an application to computation of the order of the Weyl groups and the structure of the \(W\)-invariant subalgebra of the group algebra \(A[P]\) of the free \(Z\)-module \(P\) over \(A\) on which \(W\) operates. Finally the finite Coxeter groups and the root systems are classified.

At the end of each chapter, there are many remarkable exercises which contains the work of Tits on groups with a \(BN\)-pair and associated geometrical-combinatorial structures (not yet published), Hecke ring of a Tits system, Coxeter groups of hyperbolic types, formulas on the exponents of the Weyl groups which are related to the topology of Lie groups and the orders of finite Chevalley groups.

As an appendix, there is a useful list of detailed data of the root system of each type.

Chapter IV. Coxeter groups and Tits systems:

Definitions and fundamental properties of the Coxeter groups and Tits systems are introduced. The Coxeter group is by definition a group which has a presentation by involutive generators, \(S = \{s_i; i\in I\}\) with the relations of the form \((s_is_j)^{m_{ij}} =1\) for any pair \((i,j)\in I\times I\), where \(m_{ij}\) is a positive integer or \(\infty\). (The matrix \(M = (m_{ij})\) is called Coxeter matrix of the group.) The group is also characterized by the so called cancellation axiom. The Tits system is a group \(G\) with two subgroups \(B\) and \(N\) of \(G\) which satisfies some axioms derived from the structure of a Borel subgroup \(B\) and the normalizer \(N\) of a maximal torus of \(B\) in a connected algebraic group. Then \(B/B\cap N\) has a structure of a Coxeter group. Especially, we have a useful criterion of simplicity of the group which is applicable to all simple groups of Lie types.

Chapter V. Groups generated by reflections:

The Coxeter group can be identified with a subgroup of \(\mathrm{GL}(E)\), where \(E=R(S)\) is a vector space over \(R\) with the canonical base \(\{e_i; i\in I\}\). Let \(B_H\) be the bilinear form on \(E\) defined by \(B_H(e_i,e_j) = -\cos(\pi/m_{ij})\). Thus we have a geometric interpretation of the groups. The group is finite if and only if \(B_H\) is positive, non-degenerate (this fact plays an important role in the classification of the finite Coxeter groups), and also we have a structure of the group such that \(B_H\) is positive, degenerate. Further, a finite group in \(\mathrm{GL}(V)\), where \(V\) is a finite dimensional vector space over a field, generated by pseudo-reflections is characterized by the properties of the \(G\)-invariant subalgebra \(R\) of the symmetric algebra \(S\) of \(V\) or the \(R\)-module \(S\) manipulating Poincaré series. The exponents of the finite Coxeter group are defined using the eigenvalues of the Coxeter transformation and the order of the group can be compute by them.

Chapter VI. Root systems:

Definitions and fundamental properties of the root systems and affine Weyl groups are given with an application to computation of the order of the Weyl groups and the structure of the \(W\)-invariant subalgebra of the group algebra \(A[P]\) of the free \(Z\)-module \(P\) over \(A\) on which \(W\) operates. Finally the finite Coxeter groups and the root systems are classified.

At the end of each chapter, there are many remarkable exercises which contains the work of Tits on groups with a \(BN\)-pair and associated geometrical-combinatorial structures (not yet published), Hecke ring of a Tits system, Coxeter groups of hyperbolic types, formulas on the exponents of the Weyl groups which are related to the topology of Lie groups and the orders of finite Chevalley groups.

As an appendix, there is a useful list of detailed data of the root system of each type.

Reviewer: Eiichi Abe (Ibaraki)

### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

22Exx | Lie groups |

17Bxx | Lie algebras and Lie superalgebras |

20E42 | Groups with a \(BN\)-pair; buildings |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

17B22 | Root systems |

### Keywords:

Lie groups; Lie algebras; Coxeter groups; Tits systems; groups generated by reflections; root systems### Online Encyclopedia of Integer Sequences:

Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).Coordination sequence T1 for Zeolite Code LTA and RHO.

Coordination sequence for planar net 4.8.8.

Dimensions of the five sporadic Lie groups.

Triangle read by rows in which row n (n >= 1) gives coefficients in expansion of the polynomial f(n) * Product_{i=1..n-1} f(2i), where f(k) = (1 - x^k)/(1-x).

Number of reduced words of length n in the Weyl group D_4.

Number of reduced words of length n in the Weyl group D_6.

Number of reduced words of length n in the Weyl group D_7.

Number of reduced words of length n in the Weyl group D_8.

Number of reduced words of length n in the Weyl group D_9.

Number of reduced words of length n in the Weyl group D_10.

Number of reduced words of length n in the Weyl group D_11.

Number of reduced words of length n in the Weyl group D_12.

Number of reduced words of length n in the Weyl group D_13.

Number of reduced words of length n in the Weyl group D_14.

Number of reduced words of length n in the Weyl group D_15.

Number of reduced words of length n in the Weyl group D_16.

Number of reduced words of length n in the Weyl group D_17.

Number of reduced words of length n in the Weyl group D_18.

Number of reduced words of length n in the Weyl group D_19.

Number of reduced words of length n in the Weyl group D_20.

Number of reduced words of length n in the Weyl group D_21.

Number of reduced words of length n in the Weyl group D_22.

Number of reduced words of length n in the Weyl group D_23.

Number of reduced words of length n in the Weyl group D_24.

Number of reduced words of length n in the Weyl group D_25.

Number of reduced words of length n in the Weyl group D_26.

Number of reduced words of length n in the Weyl group D_27.

Number of reduced words of length n in the Weyl group D_28.

Number of reduced words of length n in the Weyl group D_29.

Number of reduced words of length n in the Weyl group D_30.

Number of reduced words of length n in the Weyl group D_31.

Number of reduced words of length n in the Weyl group D_32.

Number of reduced words of length n in the Weyl group D_33.

Number of reduced words of length n in the Weyl group D_34.

Number of reduced words of length n in the Weyl group D_35.

Number of reduced words of length n in the Weyl group D_36.

Number of reduced words of length n in the Weyl group D_37.

Number of reduced words of length n in the Weyl group D_38.

Number of reduced words of length n in the Weyl group D_39.

Number of reduced words of length n in the Weyl group D_40.

Number of reduced words of length n in the Weyl group D_41.

Number of reduced words of length n in the Weyl group D_42.

Number of reduced words of length n in the Weyl group D_43.

Number of reduced words of length n in the Weyl group D_44.

Number of reduced words of length n in the Weyl group D_45.

Number of reduced words of length n in the Weyl group D_46.

Number of reduced words of length n in the Weyl group D_47.

Number of reduced words of length n in the Weyl group D_48.

Number of reduced words of length n in the Weyl group D_49.

Number of reduced words of length n in the Weyl group D_50.

Number of reduced words of length n in the Weyl group E_7 on 7 generators and order 2903040.

Growth series for affine Coxeter group (or affine Weyl group) D_4.

Growth series for affine Coxeter group (or affine Weyl group) D_5.

Growth series for affine Coxeter group (or affine Weyl group) D_6.

Growth series for affine Coxeter group (or affine Weyl group) D_7.

Growth series for affine Coxeter group (or affine Weyl group) D_8.

Growth series for affine Coxeter group (or affine Weyl group) D_9.

Growth series for affine Coxeter group (or affine Weyl group) D_10.

Growth series for affine Coxeter group (or affine Weyl group) D_11.

Growth series for affine Coxeter group (or affine Weyl group) D_12.

The growth series for the affine Coxeter (or Weyl) group [3,5] (or H_3).

The growth series for the affine Coxeter (or Weyl) group [3,3,5] (or H_4).

The growth series for the affine Weyl group F_4.

The growth series for the affine Weyl group E_7.

Growth series for affine Coxeter group B_4.

Growth series for affine Coxeter group B_5.

Growth series for affine Coxeter group B_6.

Growth series for affine Coxeter group B_7.

Growth series for affine Coxeter group B_8.

Growth series for affine Coxeter group B_9.

Growth series for affine Coxeter group B_10.

Growth series for affine Coxeter group B_11.

Growth series for affine Coxeter group B_12.

The growth series for the affine Weyl group E_8.