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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 |