Characters of finite Coxeter groups and Iwahori-Hecke algebras.

*(English)*Zbl 0996.20004
London Mathematical Society Monographs. New Series. 21. Oxford: Clarendon Press. xv, 446 p. (2000).

This is an important book which gives a largely self-contained account of the representations of Coxeter groups and their Iwahori-Hecke algebras. The authors make full use of recent advances and also explain the role that Coxeter groups and their Iwahori-Hecke algebras play in the representation theory of the groups of Lie type. Each chapter concludes with bibliographical remarks and numerous examples (there are hints for many of the problems). The book is both a valuable resource for the expert and good starting point for the beginning researcher in this field.

The class of Coxeter groups were first studied by H. S. M. Coxeter when he considered the finite groups which have a faithful representation as a group generated by reflections acting on a finite dimensional Euclidean vector space. Coxeter proved the remarkable result that a finite group is a finite reflection group if and only if it has a presentation of the form \[ \langle s\in S\mid s^2=1\text{ and }(rs\ldots)_{m_{rs}}=(sr\ldots)_{m_{sr}}\text{ for all }r\neq s\in S\rangle, \] where \(m_{rs}=m_{sr}\geq 1\) and \((ab\ldots)_m=(ab)^{m/2}\) if \(m\) is even and \((ab\ldots)_m=a(ba\ldots)_{m-1}\) if \(m\) is odd. The homogeneous relations are known as the braid relations. Moreover, Coxeter classified these groups showing that they fall into four infinite families: \({\mathbf A}_n\) (the symmetric groups), \({\mathbf B}_n\) (the signed permutation groups), \({\mathbf D}_n\) (the subgroup of \({\mathbf B}_n\) consisting of permutations with an even number of sign changes) and \({\mathbf I}_2(p)\) (the dihedral groups); with six additional exceptional groups \({\mathbf E}_6\), \({\mathbf E}_7\) and \({\mathbf E}_8\), \({\mathbf F}_4\), \({\mathbf H}_3\) and \({\mathbf H}_4\).

The importance of Coxeter groups in modern mathematics derives from their close connections with Lie algebras and the groups of Lie type. In many respects the Weyl groups (i.e. a Coxeter group of type \({\mathbf A}_n\), \({\mathbf B}_n\), \({\mathbf D}_n\), \({\mathbf E}_6\), \({\mathbf E}_7\), \({\mathbf E}_8\), \({\mathbf F}_4\) or \({\mathbf G}_2={\mathbf I}_2(3)\)) look like finite groups of Lie type which are “defined over fields of characteristic one”. To explain what is meant by this, let \(G\) be a finite group of Lie type defined over a finite field with \(q\) elements, let \(B\) be a Borel subgroup of \(G\) and let \(H_q(G,B)\) be the algebra \(H_q(G,B)=\text{End}_{\mathbb{C} G}(\text{Ind}_B^G(1))\), where \(\text{Ind}_B^G(1)\) is the trivial representation of \(B\) induced up to \(G\) (that is, the permutation representation of \(G\) upon the right cosets of \(B\) in \(G\)). Let \((W,S)\) be the Coxeter system of \(G\). (For example, if \(G=\text{GL}_n(q)\) then \(B\) is the subgroup of upper triangular matrices in \(G\) and \(W\) is the symmetric group \({\mathfrak S}_n\) with generating set \(S=\{(1,2),(2,3),\dots,(n-1,n)\}\).) Iwahori and Matsumoto proved that \(H_q(G,B)\) is isomorphic to the associative unital \(\mathbb{C}\)-algebra with generators \(\{T_s\mid s\in S\}\) and relations \((T_s-q)(T_s+1)=0\) and \((T_rT_s\ldots)_{m_{rs}}=(T_sT_r\ldots)_{m_{sr}}\), for \(r,s\in S\) with \(r\neq s\). (Iwahori gave a case by case argument for the untwisted Chevalley groups; soon afterwards Matsumoto gave a uniform proof covering all cases.) Notice, in particular, that if we set \(q=1\) then \(H_q(G,B)\) is isomorphic to the group algebra of \(W\). Geck and Pfeiffer prove these results in Chapter 8.

The algebras \(H_q(G,B)\) are special instances of the Iwahori-Hecke algebras of the book’s title. These algebras have applications in a diverse range of fields from the representation theory of Lie groups, to knot theory and statistical mechanics. The Iwahori-Hecke algebras are deformations of the group algebras of Coxeter groups, and they are the forerunners of quantum groups. Their main successes have been in their connections with knot theory and in explaining certain “generic” features of representation theory of the groups of Lie type. For example, it is not hard to show that there is a one-to-one correspondence between the irreducible constituents of \(\text{Ind}_B^G(1)\) and the irreducible representations of \(H_q(G,B)\); this yields a one-to-one correspondence \(\chi_q\leftrightarrow\chi\) between the irreducible constituents of \(\text{Ind}_B^G(1)\) and the irreducible representations of \(W\). Moreover, for each \(\chi\) there is a polynomial \(d_\chi(x)\) such that \(d_\chi(1)\) is the dimension of \(\chi\) and \(d_\chi(q)\) is the dimension of \(\chi_q\) for all \(q\). The polynomials \(d_\chi(q)\) are the “generic degrees” of \(W\) and they are easily computed using the Iwahori-Hecke algebra.

The aim of the book under review is to develop “the theory of conjugacy classes and irreducible characters, both for finite Coxeter groups and the associated Iwahori-Hecke algebras…in a systematic way”. The book places a strong emphasis on algorithms; indeed, in some respects this book grew out of the computer package CHEVIE [M. Geck, G. Hiss, F. Lübeck, G. Malle, G. Pfeiffer, Appl. Algebra Eng. Commun. Comput. 7, No. 3, 175-210 (1996; Zbl 0847.20006)] which implements many of the (computational) results contained in this book.

The book begins by defining a Coxeter group to be a group with a presentation determined by a “Coxeter matrix” (the relationship between a Coxeter matrix \((c_{rs})\) and the presentation above is that \(c_{rs}c_{sr}=4\cos^2(\tfrac\pi{m_{rs}})\)). Basic facts about root systems, the length function, parabolic subgroups and distinguished coset representatives are established. The first major result is Matsumoto’s monoid lemma. Next parabolic subgroups and their distinguished coset representatives are introduced and these are used to study Solomon’s descent algebra, the parabolic Burnside ring and the associated parabolic table of marks.

Chapter 3 is a detailed description of the conjugacy classes of Coxeter groups, which were classified by R. W. Carter [Compos. Math. 25, 1-59 (1972; Zbl 0254.17005)]. Geck and Pfeiffer classify the conjugacy classes of Coxeter groups inductively in terms of cuspidal conjugacy classes: a conjugacy class is cuspidal if no element in the class is contained in a proper parabolic subgroup. Case-by-case arguments are used to describe the cuspidal classes.

As a prelude to introducing the Iwahori-Hecke algebra of \(W\), Chapter 4 looks at the braid group associated to a Coxeter group; again this is defined by a presentation. The authors show that the positive braid monoid embeds in the braid group and that every element of the braid group has a normal form. The Iwahori-Hecke algebra of \((W,S)\) is defined as a quotient of the group algebra of the braid group by the ideal generated by the skein relations. It is also shown that this definition of the Iwahori-Hecke algebra agrees with the algebra \(H_q(G,B)\) defined above by Iwahori’s presentation (the authors also consider multiparameter Hecke algebras). The authors also prove Ocneanu’s trace formula and use this to construct Jones’ HOMFLY polynomial using the Iwahori-Hecke algebras of type \(\mathbf A\).

In Chapter 5 the irreducible characters of the finite Coxeter groups are constructed. Ultimately, case-by-case arguments are needed; however, the authors first use Macdonald-Lusztig-Spaltenstein \(j\)-induction and Lusztig’s \(a\)-function to develop the theory as far as possible using general methods. One consequence of these constructions is the important result that the rationals are a splitting field for the finite Weyl groups. Chapters 6 continues to study the representations of finite Coxeter groups looking at Curtis, Iwahori and Kilmoyer’s theory of characters of parabolic type and Lusztig’s families.

Chapter 7 develops the representation theory of symmetric algebras and of decomposition maps. In particular, the following results are established for symmetric algebras: a general Tits’ deformation theorem (in particular, this shows that generically the Iwahori-Hecke algebra is isomorphic to the group algebra of the corresponding Coxeter group); a semisimplicity criterion; a description of the centre of a “reduction stable” symmetric algebra; a description of the blocks of defect \(0\).

Chapter 8 begins the study of the representation theory of the Iwahori-Hecke algebras by first showing that they are symmetric algebras (so all of the results of Chapter 7 apply). Using the description of conjugacy classes from Chapter 3, the character tables of the Iwahori-Hecke algebras are defined (in terms of elements of minimal length in each conjugacy class), and computed in the dihedral case. Chapter 9 studies induction and restriction of irreducible characters of parabolic subalgebras of Iwahori-Hecke algebras, culminating with an explicit description of the splitting field of an Iwahori-Hecke algebra.

Finally, in Chapter 10 the character tables and generic degrees of the Iwahori-Hecke algebras of type \({\mathbf A}_n\), \({\mathbf B}_n\) and \({\mathbf D}_n\) are explicitly computed and in chapter 11 the exceptional cases are covered partly using Kazhdan and Lusztig’s notion of \(W\)-graphs and partly via computer calculations based on the algorithms given in the text, explicit tables of which are given in the appendices. There are also some new results describing the two parameter \(W\)-graphs of type \({\mathbf F}_4\). The character tables are also used to determine the blocks of the exceptional Iwahori-Hecke algebras at complex roots of unity.

All up this is a very nice book which comprehensively covers all of the main features of the semisimple representation theory of finite Coxeter groups and their Iwahori-Hecke algebras. The only notable omission from the text is a proof of the classification of finite Coxeter groups. The authors remark that this already appears in a number of books; however, as the classification can be proved in only a few pages I would have liked to see it included. Similarly, it would have been nice to have Lusztig’s isomorphism theorem and a more detailed discussion of cell representations and Lusztig’s work on the \(a\)-function included; however, this omission is understandable because a proper treatment of these topics requires some deep geometry which is beyond the scope of the book.

In summary, this is a very fine book which belongs on the shelves of anyone who is interested in the representation theory of Coxeter groups, Iwahori-Hecke algebras and, more generally, the groups of Lie type.

The class of Coxeter groups were first studied by H. S. M. Coxeter when he considered the finite groups which have a faithful representation as a group generated by reflections acting on a finite dimensional Euclidean vector space. Coxeter proved the remarkable result that a finite group is a finite reflection group if and only if it has a presentation of the form \[ \langle s\in S\mid s^2=1\text{ and }(rs\ldots)_{m_{rs}}=(sr\ldots)_{m_{sr}}\text{ for all }r\neq s\in S\rangle, \] where \(m_{rs}=m_{sr}\geq 1\) and \((ab\ldots)_m=(ab)^{m/2}\) if \(m\) is even and \((ab\ldots)_m=a(ba\ldots)_{m-1}\) if \(m\) is odd. The homogeneous relations are known as the braid relations. Moreover, Coxeter classified these groups showing that they fall into four infinite families: \({\mathbf A}_n\) (the symmetric groups), \({\mathbf B}_n\) (the signed permutation groups), \({\mathbf D}_n\) (the subgroup of \({\mathbf B}_n\) consisting of permutations with an even number of sign changes) and \({\mathbf I}_2(p)\) (the dihedral groups); with six additional exceptional groups \({\mathbf E}_6\), \({\mathbf E}_7\) and \({\mathbf E}_8\), \({\mathbf F}_4\), \({\mathbf H}_3\) and \({\mathbf H}_4\).

The importance of Coxeter groups in modern mathematics derives from their close connections with Lie algebras and the groups of Lie type. In many respects the Weyl groups (i.e. a Coxeter group of type \({\mathbf A}_n\), \({\mathbf B}_n\), \({\mathbf D}_n\), \({\mathbf E}_6\), \({\mathbf E}_7\), \({\mathbf E}_8\), \({\mathbf F}_4\) or \({\mathbf G}_2={\mathbf I}_2(3)\)) look like finite groups of Lie type which are “defined over fields of characteristic one”. To explain what is meant by this, let \(G\) be a finite group of Lie type defined over a finite field with \(q\) elements, let \(B\) be a Borel subgroup of \(G\) and let \(H_q(G,B)\) be the algebra \(H_q(G,B)=\text{End}_{\mathbb{C} G}(\text{Ind}_B^G(1))\), where \(\text{Ind}_B^G(1)\) is the trivial representation of \(B\) induced up to \(G\) (that is, the permutation representation of \(G\) upon the right cosets of \(B\) in \(G\)). Let \((W,S)\) be the Coxeter system of \(G\). (For example, if \(G=\text{GL}_n(q)\) then \(B\) is the subgroup of upper triangular matrices in \(G\) and \(W\) is the symmetric group \({\mathfrak S}_n\) with generating set \(S=\{(1,2),(2,3),\dots,(n-1,n)\}\).) Iwahori and Matsumoto proved that \(H_q(G,B)\) is isomorphic to the associative unital \(\mathbb{C}\)-algebra with generators \(\{T_s\mid s\in S\}\) and relations \((T_s-q)(T_s+1)=0\) and \((T_rT_s\ldots)_{m_{rs}}=(T_sT_r\ldots)_{m_{sr}}\), for \(r,s\in S\) with \(r\neq s\). (Iwahori gave a case by case argument for the untwisted Chevalley groups; soon afterwards Matsumoto gave a uniform proof covering all cases.) Notice, in particular, that if we set \(q=1\) then \(H_q(G,B)\) is isomorphic to the group algebra of \(W\). Geck and Pfeiffer prove these results in Chapter 8.

The algebras \(H_q(G,B)\) are special instances of the Iwahori-Hecke algebras of the book’s title. These algebras have applications in a diverse range of fields from the representation theory of Lie groups, to knot theory and statistical mechanics. The Iwahori-Hecke algebras are deformations of the group algebras of Coxeter groups, and they are the forerunners of quantum groups. Their main successes have been in their connections with knot theory and in explaining certain “generic” features of representation theory of the groups of Lie type. For example, it is not hard to show that there is a one-to-one correspondence between the irreducible constituents of \(\text{Ind}_B^G(1)\) and the irreducible representations of \(H_q(G,B)\); this yields a one-to-one correspondence \(\chi_q\leftrightarrow\chi\) between the irreducible constituents of \(\text{Ind}_B^G(1)\) and the irreducible representations of \(W\). Moreover, for each \(\chi\) there is a polynomial \(d_\chi(x)\) such that \(d_\chi(1)\) is the dimension of \(\chi\) and \(d_\chi(q)\) is the dimension of \(\chi_q\) for all \(q\). The polynomials \(d_\chi(q)\) are the “generic degrees” of \(W\) and they are easily computed using the Iwahori-Hecke algebra.

The aim of the book under review is to develop “the theory of conjugacy classes and irreducible characters, both for finite Coxeter groups and the associated Iwahori-Hecke algebras…in a systematic way”. The book places a strong emphasis on algorithms; indeed, in some respects this book grew out of the computer package CHEVIE [M. Geck, G. Hiss, F. Lübeck, G. Malle, G. Pfeiffer, Appl. Algebra Eng. Commun. Comput. 7, No. 3, 175-210 (1996; Zbl 0847.20006)] which implements many of the (computational) results contained in this book.

The book begins by defining a Coxeter group to be a group with a presentation determined by a “Coxeter matrix” (the relationship between a Coxeter matrix \((c_{rs})\) and the presentation above is that \(c_{rs}c_{sr}=4\cos^2(\tfrac\pi{m_{rs}})\)). Basic facts about root systems, the length function, parabolic subgroups and distinguished coset representatives are established. The first major result is Matsumoto’s monoid lemma. Next parabolic subgroups and their distinguished coset representatives are introduced and these are used to study Solomon’s descent algebra, the parabolic Burnside ring and the associated parabolic table of marks.

Chapter 3 is a detailed description of the conjugacy classes of Coxeter groups, which were classified by R. W. Carter [Compos. Math. 25, 1-59 (1972; Zbl 0254.17005)]. Geck and Pfeiffer classify the conjugacy classes of Coxeter groups inductively in terms of cuspidal conjugacy classes: a conjugacy class is cuspidal if no element in the class is contained in a proper parabolic subgroup. Case-by-case arguments are used to describe the cuspidal classes.

As a prelude to introducing the Iwahori-Hecke algebra of \(W\), Chapter 4 looks at the braid group associated to a Coxeter group; again this is defined by a presentation. The authors show that the positive braid monoid embeds in the braid group and that every element of the braid group has a normal form. The Iwahori-Hecke algebra of \((W,S)\) is defined as a quotient of the group algebra of the braid group by the ideal generated by the skein relations. It is also shown that this definition of the Iwahori-Hecke algebra agrees with the algebra \(H_q(G,B)\) defined above by Iwahori’s presentation (the authors also consider multiparameter Hecke algebras). The authors also prove Ocneanu’s trace formula and use this to construct Jones’ HOMFLY polynomial using the Iwahori-Hecke algebras of type \(\mathbf A\).

In Chapter 5 the irreducible characters of the finite Coxeter groups are constructed. Ultimately, case-by-case arguments are needed; however, the authors first use Macdonald-Lusztig-Spaltenstein \(j\)-induction and Lusztig’s \(a\)-function to develop the theory as far as possible using general methods. One consequence of these constructions is the important result that the rationals are a splitting field for the finite Weyl groups. Chapters 6 continues to study the representations of finite Coxeter groups looking at Curtis, Iwahori and Kilmoyer’s theory of characters of parabolic type and Lusztig’s families.

Chapter 7 develops the representation theory of symmetric algebras and of decomposition maps. In particular, the following results are established for symmetric algebras: a general Tits’ deformation theorem (in particular, this shows that generically the Iwahori-Hecke algebra is isomorphic to the group algebra of the corresponding Coxeter group); a semisimplicity criterion; a description of the centre of a “reduction stable” symmetric algebra; a description of the blocks of defect \(0\).

Chapter 8 begins the study of the representation theory of the Iwahori-Hecke algebras by first showing that they are symmetric algebras (so all of the results of Chapter 7 apply). Using the description of conjugacy classes from Chapter 3, the character tables of the Iwahori-Hecke algebras are defined (in terms of elements of minimal length in each conjugacy class), and computed in the dihedral case. Chapter 9 studies induction and restriction of irreducible characters of parabolic subalgebras of Iwahori-Hecke algebras, culminating with an explicit description of the splitting field of an Iwahori-Hecke algebra.

Finally, in Chapter 10 the character tables and generic degrees of the Iwahori-Hecke algebras of type \({\mathbf A}_n\), \({\mathbf B}_n\) and \({\mathbf D}_n\) are explicitly computed and in chapter 11 the exceptional cases are covered partly using Kazhdan and Lusztig’s notion of \(W\)-graphs and partly via computer calculations based on the algorithms given in the text, explicit tables of which are given in the appendices. There are also some new results describing the two parameter \(W\)-graphs of type \({\mathbf F}_4\). The character tables are also used to determine the blocks of the exceptional Iwahori-Hecke algebras at complex roots of unity.

All up this is a very nice book which comprehensively covers all of the main features of the semisimple representation theory of finite Coxeter groups and their Iwahori-Hecke algebras. The only notable omission from the text is a proof of the classification of finite Coxeter groups. The authors remark that this already appears in a number of books; however, as the classification can be proved in only a few pages I would have liked to see it included. Similarly, it would have been nice to have Lusztig’s isomorphism theorem and a more detailed discussion of cell representations and Lusztig’s work on the \(a\)-function included; however, this omission is understandable because a proper treatment of these topics requires some deep geometry which is beyond the scope of the book.

In summary, this is a very fine book which belongs on the shelves of anyone who is interested in the representation theory of Coxeter groups, Iwahori-Hecke algebras and, more generally, the groups of Lie type.

Reviewer: Andrew Mathas (Sydney)

##### MSC:

20C15 | Ordinary representations and characters |

20C08 | Hecke algebras and their representations |

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

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

20C33 | Representations of finite groups of Lie type |

20C40 | Computational methods (representations of groups) (MSC2010) |

20E45 | Conjugacy classes for groups |

20F36 | Braid groups; Artin groups |