Explicit Brauer induction. With applications to algebra and number theory.

*(English)*Zbl 0991.20005
Cambridge Studies in Advanced Mathematics. 40. Cambridge: Cambridge University Press. xii, 409 p. (1994).

The purpose of this book is to give an algebraic construction for explicit Brauer induction and to explore some of the consequences of the construction. First, some background.

The classical Brauer induction theorem states the following: Given a complex representation \(\rho\) of a finite group \(G\), there are one-dimensional representations \(\phi_i\) of the elementary subgroups \(H_i\) of \(G\) such that \(\rho=\sum_in_i\text{Ind}^G_{H_i}(\phi_i)\) for some integers \(n_i\). This result may be reformulated in an apparently weaker form as follows. Let \(R(G)\) be the representation ring of \(G\), that is, the free Abelian group generated by the irreducible complex characters of \(G\), with multiplication arising from the tensor product of representations. Further, let \(R_+(G)\) be the free Abelian group generated by the \(G\)-conjugacy classes \((H,\phi)^G\) of the one-dimensional representations \(\phi\) of the subgroups \(H\) of \(G\), where the “\(G\)-conjugacy” action of \(g\in G\) on \((H,\phi)\) is \(g(H,\phi)=(gHg^{-1},(g^{-1})^\times\phi)\), with \(((g^{-1})^\times\phi)(u)=\phi(g^{-1}ug)\). Then there is an additive homomorphism \(b_G\colon R_+(G)\to R(G)\), \(b_G((H,\phi)^G)=\text{Ind}^G_H(\phi)\), and the Brauer induction theorem now states that \(b_G\) is surjective. At first sight, the surjectivity of \(b_G\) is a weaker assertion than that in the original form of Brauer induction since \(H\) need no longer be elementary, but one can iterate to express an element of \(R(G)\) in terms of characters of elementary subgroups.

Leaving trivialities aside, an element \(x\) of \(R(G)\) will have many pre-images in \(R_+(G)\). The aim of explicit Brauer induction is to find a canonical pre-image for each \(x\), which is achieved in this text by constructing an additive homomorphism \(a_G\colon R(G)\to R_+(G)\) that splits \(b_G\), that is, \(b_Ga_G\) is the identity map on \(R(G)\).

After a preliminary review of representation theory in Chapter 1, the author establishes the existence of explicit Brauer induction in Chapter 2, following the algebraic method of R. Boltje [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)], rather than his own topological construction, which is reviewed briefly.

The remainder of the text is devoted to applications, each of which is placed in context by a full exposition of the background results that are needed to appreciate it. The first application of explicit Brauer induction concerns the representation theory of \(\text{GL}_m(\mathbb{F}(q^n))\), where \(\mathbb{F}(q^n)\) is the Galois field of \(q^n\) elements. T. Shintani [J. Math. Soc. Japan 28, 396-414 (1976; Zbl 0323.20041)] constructed a canonical bijection between, on the one hand, irreducible representations of \(\text{GL}_m(\mathbb{F}(q^n))\) which are invariant under the Frobenius of \(\mathbb{F}(q^n)/\mathbb{F}(q)\), and, on the other, irreducible representations of \(\text{GL}_m(\mathbb{F}(q))\). Explicit Brauer induction recovers this correspondence in the case \(m=2\), and conjecturally should do so for all \(m\). The result is based on an explicit calculation of the cuspidal (Weil) representations of \(\text{GL}_2(\mathbb{F}(q^n))\).

The remaining applications are number-theoretic in nature, being mainly concerned with the detection of elements of the class group \(\text{Cl}(\mathbb{Z} G)\) of the integral group ring of \(G\). The aim here is to shed light on results and conjectures of Chinburg, Fröhlich and Taylor in Galois module theory, that is, the determination of the structure of the \(\mathbb{Z} G\)-modules which arise from a Galois extension \(L/K\) of number fields with Galois group \(G\). To start, the author shows that explicit Brauer induction behaves well with respect to Adams operations on \(R(G)\). Taking this result in combination with Fröhlich’s Hom description of the class group, he obtains new determinantal congruences on the class group. These include confirmation of a conjecture by M. J. Taylor [J. Algebra 50, 463-487 (1978; Zbl 0377.20006)], and consequent simplifications in the treatment of the \(l\)-adic logarithm on the local group ring.

Next, he constructs new functions from \(R(G)\) to \(\text{Cl}(G)\), called restricted determinants, by applying the determinant to the components of the explicit Brauer induction formula. A modification of this definition gives instead functions from \(R(G)\) to \(\text{Cl}({\mathcal M})\), the locally free class group of a maximal order \(\mathcal M\) containing \(\mathbb{Z} G\). The main application of these results is to the computation of the Swan subgroup of the class group of \(G\), particularly when \(G\) is nilpotent. The author conjectures that the Swan subgroup of an arbitrary nilpotent group can be detected on its elementary Abelian subquotients, and he verifies this result in some cases involving small primes.

The next application of explicit Brauer induction is to the conductors of local fields. Let \(L/K\) be a finite Galois extension of complete discrete local fields, with Galois group \(G\). When the corresponding residue field extension is separable, Swan has defined a conductor \(\text{sw}_ K\colon R(G)\to\mathbb{Z}\). The author provides a generalization of the Swan conductor to the case in which the residue field extension is not separable. This generalization includes as a special case the conductor found by Kato when \(G\) is Abelian. Some explicit calculations of Swan and Kato conductors show that any nonabelian, nonseparable generalization of them is inevitably lacking some desirable property.

The final chapter is concerned with the Chinburg invariants. In the case in which \(L/K\) is a Galois extension of local fields, there is a new proof of Chinburg’s theorem that \(\Omega(L/K,U^1_L)\) is trivial when \(L/K\) is tamely ramified. In the global case, that is, when \(L/K\) is a Galois extension of number fields, the Fröhlich-Chinburg conjecture states that \(\Omega(L/K,2)=W_{L/K}\) in \(\text{Cl}(G)\), where \(\Omega(L/K,2)\) is constructed from a resolution of a certain natural \(\mathbb{Z} G\)-module, while \(W_{L/K}\) is defined in terms of the root numbers of Artin \(L\)-series of symplectic characters of \(G\). In the case in which \(L/K\) is tame, this becomes the classical conjecture of Fröhlich, which was confirmed by M. J. Taylor [Invent. Math. 63, 41-79 (1981; Zbl 0469.12003)]. When \(L/K\) is wild, a maximal order version of the conjecture has been proved by D. Holland [J. Reine Angew. Math. 425, 193-218 (1992; Zbl 0737.11033)]. The new result in this chapter is a derivation of Holland’s result through the explicit Brauer induction machine. To round off, the author gives some related results on real cyclotomic Galois module structure that do not depend on the Brauer machinery.

The author makes a convincing argument that explicit Brauer induction is a valuable addition to the weaponry of representation theorists. The book is pleasant to read, particularly as the author provides informative and discursive introductions, both to the whole text and to each chapter, which place the results in context and provide motivation for what follows. There are many explicit examples of computations for “small” groups (that is, cyclic, dihedral, quaternion of order \(8\),…), and exercises are provided at all levels, some of them being suggestions for further research.

The level of the text is advanced postgraduate, since the reader needs to have some familiarity with representation theory and algebraic number theory, particularly in its non-commutative manifestation, the theory of orders. The author is careful to sketch the results which he invokes, and provides adequate references for them. However, it would be useful to have an index of notation.

There are also one or two points in the presentation which may make the text hard going for the inexperienced. For example, in the definition of \(R_+(G)\), a fundamental point in this text, the author uses “character with nonzero value” as a synonym for “one-dimensional representation”, which is not quite accurate. He also introduces the term “\(G\)-conjugacy” before he defines it (at least, as far as I could tell). There is a gap in the author’s proof of the classical Brauer induction theorem, which (he assures me) is not too hard to fill.

These quibbles apart, the book is a good introduction to explicit Brauer induction and its arithmetic applications, particularly to Galois module structure, and it will be a valuable addition to the library of anyone working on these topics.

The classical Brauer induction theorem states the following: Given a complex representation \(\rho\) of a finite group \(G\), there are one-dimensional representations \(\phi_i\) of the elementary subgroups \(H_i\) of \(G\) such that \(\rho=\sum_in_i\text{Ind}^G_{H_i}(\phi_i)\) for some integers \(n_i\). This result may be reformulated in an apparently weaker form as follows. Let \(R(G)\) be the representation ring of \(G\), that is, the free Abelian group generated by the irreducible complex characters of \(G\), with multiplication arising from the tensor product of representations. Further, let \(R_+(G)\) be the free Abelian group generated by the \(G\)-conjugacy classes \((H,\phi)^G\) of the one-dimensional representations \(\phi\) of the subgroups \(H\) of \(G\), where the “\(G\)-conjugacy” action of \(g\in G\) on \((H,\phi)\) is \(g(H,\phi)=(gHg^{-1},(g^{-1})^\times\phi)\), with \(((g^{-1})^\times\phi)(u)=\phi(g^{-1}ug)\). Then there is an additive homomorphism \(b_G\colon R_+(G)\to R(G)\), \(b_G((H,\phi)^G)=\text{Ind}^G_H(\phi)\), and the Brauer induction theorem now states that \(b_G\) is surjective. At first sight, the surjectivity of \(b_G\) is a weaker assertion than that in the original form of Brauer induction since \(H\) need no longer be elementary, but one can iterate to express an element of \(R(G)\) in terms of characters of elementary subgroups.

Leaving trivialities aside, an element \(x\) of \(R(G)\) will have many pre-images in \(R_+(G)\). The aim of explicit Brauer induction is to find a canonical pre-image for each \(x\), which is achieved in this text by constructing an additive homomorphism \(a_G\colon R(G)\to R_+(G)\) that splits \(b_G\), that is, \(b_Ga_G\) is the identity map on \(R(G)\).

After a preliminary review of representation theory in Chapter 1, the author establishes the existence of explicit Brauer induction in Chapter 2, following the algebraic method of R. Boltje [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)], rather than his own topological construction, which is reviewed briefly.

The remainder of the text is devoted to applications, each of which is placed in context by a full exposition of the background results that are needed to appreciate it. The first application of explicit Brauer induction concerns the representation theory of \(\text{GL}_m(\mathbb{F}(q^n))\), where \(\mathbb{F}(q^n)\) is the Galois field of \(q^n\) elements. T. Shintani [J. Math. Soc. Japan 28, 396-414 (1976; Zbl 0323.20041)] constructed a canonical bijection between, on the one hand, irreducible representations of \(\text{GL}_m(\mathbb{F}(q^n))\) which are invariant under the Frobenius of \(\mathbb{F}(q^n)/\mathbb{F}(q)\), and, on the other, irreducible representations of \(\text{GL}_m(\mathbb{F}(q))\). Explicit Brauer induction recovers this correspondence in the case \(m=2\), and conjecturally should do so for all \(m\). The result is based on an explicit calculation of the cuspidal (Weil) representations of \(\text{GL}_2(\mathbb{F}(q^n))\).

The remaining applications are number-theoretic in nature, being mainly concerned with the detection of elements of the class group \(\text{Cl}(\mathbb{Z} G)\) of the integral group ring of \(G\). The aim here is to shed light on results and conjectures of Chinburg, Fröhlich and Taylor in Galois module theory, that is, the determination of the structure of the \(\mathbb{Z} G\)-modules which arise from a Galois extension \(L/K\) of number fields with Galois group \(G\). To start, the author shows that explicit Brauer induction behaves well with respect to Adams operations on \(R(G)\). Taking this result in combination with Fröhlich’s Hom description of the class group, he obtains new determinantal congruences on the class group. These include confirmation of a conjecture by M. J. Taylor [J. Algebra 50, 463-487 (1978; Zbl 0377.20006)], and consequent simplifications in the treatment of the \(l\)-adic logarithm on the local group ring.

Next, he constructs new functions from \(R(G)\) to \(\text{Cl}(G)\), called restricted determinants, by applying the determinant to the components of the explicit Brauer induction formula. A modification of this definition gives instead functions from \(R(G)\) to \(\text{Cl}({\mathcal M})\), the locally free class group of a maximal order \(\mathcal M\) containing \(\mathbb{Z} G\). The main application of these results is to the computation of the Swan subgroup of the class group of \(G\), particularly when \(G\) is nilpotent. The author conjectures that the Swan subgroup of an arbitrary nilpotent group can be detected on its elementary Abelian subquotients, and he verifies this result in some cases involving small primes.

The next application of explicit Brauer induction is to the conductors of local fields. Let \(L/K\) be a finite Galois extension of complete discrete local fields, with Galois group \(G\). When the corresponding residue field extension is separable, Swan has defined a conductor \(\text{sw}_ K\colon R(G)\to\mathbb{Z}\). The author provides a generalization of the Swan conductor to the case in which the residue field extension is not separable. This generalization includes as a special case the conductor found by Kato when \(G\) is Abelian. Some explicit calculations of Swan and Kato conductors show that any nonabelian, nonseparable generalization of them is inevitably lacking some desirable property.

The final chapter is concerned with the Chinburg invariants. In the case in which \(L/K\) is a Galois extension of local fields, there is a new proof of Chinburg’s theorem that \(\Omega(L/K,U^1_L)\) is trivial when \(L/K\) is tamely ramified. In the global case, that is, when \(L/K\) is a Galois extension of number fields, the Fröhlich-Chinburg conjecture states that \(\Omega(L/K,2)=W_{L/K}\) in \(\text{Cl}(G)\), where \(\Omega(L/K,2)\) is constructed from a resolution of a certain natural \(\mathbb{Z} G\)-module, while \(W_{L/K}\) is defined in terms of the root numbers of Artin \(L\)-series of symplectic characters of \(G\). In the case in which \(L/K\) is tame, this becomes the classical conjecture of Fröhlich, which was confirmed by M. J. Taylor [Invent. Math. 63, 41-79 (1981; Zbl 0469.12003)]. When \(L/K\) is wild, a maximal order version of the conjecture has been proved by D. Holland [J. Reine Angew. Math. 425, 193-218 (1992; Zbl 0737.11033)]. The new result in this chapter is a derivation of Holland’s result through the explicit Brauer induction machine. To round off, the author gives some related results on real cyclotomic Galois module structure that do not depend on the Brauer machinery.

The author makes a convincing argument that explicit Brauer induction is a valuable addition to the weaponry of representation theorists. The book is pleasant to read, particularly as the author provides informative and discursive introductions, both to the whole text and to each chapter, which place the results in context and provide motivation for what follows. There are many explicit examples of computations for “small” groups (that is, cyclic, dihedral, quaternion of order \(8\),…), and exercises are provided at all levels, some of them being suggestions for further research.

The level of the text is advanced postgraduate, since the reader needs to have some familiarity with representation theory and algebraic number theory, particularly in its non-commutative manifestation, the theory of orders. The author is careful to sketch the results which he invokes, and provides adequate references for them. However, it would be useful to have an index of notation.

There are also one or two points in the presentation which may make the text hard going for the inexperienced. For example, in the definition of \(R_+(G)\), a fundamental point in this text, the author uses “character with nonzero value” as a synonym for “one-dimensional representation”, which is not quite accurate. He also introduces the term “\(G\)-conjugacy” before he defines it (at least, as far as I could tell). There is a gap in the author’s proof of the classical Brauer induction theorem, which (he assures me) is not too hard to fill.

These quibbles apart, the book is a good introduction to explicit Brauer induction and its arithmetic applications, particularly to Galois module structure, and it will be a valuable addition to the library of anyone working on these topics.

Reviewer: M.E.Keating (MR 96e:20012)

##### MSC:

20C15 | Ordinary representations and characters |

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

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11R42 | Zeta functions and \(L\)-functions of number fields |

11R65 | Class groups and Picard groups of orders |

19A31 | \(K_0\) of group rings and orders |

19A22 | Frobenius induction, Burnside and representation rings |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |