Brauer's Theorem On Induced Characters
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Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after
Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...
, is a basic result in the branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
known as
character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...
, which is part of the
representation theory of finite groups The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
.


Background

A precursor to Brauer's induction theorem was Artin's induction theorem, which states that , ''G'', times the trivial character of ''G'' is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of ''G.'' Brauer's theorem removes the factor , ''G'', , but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups. Another result between Artin's induction theorem and Brauer's induction theorem, also due to Brauer and also known as ''Brauer's theorem'' or ''Brauer's lemma'' is the fact that the regular representation of ''G'' can be written as 1+\sum\lambda_i\rho_i where the \lambda_i are ''positive rationals'' and the \rho_i are induced from characters of cyclic subgroups of ''G''. Note that in Artin's theorem the characters are induced from the trivial character of the cyclic group, while here they are induced from arbitrary characters (in applications to Artin's ''L'' functions it is important that the groups are cyclic and hence all characters are linear giving that the corresponding ''L'' functions are analytic).


Statement

Let ''G'' be a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
and let Char(''G'') denote the subring of the ring of complex-valued
class function In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugati ...
s of ''G'' consisting of integer combinations of irreducible characters. Char(''G'') is known as the character ring of ''G'', and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of ''G'' is again a character of ''G.'' Its multiplication is given by the elementwise product of class functions. Brauer's induction theorem shows that the character ring can be generated (as an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
) by characters of representations induced from one-dimensional representations of subgroups of ''G''. In fact, Brauer showed that these subgroups could be chosen from a very restricted collection, now called elementary subgroups. By definition, a subgroup is elementary if it is a direct product of a cyclic group and a group whose order is a power of a prime.


Proofs

The proof of Brauer's induction theorem exploits the ring structure of Char(''G'') (most proofs also make use of a slightly larger ring, Char*(G), which consists of \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math>-combinations of irreducible characters, where ω is a primitive complex , ''G'', -th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal ''I''(''G'') of Char(''G''), so the proof reduces to showing that the trivial character is in ''I''(''G''). Several proofs of the theorem, beginning with a proof due to Brauer and John Tate, show that the trivial character is in the analogously defined ideal ''I''*(''G'') of Char*(''G'') by concentrating attention on one prime ''p'' at a time, and constructing integer-valued elements of ''I''*(''G'') which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of ''p.'' Once this is achieved for every prime divisor of , ''G'', , some manipulations with congruences and
algebraic integer In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s, again exploiting the fact that ''I''*(''G'') is an ideal of Char*(''G''), place the trivial character in ''I''(''G''). An auxiliary result here is that a \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math>-valued class function lies in the ideal ''I''*(''G'') if its values are all divisible (in \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math>) by , ''G'', . Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaith gave a proof by a radically different approach, topological in nature (an application of the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is name ...
). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably by Robert Boltje.


Applications

Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization of characters, which asserts that a complex-valued class function of ''G'' is a virtual character if and only if its restriction to each Brauer elementary subgroup of ''G'' is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character if and only if θ(1) ''> 0'' and \langle \theta,\theta \rangle =1 (where \langle,\rangle is the usual inner product on the ring of complex-valued class functions) gives a means of constructing irreducible characters without explicitly constructing the associated representations. An initial motivation for Brauer's induction theorem was application to
Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theo ...
s. It shows that those are built up from
Dirichlet L-function In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By anal ...
s, or more general Hecke L-functions. Highly significant for that application is whether each character of ''G'' is a ''non-negative'' integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of ''G'' are so expressible, then ''G'' must be a
solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminat ...
(although solvability alone does not guarantee such expressions—for example, the solvable group SL(2,3) has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when ''G'' is a finite
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . I ...
, every complex irreducible character of ''G'' is induced from a linear character of some subgroup.


References

* Corrected reprint of the 1976 original, published by Academic Press. *


Notes

{{reflist Representation theory of finite groups Theorems in representation theory