Character theory

# Character theory

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, the character of a group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

on the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

which associates to each group element the trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

of the corresponding matrix.
The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups
Representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...

entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

, so-called "modular representations", is more delicate, but Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...

developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...

.

## Applications

Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

. Close to half of the proof of the Feit–Thompson theorem
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...

involves intricate calculations with character values. Easier, but still essential, results that use character theory include the Burnside theorem
Burnside theorem
In mathematics, Burnside's theorem in group theory states that if G is a finite group of orderp^a q^b\ where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each...

(a purely group-theoretic proof of the Burnside theorem now exists, but that proof came over half a century
after Burnside's original proof), and a theorem of Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...

and Michio Suzuki
Michio Suzuki
was a Japanese mathematician who studied group theory.-Biography:He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago , the Institute for Advanced Study , the University of Tokyo , and the...

stating that a finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

cannot have a generalized quaternion group
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

as its Sylow 2 subgroup.

## Definitions

Let V be a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

F and let ρ:G → GL(V) be a representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

of a group G on V. The character of ρ is the function χρ: GF given by
where is the trace.

A character χρ is called irreducible if ρ is an irreducible representation. It is called linear if the dimension of ρ is 1. When G is finite and F has characteristic zero, the kernel of the character χρ is the normal subgroup: , which is precisely the kernel of the representation ρ.

## Properties

• Characters are class function
Class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...

s, that is, they each take a constant value on a given conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...

.

• Isomorphic representations have the same characters. Over a field of characteristic 0, representations are isomorphic if and only if they have the same character.

• If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations.

• If a character of the finite group G is restricted to a subgroup H, then the result is also a character of H.

• Every character value is a sum of n mth roots of unity, where n is the degree (that is, the dimension of the associated vector space) of the representation with character χ and m is the order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...

.

• If F is the field of complex numbers, and is irreducible, then is an algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...

for each x in G.

• If F is algebraically closed and char(F) does not divide |G|, then the number of irreducible characters of G is equal to the number of conjugacy classes of G. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of G.

### Arithmetic properties

Let ρ and σ be representations of G. Then the following identities hold:

where is the direct sum, is the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

, denotes the conjugate transpose
Conjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...

of ρ, and Alt2 is the alternating product
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

Alt2 (ρ) = and Sym2 is the symmetric square, which is determined by
.

## Character tables

The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on the representatives of the respective conjugacy class of G. The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degree
Degree (mathematics)
In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...

s of the irreducible characters. Characters of degree 1 are known as linear characters.

Here is the character table of , the cyclic group with three elements and generator u:
 (1) (u) (u2) 1 1 1 1 χ1 1 ω ω2 χ2 1 ω2 ω

where ω is a primitive third root of unity.

The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes. The first row of the character table always consists of 1s, and corresponds to the trivial representation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...

(the 1-dimensional representation consisting of 1×1 matrices containing the entry 1).

### Orthogonality relations

The space of complex-valued class function
Class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G...

s of a finite group G has a natural inner-product:

where means the complex conjugate of the value of on g. With respect to this inner product, the irreducible characters form an orthonormal basis
for the space of class-functions, and this yields the orthogonality relation for the rows of the character
table:

For the orthogonality relation for columns is as follows:

where the sum is over all of the irreducible characters of G and the symbol denotes the order of the centralizer of .

The orthogonality relations can aid many computations including:
• Decomposing an unknown character as a linear combination of irreducible characters.
• Constructing the complete character table when only some of the irreducible characters are known.
• Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
• Finding the order of the group.

### Character table properties

Certain properties of the group G can be deduced from its character table:
• The order of G is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). (See Representation theory of finite groups#Applying Schur's lemma.) More generally, the sum of the squares of the absolute values of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
• All normal subgroups of G (and thus whether or not G is simple) can be recognised from its character table. The kernel
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...

of a character χ is the set of elements g in G for which χ(g) = χ(1); this is a normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G.
• The derived subgroup of G is the intersection of the kernels of the linear characters of G. In particular, G is Abelian if and only if all its irreducible characters are linear.
• It follows, using some results of Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...

from modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...

, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of Graham Higman
Graham Higman
Graham Higman FRS was a leading British mathematician. He is known for his contributions to group theory....

).

The character table does not in general determine the group up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

isomorphism
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...

: for example, the quaternion group
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...

Q and the dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

of 8 elements (D4) have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by E. C. Dade.

The linear characters form a character group
Character group
In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters which arises in the related context of character theory...

, which has important number theoretic
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

connections.

## Induced characters and Frobenius reciprocity

The characters discussed in this section are assumed to be complex-valued. Let H be a subgroup of the finite group G. Given a character of G, let denote its restriction to H.
Let be a character of H. Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...

showed how to construct a character of G from , using what is now known as Frobenius reciprocity. Since the irreducible
characters of G form an orthonormal basis for the space of complex-valued class functions of G,
there is a unique class function of G with the property that

for each irreducible character
of G (the leftmost inner product is for class functions of G and the rightmost inner product
is for class functions of H). Since the restriction of a character of G to the subgroup H
is again a character of H, this definition makes it clear that is a
non-negative integer combination of irreducible characters of G, so is indeed a character of G.
It is known as the character of G induced from θ. The defining formula of Frobenius reciprocity
can be extended to general complex-valued class functions.

Given a matrix representation ρ of H, Frobenius later gave an explicit way to construct a matrix representation of G, known as the representation induced from
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...

ρ, and written analogously as . This led to an alternative description of the induced
character . This induced character vanishes on all elements of G which are
not conjugate to any element of H. Since the induced character is a class function of G, it is only now necessary to describe its values on elements of H. Writing G as a disjoint union of right cosets
of H, say

and given an element h of H, the value
is precisely the sum of those for which
the conjugate is also in H. Because θ is a class function of H, this value does not depend on the particular choice of coset representatives.

This alternative description of the induced character sometimes allows explicit computation from relatively
little information about the embedding of H in G, and is often useful for calculation of
particular character tables. When θ is the trivial character of H, the induced character
obtained is known as the permutation character of G (on the cosets of H).

The general technique of character induction and later refinements found numerous applications in finite group theory and elsewhere in mathematics, in the hands of mathematicians such as Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

, Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...

, Walter Feit
Walter Feit
Walter Feit was a Jewish Austrian-American mathematician who worked in finite group theory and representation theory....

and Michio Suzuki
Michio Suzuki
was a Japanese mathematician who studied group theory.-Biography:He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago , the Institute for Advanced Study , the University of Tokyo , and the...

, as well as Frobenius himself.

## Mackey decomposition

Mackey decomposition was defined and explored by George Mackey in the context of Lie groups, but is a powerful tool in the character theory and representation theory of finite groups. Its basic form concerns the way a character (or module) induced from a subgroup H of a finite group G behaves on restriction back to a (possibly different) subgroup K of G, and makes use of the decomposition of G into (H,K)-double cosets.

If

is a disjoint union, and is a complex class function of H, then Mackey's formula states that

where is the class function of defined by for each h in H. There is a similar formula for the restriction of an induced module to a subgroup, which holds for representations over any ring, and has applications in a wide variety of algebraic and topological contexts.

Mackey decomposition, in conjunction with Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functions θ and ψ induced from respective subgroups H and K, whose utility lies in the fact that it only depends on how conjugates of H and K intersect each other. The formula (with its derivation) is:

(where T is a full set of (H,K)- double coset representatives, as before). This formula is often used when θ and ψ are linear characters, in which case all the inner products appearing in the right hand sum are either 1 or 0, depending on whether or not the linear characters θt and ψ have the same restriction to . If θ and ψ are both trivial characters, then the inner product simplifies to |T|.

## "Twisted" dimension

One may interpret the character of a representation as the "twisted" dimension of a vector space
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

. Treating the character as a function of the elements of the group , its value at the identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

is the dimension of the space, since Accordingly, one can view the other values of the character as "twisted" dimensions.

One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...

: the j-invariant
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...

is the graded dimension of an infinite-dimensional graded representation of the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...

, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.