In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
group theory
In 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 ...
, the character of a
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
is a
function on the
group that associates to each group element the
trace of the corresponding
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. The character carries the essential information about the representation in a more condensed form.
Georg Frobenius initially developed
representation theory of finite groups 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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
is determined (up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
) by its character. The situation with representations over a
field of positive
characteristic, so-called "modular representations", is more delicate, but
Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
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.
Applications
Characters of
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
s 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. Close to half of the
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
of the
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. suggested using t ...
involves intricate calculations with character values. Easier, but still essential, results that use character theory include
Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of
Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
and
Michio Suzuki stating that a finite
simple group cannot have a
generalized quaternion group as its
Sylow -subgroup.
Definitions
Let be a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a
field and let be a
representation of a group on . The character of is the function given by
:
where is the
trace.
A character is called irreducible or simple if is an
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
. The degree of the character is the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of ; in characteristic zero this is equal to the value . A character of degree 1 is called linear. When is finite and has characteristic zero, the kernel of the character is the
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
:
:
which is precisely the kernel of the representation . However, the character is ''not'' a group homomorphism in general.
Properties
* Characters are
class functions, that is, they each take a constant value on a given
conjugacy class. More precisely, the set of irreducible characters of a given group into a field form a
basis of the -vector space of all class functions .
*
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
representations have the same characters. Over a field of
characteristic , two representations are isomorphic
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
they have the same character.
* If a representation is the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of
subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations.
* If a character of the finite group is restricted to a
subgroup , then the result is also a character of .
* Every character value is a sum of -th
roots of unity, where is the degree (that is, the dimension of the associated vector space) of the representation with character and is the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of . In particular, when , every such character value is an
algebraic integer.
* If and is irreducible, then
is an
algebraic integer for all in .
* If is
algebraically closed and does not divide the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of , then the number of irreducible characters of is equal to the number of
conjugacy classes of . Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of (and they even divide if ).
Arithmetic properties
Let ρ and σ be representations of . Then the following identities hold:
*
*
*
*