In
mathematics, a character is (most commonly) a special kind of
function from a
group to a
field (such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s). There are at least two distinct, but overlapping meanings.
Other uses of the word "character" are almost always qualified.
Multiplicative character
A multiplicative character (or linear character, or simply character) on a group ''G'' is a
group homomorphism from ''G'' to the
multiplicative group of a field , usually the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms 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 com ...
under pointwise multiplication.
This group is referred to as the
character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
); other such homomorphisms are then called ''quasi-characters''.
Dirichlet characters can be seen as a special case of this definition.
Multiplicative characters are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
, i.e. if
are different characters on a group ''G'' then from
it follows that
.
Character of a representation
The character
of a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of a group ''G'' on a
finite-dimensional 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 ...
''V'' over a field ''F'' is the
trace of the representation
, i.e.
:
for
In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "
character theory" and one-dimensional characters are also called "linear characters" within this context.
Alternative definition
If restricted to
finite
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
Traditionally, a finite verb (from la, fīnītus, past partici ...
abelian group with
representation in
(i.e.
), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a
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 mo ...
of
representations. For non-abelian groups, the original definition would be more general than this one):
A character
of group
is a group homomorphism
i.e.
for all
If
is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by
where
is the
circle group.
See also
*
Character group
*
Dirichlet character
*
Harish-Chandra character
*
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ...
*
Infinitesimal character
*
Alternating character
*
Characterization (mathematics)
In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X' ...
*
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
References
* Lectures Delivered at the University of Notre Dame
*
External links
* {{springer, title=Character of a group, id=p/c021560
Representation theory