Character (math)
In the mathematical branch of the representation theory of groups , characters are certain representations of the group in a body , usually in the body of complex numbers .
Characters as group homomorphisms
Abstract and topological groups
Be it a group or a topological group . A character of is a group homomorphism
into the multiplicative group of complex numbers ; In the case of topological groups, continuity of character is required. A unitary character is a character whose images lie on the unit circle in the complex number plane , i.e. i.e., which is a homomorphism in the circle group (these numbers correspond precisely to the unitary mappings of the complex numbers in themselves). A unitary character whose images are even real, that is, are located in, is called a square character . Characters that are constant, i.e. whose images are always 1, are called trivial, all others are nontrivial.
The nontrivial quadratic characters of the multiplicative group of an oblique body play a key role in synthetic geometry in introducing a weak arrangement on the affine plane above this oblique body.
Note: Often, general characters are referred to as quasi-characters and unitary characters as characters (without an addition).
properties
- The characters of the train with the
- declared group connection an Abelian group , the character group.
- Pontryagin duality : For locally compact Abelian groups, the group of unitary characters with the compact-open topology is in turn a locally compact group; it is also called the dual group . The dual group of is naturally isomorphic to the parent group .
- The characters of FIG. 4 correspond to the one-dimensional complex representations of FIG. 4, the unitary characters correspond to the unitary one-dimensional representations.
- A character is unitary if and only if applies to all .
- If finite, every character is unitary.
- For a character of a finite group :
- 1 stands for the trivial character with for all . An analogous statement applies to compact topological groups; the sum is to be replaced by an integral based on the Haar measure .
Example S _{3}
On the symmetrical group S _{3 of the} third degree there are exactly two group homomorphisms with values in , namely the trivial group homomorphism and the sign function . This example shows that for non-Abelian groups the characters defined here are not sufficient to reconstruct the group, that is, there is no Pontryagin duality.
For the investigation of non-Abelian groups one uses the more general concept of the character of a representation presented below.
Dirichlet characters
In number theory , a Dirichlet character is a character on the group
For such a character one defines a function also called a Dirichlet character
- ,
- .
Dirichlet characters play an important role in proving Dirichlet's theorem about the existence of an infinite number of prime numbers in arithmetic progressions . One looks at so-called L series, which are Dirichlet series with a Dirichlet character as coefficients.
As for finite Abelian groups, the character group isomorphic to the original group, there are different characters in the group , it is the Euler phi function .
For example , i. In other words , there are three other characters in addition to the main or trivial character:
k | 1 | 2 | 3 | 4th |
---|---|---|---|---|
1 | 1 | 1 | 1 | |
1 | −1 | −1 | 1 | |
1 | i | -i | −1 | |
1 | -i | i | −1 |
For a Dirichlet character :
For a solid applies
where the sum is taken across all characters .
A Dirichlet character is a fully multiplicative number theoretic function .
Algebraic groups
If an algebraic group , then a character of is a homomorphism ; where is the multiplicative group . The characters of form an (abstract) Abelian group, which is denoted by or .
Characters from representations
definition
The following concept of a character comes from the representation theory of groups and is an extension of the character concept defined above.
If a group, a body and a finite-dimensional linear representation of , then the mapping is called
which assigns the trace of the corresponding -linear automorphism to a group element , the character of . In the one-dimensional case, representation and character are practically identical and it is a character of in the sense defined above. In the multi-dimensional case, however, it is usually not multiplicative. If the characteristic is finite and algebraically closed, then the theory can be completely reduced to the one-dimensional case if and only if it is abelian.
Irreducible characters
The characters of irreducible representations are also called irreducible. The one-dimensional representations are precisely the group homomorphisms considered above, which, because of their one-dimensionality, agree with their characters.
For representations of finite groups and when the characteristic of the body is not a divisor of the group order , which is always fulfilled in particular with characteristic 0, i.e. with bodies like or , all representations are sums of irreducible representations according to Maschke's theorem . Because the trace with regard to the formation of the direct sum is additive, all characters are then sums of irreducible characters. See representation theory of finite groups .
properties
- Equivalent representations have the same character. The inversion - if two characters are identical, the corresponding representations are also equivalent - does not always apply, but always, for example, if the characteristic of the body is 0 and the representation is irreducible.
- If the field of complex numbers is finite, then the values of the characters are always finite sums of roots of unity , especially algebraic numbers , and it is true again .
- Characters are constant on conjugation classes . A tabular listing of the values of the characters of the irreducible representations of a finite group on the individual conjugation classes is called a character table . A practical property for finding irreducible representations are the Schursch orthogonality relations for characters.
- Each character is the neutral element to the dimension of the presentation space from, because the neutral element in a matrix representation of the identity matrix displayed and this has as trace the sum of the diagonal elements, which is the dimension of the presentation space.
- The following applies to the character of any representation:
- is the sum of the eigenvalues of with multiplicity.
- Let be the character of a unitary representation of the dimension Then:
- The following applies to the order :
- is the sum of -th roots of unity .
- is a normal divisor in
- Let be two linear representations of and be the associated characters. Then:
- The character of the dual representation of is given by
- The character of the direct sum corresponds
- The character of the tensor product corresponds to
- The character of the related representation is
- Be the character to the character then the character of given by
- Let be a linear representation of and be the corresponding character. Let be the character of the symmetrical square and be the character of the alternating square. For each applies:
Examples
The character of a -dimensional representation is
For the permutation representation of associated to the left operation of on a finite set is
In addition to the two group homomorphisms already mentioned above, there is another irreducible character of group S _{3} . This comes from the two-dimensional irreducible representation of this group . It maps the neutral element to 2, the dimension of the representation space, the three elements of order 2 are mapped to 0 and the two nontrivial rotations to .
Another example is the character of the regular representation He is given by
Here it makes sense to only speak of the regular representation and not to differentiate between left and right regular, as they are isomorphic to each other and therefore have the same character.
As a final example, consider Be defined by:
Then the character is given by
As can be seen from this example, the character is generally not a group homomorphism.
Dot product and characters
Class functions
To prove some interesting results about characters, it is worth looking at a more general set of functions on a group:
The class functions:
A function that fulfills is called a class function.
The set of all class functions is an algebra, the dimension of which corresponds to the number of conjugation classes of .
sentence
Let be the different irreducible characters of A class function on is a character of if and only if it can be represented as a linear combination with non-negative coefficients.
proof
Be so that with for everyone Then the character is the direct sum of the representations that belong to the. Conversely, a character can always be written as the sum of irreducible characters.
Scalar product
Evidence for the following results from this section can be found in
To do this, however, we first need a few definitions:
A scalar product can be defined on the set of all complex-valued functions on a finite group :
In addition, one can define a symmetrical bilinear form:
Both forms match on the characters. The index for both forms and can be omitted if there is no risk of confusion with regard to the underlying group.
For two modules we define where is the vector space of all -linear mappings. This form is bilinear with respect to the direct sum.
Decomposition and irreducibility of characters
In the following, these bilinear forms enable us to obtain some important results with regard to the decomposition and irreducibility of representations.
sentence
If the characters of two non-isomorphic irreducible representations of a finite group then hold
- d. h., has "norm"
Corollary
If the characters are then:
This corollary is a direct consequence of the above theorem, the lemma of Schur and the complete reducibility of the representations of finite groups.
sentence
Let be a linear representation of with character . Let the irreducible apply . Let now be an irreducible representation of with character Then the following applies:
The number of partial representations that are too equivalent does not depend on the given decomposition and corresponds to the scalar product
, i.e. the isotype of is independent of the choice of decomposition and it applies
and thus
Corollary
Two representations with the same character are isomorphic. That is, every representation of a finite group is determined by its character.
Now we get a very practical result for examining representations:
- Irreducibility criterion
Let the character of a representation then be and it applies if and only if is irreducible.
Together with the first set thus form the characters of irreducible representations with respect to this scalar product an orthonormal system on
Corollary
Let be a vector space with Every irreducible representation of is -mal in the regular representation . That means, for the regular representation of : where describes the set of all irreducible representations of that are pairwise not isomorphic to one another.
In words of group algebra we get as algebras.
As a numerical result we get:
where the regular representation denotes and / or the characters belonging to / . In addition, it should be mentioned that the neutral element denotes the group.
This formula is a necessary and sufficient condition for all irreducible representations of a group except for isomorphism and provides a possibility to check whether one has found all irreducible representations of a group apart from isomorphism.
Likewise, we get, again about the character of the regular representation, but this time for equality:
By describing the representations with the convolutionalgebra , we get equivalent formulations of these last two equations:
The Fourier inversion formula:
You can also show the Plancherel formula :
In both formulas there is a linear representation of the group and
The above corollary has another consequence:
lemma
Be a group. Then are equivalent:
- is abelian .
- Every function on is a class function.
- All irreducible representations of have degrees
Finally, let us remind you once again of the definition of the class functions in order to recognize what a special position the characters occupy among them:
Orthonormal property
Be a finite group. The pairwise non-isomorphic irreducible characters of form an orthonormal basis of with respect to the scalar product defined at the beginning of the section.
That means for irreducible characters and the following applies:
The proof is based on the proof that there is no class function other than that which is orthogonal to the irreducible characters.
Equivalent to the orthonormal property, the following applies:
The number of all irreducible representations of a finite group except for isomorphism corresponds exactly to the number of all conjugation classes of
In words of group algebra this means that there are just as many simple modules (except for isomorphism) as there are conjugation classes of
literature
Character of a finite group
- JH Conway : Atlas of Finite Groups. Maximum Subgroups and Ordinary Characters for Simple Groups . Clarendon Press, Oxford 1985, ISBN 0-19-853199-0 .
Dirichlet character
- Jörg Brüdern : Introduction to analytical number theory . Springer, Berlin a. a. 1995, ISBN 3-540-58821-3 .
Further literature
- ^ Jean-Pierre Serre: Linear Representations of Finite Groups. Springer Verlag, New York 1977, ISBN 0-387-90190-6 .
- ^ William Fulton, Joe Harris: Representation Theory A First Course. Springer-Verlag, New York 1991, ISBN 0-387-97527-6 .
- ^ JL Alperin, Rowen B. Bell: Groups and Representations. Springer-Verlag, New York 1995, ISBN 0-387-94525-3 .