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 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 is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer 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 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. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem, and a theorem of Richard Brauer 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 vector space 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. The degree of the character is the dimension 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:
which is precisely the kernel of the representation. However, the character is not a group homomorphism in general.

Properties

Let ρ and σ be representations of. Then the following identities hold:
where is the direct sum, is the tensor product, denotes the conjugate transpose of, and is the alternating product and 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 in a compact form. Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of. The columns are labelled by the conjugacy classes of. It is customary to label the first row by the character of the trivial representation, which is the trivial action of on a 1-dimensional vector space by for all. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the identity. Therefore, the first column contains the degree of each irreducible character.
Here is the character table of
the cyclic group with three elements and generator u:

where 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.

Orthogonality relations

The space of complex-valued class functions of a finite group has a natural inner-product:
where is the complex conjugate of. 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 in, applying the same inner product to the columns of the character table yields:
where the sum is over all of the irreducible characters of and the symbol denotes the order of the centralizer of. Note that since and are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
The orthogonality relations can aid many computations including:
Certain properties of the group can be deduced from its character table:
The character table does not in general determine the group up to isomorphism: for example, the quaternion group and the dihedral group of elements,, 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 representations of are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if and are linear representations, then defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation. This group is connected to Dirichlet characters and Fourier analysis.

Induced characters and Frobenius reciprocity

The characters discussed in this section are assumed to be complex-valued. Let be a subgroup of the finite group. Given a character of, let denote its restriction to. Let be a character of. Ferdinand Georg Frobenius showed how to construct a character of from, using what is now known as Frobenius reciprocity. Since the irreducible characters of form an orthonormal basis for the space of complex-valued class functions of, there is a unique class function of with the property that
for each irreducible character of . Since the restriction of a character of to the subgroup is again a character of, this definition makes it clear that is a non-negative integer combination of irreducible characters of, so is indeed a character of. It is known as the character of induced from. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.
Given a matrix representation of, Frobenius later gave an explicit way to construct a matrix representation of, known as the representation induced from, and written analogously as. This led to an alternative description of the induced character. This induced character vanishes on all elements of which are not conjugate to any element of. Since the induced character is a class function of, it is only now necessary to describe its values on elements of. If one writes as a disjoint union of right cosets of, say
then, given an element of, we have:
Because is a class function of, 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 in, and is often useful for calculation of particular character tables. When is the trivial character of, the induced character obtained is known as the permutation character of .
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, Richard Brauer, Walter Feit and Michio Suzuki, as well as Frobenius himself.

Mackey decomposition

The 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 induced from a subgroup of a finite group behaves on restriction back to a subgroup of, and makes use of the decomposition of into -double cosets.
If
is a disjoint union, and is a complex class function of, then Mackey's formula states that
where is the class function of defined by for all in. 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 and induced from respective subgroups and, whose utility lies in the fact that it only depends on how conjugates of and intersect each other. The formula is:
. This formula is often used when and are linear characters, in which case all the inner products appearing in the right hand sum are either or, depending on whether or not the linear characters and have the same restriction to. If and are both trivial characters, then the inner product simplifies to.

"Twisted" dimension

One may interpret the character of a representation as the "twisted" dimension of a vector space. Treating the character as a function of the elements of the group, its value at the identity 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: the -invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.

Characters of Lie groups and Lie algebras

If is a Lie group and a finite dimensional representation of, the character of is defined precisely as for any group as
Meanwhile, if is a Lie algebra and a finite dimensional representation of, we can define the character by
The character will satisfy for all in the associated Lie group and all. If we have a Lie group representation and an associated Lie algebra representation, the character of the Lie algebra representation is related to the character of the group representation by the formula
Suppose now that is a complex semisimple Lie algebra with Cartan subalgebra. The value of the character of an irreducible representation of is determined by its values on. The restriction of the character to can easily be computed in terms of the weight spaces, as follows:
where the sum is over all weights of and where is the multiplicity of.
The character can be computed more explicitly by the Weyl character formula.