Character theory explained

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 (up to isomorphism) 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 (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 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

\chi _{\rho} (g) = \operatorname{Tr}(\rho(g))

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:

\ker \chi _{\rho} : = \left \lbrace g \in G \mid \chi _{\rho} (g) = \chi _{\rho} ( 1 ) \right \rbrace,

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 representations have the same characters. Over a field of characteristic, two representations are isomorphic if and only if they have the same character.^[1]
• 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 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 of . In particular, when, every such character value is an algebraic integer.
• If and is irreducible, then $$[G:C\_G(x)]\backslash frac$$ is an algebraic integer for all in .
• If is algebraically closed and does not divide the order 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:

\chi _{\rho} = \chi _{\rho} + \chi _{\sigma} \chi _{\rho} = \chi _{\rho} ⋅ \chi _{\sigma} \chi _{\rho ^{*}} = \overline {\chi _{\rho}} \chi ^{2 }} _{{\scriptscriptstyle \rm{Alt} \rho}(g) = \tfrac{ 1 }{ 2 } \left[ \left(\chi _{\rho} (g) \right) ^{2} - \chi _{\rho} (g ^{2)} \right] \chi ^{2 }} _{{\scriptscriptstyle \rm{Sym} \rho}(g) = \tfrac{ 1 }{ 2 } \left[ \left(\chi _{\rho} (g) \right) ^{2} + \chi _{\rho} (g ^{2)} \right] 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$$\backslash rho\; \backslash otimes\; \backslash rho\; =\; \backslash left(\backslash rho\; \backslash wedge\; \backslash rho\; \backslash right)\; \backslash oplus\; \backslash textrm^2\; \backslash rho.$$

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 (representatives of) 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

\rho(g) =1 for all g\in G . 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

C _{3} = \langle u \mid u ^{3} = 1 \rangle,

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.^[2]

Orthogonality relations

See main article: Schur orthogonality relations. The space of complex-valued class functions of a finite group has a natural inner product:

\left \langle \alpha, \beta\right \rangle : = (1)/(|G|) \sum _{g} \alpha(g) \overline{\beta(g)}

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:

\left \langle \chi _{i,} \chi _{j} \right \rangle = \begin{cases} 0 & if i \ne j, \ 1 & if i = j. \end{cases}

For in, applying the same inner product to the columns of the character table yields:

\sum _{\chi _{i}} \chi _{i(g)} \overline{\chi _{i(h)}} = \begin{cases} \left | C _{G(g)} \right |, & if g, h areconjugate \ 0 & otherwise. \end{cases}

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:

• 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 can be deduced from its character table:

• The order of is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). 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 (and thus whether or not is simple) can be recognised from its character table. The kernel of a character is the set of elements in for which ; this is a normal subgroup of . Each normal subgroup of is the intersection of the kernels of some of the irreducible characters of .
• The commutator subgroup of is the intersection of the kernels of the linear characters of .
• If is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that is abelian iff each conjugacy class is a singleton iff the character table of is

|G| x |G| iff each irreducible character is linear.

• It follows, using some results of Richard Brauer from modular representation theory, 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).

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

\rho _{1:G\to} V _{1} and \rho _{2:G\to} V _{2} are linear representations, then \rho _{1 ⊗ \rho} _{2} (g) = (\rho _{1(g) ⊗ \rho} _{2(g))} defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation [\chi _{1*\chi} _{2](g)=\chi} _{1(g)\chi} _{2(g)} . This group is connected to Dirichlet characters and Fourier analysis.

Induced characters and Frobenius reciprocity

See main article: Induced character 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

\langle \theta ^{G} , \chi \rangle _{G} = \langle \theta,\chi _{H} \rangle _{H}

for each irreducible character of (the leftmost inner product is for class functions of and the rightmost inner product is for class functions 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

G = Ht _{1} \cup \ldots \cup Ht _{n,}

then, given an element of, we have:

\theta ^{G(h)} = \sum _{i : t _{iht} ^{-1} ^{i} \in H} \theta \left (t _{iht} ^{-1} ^{i} \right ).

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 (on the cosets 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 (or module) induced from a subgroup of a finite group behaves on restriction back to a (possibly different) subgroup of, and makes use of the decomposition of into -double cosets.

If $G\; =\; \backslash bigcup\_\; HtK$ is a disjoint union, and is a complex class function of, then Mackey's formula states that

\left( \theta ^{G} \right) _{K} = \sum \left(\left [\theta ^{t} \right ] _{t ^{-1} Ht \cap K} \right) ^{K} ,

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 (with its derivation) is:

\begin{align} \left \langle \theta ^{G} ,\psi ^{G} \right \rangle & = \left \langle \left(\theta ^{G} \right) _{K} ,\psi \right \rangle \\ & = \sum \left \langle \left( \left [\theta ^{t} \right ] _{t ^{-1} Ht \cap K} \right) ^{K} , \psi \right \rangle \\ & = \sum _{t} \left \langle \left(\theta ^{t} \right) _{t ^{-1} Ht \cap K} ,\psi _{t ^{-1} Ht \cap K} \right \rangle, \end{align}

(where is a full set of -double coset representatives, as before). 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

See also: Weyl character formula and Algebraic character.

If

G is a Lie group and \rho a finite-dimensional representation of G , the character \chi _{\rho} of \rho is defined precisely as for any group as\chi _{\rho(g)=\operatorname{Tr}(\rho(g))} .Meanwhile, if ak g is a Lie algebra and \rho a finite-dimensional representation of ak g , we can define the character \chi _{\rho} by\chi ^{\rho(X)} _{\rho(X) = \operatorname{Tr}(e} ) .The character will satisfy \chi _{\rho(\operatorname{Ad}} _{g(X))=\chi} _{\rho(X)} for all g in the associated Lie group G and all X\in ak g . If we have a Lie group representation and an associated Lie algebra representation, the character \chi _{\rho} of the Lie algebra representation is related to the character \Chi _{\rho} of the group representation by the formula\chi _{\rho(X)=\Chi} ^{X)} ^{\rho(e} .

Suppose now that

ak g is a complex semisimple Lie algebra with Cartan subalgebra ak h . The value of the character \chi _{\rho} of an irreducible representation \rho of ak g is determined by its values on ak h . The restriction of the character to ak h can easily be computed in terms of the weight spaces, as follows:\chi _{\rho(H)} = \sum _{λ} m _{λ} e ^{λ(H)} , H\in ak h ,where the sum is over all weights λ of \rho and where m _{λ} is the multiplicity of λ .^[3]

The (restriction to

ak h of the) character can be computed more explicitly by the Weyl character formula.

See also

• • Association schemes, a combinatorial generalization of group-character theory.
• Clifford theory, introduced by A. H. Clifford in 1937, yields information about the restriction of a complex irreducible character of a finite group to a normal subgroup .
• Frobenius formula
• Real element, a group element g such that χ(g) is a real number for all characters χ

References

• Lecture 2 of online
• Book: Gannon, Terry . Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics . 2006 . 978-0-521-83531-2 .
  • Book: Isaacs, I.M. . Character Theory of Finite Groups . Dover . 1994 . 978-0-486-68014-9 . Corrected reprint of the 1976 original, published by Academic Press..
• Book: James . Gordon . Martin Liebeck. Liebeck . Martin . Representations and Characters of Groups (2nd ed.) . 2001 . Cambridge University Press . 978-0-521-00392-6.
• Book: Serre, Jean-Pierre . J.-P. Serre . Linear Representations of Finite Groups . Springer-Verlag . 1977 . 978-0-387-90190-9 . New York-Heidelberg . Graduate Texts in Mathematics . 42 . Translated from the second French edition by Leonard L. Scott . 0450380 . 10.1007/978-1-4684-9458-7 . registration .

Notes and References

1. Nicolas Bourbaki, Algèbre, Springer-Verlag, 2012, Chap. 8, p392
2. Serre, §2.5
3. Proposition 10.12