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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 the Burnside theorem (a purely group-theoretic proof of the Burnside 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 2-subgroup.

Definitions

Let V be a finite-dimensional vector space over a field F and let ρ : G → GL(V) be a representation of a group G on V. The character of ρ is the function χ_ρ : G → F given by

χρ(g)=Tr(ρ(g))displaystyle chi _rho (g)=mathrm Tr (rho (g))

where Tr is the trace.

A character χ_ρ is called irreducible or simple if ρ is an irreducible representation. The degree of the character χ is the dimension of ρ; in characteristic zero
this is equal to the value χ(1). A character of degree 1 is called linear. When G is finite and F has characteristic zero, the kernel of the character χ_ρ is the normal subgroup:

ker⁡χρ:=g∈G∣χρ(g)=χρ(1),displaystyle ker chi _rho :=leftlbrace gin Gmid chi _rho (g)=chi _rho (1)rightrbrace ,

which is precisely the kernel of the representation ρ. 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 G into a field K form a basis of the K-vector space of all class functions G → K.


• 
  Isomorphic representations have the same characters. Over an algebraically closed field of characteristic 0, semisimple representations are isomorphic if and only if they have the same character.


• 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 G is restricted to a subgroup H, then the result is also a character of H.


• Every character value χ(g) is a sum of n m-th roots of unity, where n is the degree (that is, the dimension of the associated vector space) of the representation with character χ and m is the order of g. In particular, when F = C, every such character value is an algebraic integer.


• If F = C, and χ is irreducible, then

[G:CG(x)]χ(x)χ(1)displaystyle [G:C_G(x)]frac chi (x)chi (1)

is an algebraic integer for all x in G.

• If F is algebraically closed and char(F) does not divide the order of G, then the number of irreducible characters of G is equal to the number of conjugacy classes of G. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of G (and they even divide [G : Z(G)] if F = C).

Arithmetic properties

Let ρ and σ be representations of G. Then the following identities hold:

χρ⊕σ=χρ+χσdisplaystyle chi _rho oplus sigma =chi _rho +chi _sigma

χρ⊗σ=χρ⋅χσdisplaystyle chi _rho otimes sigma =chi _rho cdot chi _sigma

χρ∗=χρ¯displaystyle chi _rho ^*=overline chi _rho

χAlt2ρ(g)=12[(χρ(g))2−χρ(g2)]displaystyle chi _scriptscriptstyle rm Alt^2rho (g)=tfrac 12left[left(chi _rho (g)right)^2-chi _rho (g^2)right]

χSym2ρ(g)=12[(χρ(g))2+χρ(g2)]displaystyle chi _scriptscriptstyle rm Sym^2rho (g)=tfrac 12left[left(chi _rho (g)right)^2+chi _rho (g^2)right]

where ρ ⊕ σ is the direct sum, ρ ⊗ σ is the tensor product, ρ^∗ denotes the conjugate transpose of ρ, and Alt² is the alternating product Alt²ρ = ρ ∧ ρ and Sym² is the symmetric square, which is determined by

ρ⊗ρ=(ρ∧ρ)⊕Sym2ρdisplaystyle rho otimes rho =left(rho wedge rho right)oplus textrm Sym^2rho .

Character tables

The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G 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 G. The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degree of the irreducible characters. Characters of degree 1 are known as linear characters.

Here is the character table of

C3=⟨u∣u3=1⟩,displaystyle C_3=langle umid u^3=1rangle ,

the cyclic group with three elements and generator u:

where ω 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.^[1] The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1 × 1 matrices containing the entry 1).

Orthogonality relations

The space of complex-valued class functions of a finite group G has a natural inner-product:

⟨α,β⟩:=1|G|∑g∈Gα(g)β(g)¯displaystyle leftlangle alpha ,beta rightrangle :=frac 1Gsum _gin Galpha (g)overline beta (g)

where β(g) is the complex conjugate of β(g). 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:

⟨χi,χj⟩={0 if i≠j,1 if i=j.displaystyle leftlangle chi _i,chi _jrightrangle =begincases0&mbox if ineq j,\1&mbox if i=j.endcases

For g, h in G the orthogonality relation for columns is as follows:

∑χiχi(g)χi(h)¯={|CG(g)|, if g,h are conjugate 0 otherwise.displaystyle sum _chi _ichi _i(g)overline chi _i(h)=,&mbox if g,hmbox are conjugate \0&mbox otherwise.endcases

where the sum is over all of the irreducible characters χ_i of G and the symbol |C_G(g)| denotes the order of the centralizer of g.

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 G can be deduced from its character table:

• The order of G is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). (See Representation theory of finite groups#Applying Schur's lemma.) 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 G (and thus whether or not G is simple) can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ(g) = χ(1); this is a normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G.


• The derived subgroup of G is the intersection of the kernels of the linear characters of G. In particular, G is Abelian if and only if all its irreducible characters are 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 Q and the dihedral group of 8 elements, D₄, 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 characters form a character group, which has important number theoretic connections.^[which?]

Induced characters and Frobenius reciprocity

The characters discussed in this section are assumed to be complex-valued. Let H be a subgroup of the finite group G. Given a character χ of G, let χ_H denote its restriction to H. Let θ be a character of H. Ferdinand Georg Frobenius showed how to construct a character of G from θ, using what is now known as Frobenius reciprocity. Since the irreducible characters of G form an orthonormal basis for the space of complex-valued class functions of G, there is a unique class function θ^G of G with the property that

⟨θG,χ⟩G=⟨θ,χH⟩Hdisplaystyle langle theta ^G,chi rangle _G=langle theta ,chi _Hrangle _H

for each irreducible character χ of G (the leftmost inner product is for class functions of G and the rightmost inner product is for class functions of H). Since the restriction of a character of G to the subgroup H is again a character of H, this definition makes it clear that θ^G is a non-negative integer combination of irreducible characters of G, so is indeed a character of G. It is known as the character of G induced from θ. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.

Given a matrix representation ρ of H, Frobenius later gave an explicit way to construct a matrix representation of G, known as the representation induced from ρ, and written analogously as ρ^G. This led to an alternative description of the induced character θ^G. This induced character vanishes on all elements of G which are not conjugate to any element of H. Since the induced character is a class function of G, it is only now necessary to describe its values on elements of H. If one writes G as a disjoint union of right cosets of H, say

G=Ht1∪…∪Htn,displaystyle G=Ht_1cup ldots cup Ht_n,

then, given an element h of H, we have:

θG(h)=∑i : tihti−1∈Hθ(tihti−1).displaystyle theta ^G(h)=sum _i : t_iht_i^-1in Htheta left(t_iht_i^-1right).

Because θ is a class function of H, 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 H in G, and is often useful for calculation of particular character tables. When θ is the trivial character of H, the induced character obtained is known as the permutation character of G (on the cosets of H).

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

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 H of a finite group G behaves on restriction back to a (possibly different) subgroup K of G, and makes use of the decomposition of G into (H, K)-double cosets.

If

G=⋃t∈THtKdisplaystyle G=bigcup _tin THtK

is a disjoint union, and θ is a complex class function of H, then Mackey's formula states that

(θG)K=∑t∈T([θt]t−1Ht∩K)K,displaystyle left(theta ^Gright)_K=sum _tin Tleft(left[theta ^tright]_t^-1Htcap Kright)^K,

where θ^t is the class function of t⁻¹Ht defined by θ^t(t⁻¹ht) = θ(h) for all h in H. 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 θ and ψ induced from respective subgroups H and K, whose utility lies in the fact that it only depends on how conjugates of H and K intersect each other. The formula (with its derivation) is:

⟨θG,ψG⟩=⟨(θG)K,ψ⟩=∑t∈T⟨([θt]t−1Ht∩K)K,ψ⟩=∑t∈T⟨(θt)t−1Ht∩K,ψt−1Ht∩K⟩,displaystyle beginalignedleftlangle theta ^G,psi ^Grightrangle &=leftlangle left(theta ^Gright)_K,psi rightrangle \&=sum _tin Tleftlangle left(left[theta ^tright]_t^-1Htcap Kright)^K,psi rightrangle \&=sum _tin Tleftlangle left(theta ^tright)_t^-1Htcap K,psi _t^-1Htcap Krightrangle ,endaligned

(where T is a full set of (H, K)-double coset representatives, as before). This formula is often used when θ and ψ are linear characters, in which case all the inner products appearing in the right hand sum are either 1 or 0, depending on whether or not the linear characters θ^t and ψ have the same restriction to t⁻¹Ht ∩ K. If θ and ψ are both trivial characters, then the inner product simplifies to |T |.

"Twisted" dimension

One may interpret the character of a representation as the "twisted" dimension of a vector space.^[2] Treating the character as a function of the elements of the group χ(g), its value at the identity is the dimension of the space, since χ(1) = Tr(ρ(1)) = Tr(I_V) = dim(V). Accordingly, one can view the other values of the character as "twisted" dimensions.^{[clarification needed]}

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 j-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.^[2]

Characters of Lie groups and Lie algebras

If Gdisplaystyle G is a Lie group and ρdisplaystyle rho a finite dimensional representation of Gdisplaystyle G, the character Xρdisplaystyle mathrm X _rho of ρdisplaystyle rho is defined precisely as for any group as

Xρ(g)=Tr(ρ(g))displaystyle mathrm X _rho (g)=mathrm Tr (rho (g)).

Meanwhile, if gdisplaystyle mathfrak g is a Lie algebra and ρdisplaystyle rho a finite dimensional representation of gdisplaystyle mathfrak g, we can define the character χρdisplaystyle chi _rho by

χρ(X)=Tr(eρ(X))displaystyle chi _rho (X)=mathrm Tr (e^rho (X)).

The character will satisfy χρ(Adg(X))=χρ(X)displaystyle chi _rho (mathrm Ad _g(X))=chi _rho (X) for all gdisplaystyle g in the associated Lie group Gdisplaystyle G and all X∈gdisplaystyle Xin mathfrak g. If we have a Lie group representation and an associated Lie algebra representation, the character χρdisplaystyle chi _rho of the Lie algebra representation is related to the character Xρdisplaystyle mathrm X _rho of the group representation by the formula

χρ(X)=Xρ(eX)displaystyle chi _rho (X)=mathrm X _rho (e^X).

Suppose now that gdisplaystyle mathfrak g is a complex semisimple Lie algebra with Cartan subalgebra hdisplaystyle mathfrak h. The value of the character χρdisplaystyle chi _rho of an irreducible representation ρdisplaystyle rho of gdisplaystyle mathfrak g is determined by its values on hdisplaystyle mathfrak h. The restriction of the character to hdisplaystyle mathfrak h can easily be computed in terms of the weight spaces, as follows:

χρ(H)=∑λmλeλ(H),H∈hdisplaystyle chi _rho (H)=sum _lambda m_lambda e^lambda (H),quad Hin mathfrak h,

where the sum is over all weights λdisplaystyle lambda of ρdisplaystyle rho and where mλdisplaystyle m_lambda is the multiplicity of λdisplaystyle lambda .^[3]

The (restriction to hdisplaystyle mathfrak 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 G to a normal subgroup N.


• Frobenius formula


• 
  Real element, a group element g such that χ(g) is a real number for all characters χ
  

References

External links

• 
  Character at PlanetMath.org.