Character theory

Template:Short description Template:About 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.

ApplicationsEdit

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 theorems|Sylow Template:Math-subgroup]].

DefinitionsEdit

Let Template:Mvar be a finite-dimensional vector space over a field Template:Mvar and let Template:Math be a representation of a group Template:Mvar on Template:Mvar. The character of Template:Mvar is the function Template:Math given by

<math>\chi_{\rho}(g) = \operatorname{Tr}(\rho(g))</math>

where Template:Math is the trace.

A character Template:Math is called irreducible or simple if Template:Mvar is an irreducible representation. The degree of the character Template:Mvar is the dimension of Template:Mvar; in characteristic zero this is equal to the value Template:Math. A character of degree 1 is called linear. When Template:Mvar is finite and Template:Mvar has characteristic zero, the kernel of the character Template:Math is the normal subgroup:

<math>\ker \chi_\rho := \left \lbrace g \in G \mid \chi_{\rho}(g) = \chi_{\rho}(1) \right \rbrace, </math>

which is precisely the kernel of the representation Template:Mvar. However, the character is not a group homomorphism in general.

PropertiesEdit

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 Template:Mvar into a field Template:Mvar form a basis of the Template:Mvar-vector space of all class functions Template:Math.
Isomorphic representations have the same characters. Over a field of characteristic Template:Math, two representations are isomorphic if and only if they have the same character.<ref>Nicolas Bourbaki, Algèbre, Springer-Verlag, 2012, Chap. 8, p392</ref>
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 Template:Mvar is restricted to a subgroup Template:Mvar, then the result is also a character of Template:Mvar.
Every character value Template:Math is a sum of Template:Mvar Template:Mvar-th roots of unity, where Template:Mvar is the degree (that is, the dimension of the associated vector space) of the representation with character Template:Mvar and Template:Mvar is the order of Template:Mvar. In particular, when Template:Math, every such character value is an algebraic integer.
If Template:Math and Template:Mvar is irreducible, then <math display="block">[G:C_G(x)]\frac{\chi(x)}{\chi(1)}</math> is an algebraic integer for all Template:Mvar in Template:Mvar.
If Template:Mvar is algebraically closed and Template:Math does not divide the order of Template:Mvar, then the number of irreducible characters of Template:Mvar is equal to the number of conjugacy classes of Template:Mvar. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of Template:Mvar (and they even divide Template:Math if Template:Math).

Arithmetic propertiesEdit

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

<math>\chi_{\rho \oplus \sigma} = \chi_\rho + \chi_\sigma</math>
<math>\chi_{\rho \otimes \sigma} = \chi_\rho \cdot \chi_\sigma</math>
<math>\chi_{\rho^*} = \overline {\chi_\rho}</math>
<math>\chi_{{\scriptscriptstyle \rm{Alt}^2} \rho}(g) = \tfrac{1}{2}\! \left[ \left(\chi_\rho (g) \right)^2 - \chi_\rho (g^2) \right]</math>
<math>\chi_{{\scriptscriptstyle \rm{Sym}^2} \rho}(g) = \tfrac{1}{2}\! \left[ \left(\chi_\rho (g) \right)^2 + \chi_\rho (g^2) \right]</math>

where Template:Math is the direct sum, Template:Math is the tensor product, Template:Math denotes the conjugate transpose of Template:Mvar, and Template:Math is the alternating product Template:Math and Template:Math is the symmetric square, which is determined by <math display="block">\rho \otimes \rho = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho.</math>

Character tablesEdit

Template:Further The irreducible complex characters of a finite group form a character table which encodes much useful information about the group Template:Mvar 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 Template:Mvar. The columns are labelled by (representatives of) the conjugacy classes of Template:Mvar. It is customary to label the first row by the character of the trivial representation, which is the trivial action of Template:Mvar on a 1-dimensional vector space by <math> \rho(g)=1</math> for all <math> g\in G </math>. 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

<math>C_3 = \langle u \mid u^{3} = 1 \rangle,</math>

the cyclic group with three elements and generator u:

	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:Mvar	Template:Math
Template:Math	Template:Math	Template:Math	Template:Mvar

where Template:Mvar 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.<ref>Serre, §2.5</ref>

Orthogonality relationsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The space of complex-valued class functions of a finite group Template:Mvar has a natural inner product:

<math>\left \langle \alpha, \beta\right \rangle := \frac{1}{|G|}\sum_{g \in G} \alpha(g) \overline{\beta(g)}</math>

where Template:Math is the complex conjugate of Template:Math. 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:

<math>\left \langle \chi_i, \chi_j \right \rangle = \begin{cases} 0 & \mbox{ if } i \ne j, \\ 1 & \mbox{ if } i = j. \end{cases}</math>

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

<math>\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases} \left | C_G(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}</math>

where the sum is over all of the irreducible characters Template:Math of Template:Mvar and the symbol Template:Math denotes the order of the centralizer of Template:Mvar. Note that since Template:Mvar and Template:Mvar 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 propertiesEdit

Certain properties of the group Template:Mvar can be deduced from its character table:

The order of Template:Mvar 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 Template:Mvar (and thus whether or not Template:Mvar is simple) can be recognised from its character table. The kernel of a character Template:Mvar is the set of elements Template:Mvar in Template:Mvar for which Template:Math; this is a normal subgroup of Template:Mvar. Each normal subgroup of Template:Mvar is the intersection of the kernels of some of the irreducible characters of Template:Mvar.
The commutator subgroup of Template:Mvar is the intersection of the kernels of the linear characters of Template:Mvar.
If Template:Mvar is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that Template:Mvar is abelian iff each conjugacy class is a singleton iff the character table of Template:Mvar is <math>|G| \!\times\! |G|</math> 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 Template:Mvar and the dihedral group of Template:Math elements, Template:Math, 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 Template:Mvar are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if <math>\rho_1:G\to V_1</math> and <math> \rho_2:G\to V_2</math> are linear representations, then <math> \rho_1\otimes\rho_2 (g)=(\rho_1(g)\otimes\rho_2(g))</math> defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation <math> [\chi_1*\chi_2](g)=\chi_1(g)\chi_2(g)</math>. This group is connected to Dirichlet characters and Fourier analysis.

Induced characters and Frobenius reciprocityEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The characters discussed in this section are assumed to be complex-valued. Let Template:Mvar be a subgroup of the finite group Template:Mvar. Given a character Template:Mvar of Template:Mvar, let Template:Math denote its restriction to Template:Mvar. Let Template:Mvar be a character of Template:Mvar. Ferdinand Georg Frobenius showed how to construct a character of Template:Mvar from Template:Mvar, using what is now known as Frobenius reciprocity. Since the irreducible characters of Template:Mvar form an orthonormal basis for the space of complex-valued class functions of Template:Mvar, there is a unique class function Template:Math of Template:Mvar with the property that

<math> \langle \theta^{G}, \chi \rangle_G = \langle \theta,\chi_H \rangle_H </math>

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

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

<math>G = Ht_1 \cup \ldots \cup Ht_n,</math>

then, given an element Template:Mvar of Template:Mvar, we have:

<math> \theta^G(h) = \sum_{i \ : \ t_iht_i^{-1} \in H} \theta \left (t_iht_i^{-1} \right ).</math>

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

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 decompositionEdit

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 Template:Mvar of a finite group Template:Mvar behaves on restriction back to a (possibly different) subgroup Template:Mvar of Template:Mvar, and makes use of the decomposition of Template:Mvar into Template:Math-double cosets.

If <math display="inline"> G = \bigcup_{t \in T} HtK </math> is a disjoint union, and Template:Mvar is a complex class function of Template:Mvar, then Mackey's formula states that

<math>\left( \theta^{G}\right)_K = \sum_{ t \in T} \left(\left [\theta^{t} \right ]_{t^{-1}Ht \cap K}\right)^{K},</math>

where Template:Math is the class function of Template:Math defined by Template:Math for all Template:Mvar in Template:Mvar. 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 Template:Mvar and Template:Mvar induced from respective subgroups Template:Mvar and Template:Mvar, whose utility lies in the fact that it only depends on how conjugates of Template:Mvar and Template:Mvar intersect each other. The formula (with its derivation) is:

<math>\begin{align}

\left \langle \theta^{G},\psi^{G} \right \rangle &= \left \langle \left(\theta^{G}\right)_{K},\psi \right \rangle \\ &= \sum_{ t \in T} \left \langle \left( \left [\theta^{t} \right ]_{t^{-1}Ht \cap K}\right)^{K}, \psi \right \rangle \\ &= \sum_{t \in T} \left \langle \left(\theta^{t} \right)_{t^{-1}Ht \cap K},\psi_{t^{-1}Ht \cap K} \right \rangle, \end{align}</math>

(where Template:Mvar is a full set of Template:Math-double coset representatives, as before). This formula is often used when Template:Mvar and Template:Mvar are linear characters, in which case all the inner products appearing in the right hand sum are either Template:Math or Template:Math, depending on whether or not the linear characters Template:Math and Template:Mvar have the same restriction to Template:Math. If Template:Mvar and Template:Mvar are both trivial characters, then the inner product simplifies to Template:Math.

"Twisted" dimensionEdit

One may interpret the character of a representation as the "twisted" dimension of a vector space.<ref name="Gannon">Template:Harv</ref> Treating the character as a function of the elements of the group Template:Math, its value at the identity is the dimension of the space, since Template:Math. Accordingly, one can view the other values of the character as "twisted" dimensions.Template:Clarify

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|Template:Mvar-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.<ref name="Gannon" />

Characters of Lie groups and Lie algebrasEdit

Template:See also

If <math>G</math> is a Lie group and <math>\rho</math> a finite-dimensional representation of <math>G</math>, the character <math>\chi_\rho</math> of <math>\rho</math> is defined precisely as for any group as

<math>\chi_\rho(g)=\operatorname{Tr}(\rho(g))</math>.

Meanwhile, if <math>\mathfrak g</math> is a Lie algebra and <math>\rho</math> a finite-dimensional representation of <math>\mathfrak g</math>, we can define the character <math>\chi_\rho</math> by

<math>\chi_\rho(X)=\operatorname{Tr}(e^{\rho(X)})</math>.

The character will satisfy <math>\chi_\rho(\operatorname{Ad}_g(X))=\chi_\rho(X)</math> for all <math>g</math> in the associated Lie group <math> G</math> and all <math>X\in\mathfrak g</math>. If we have a Lie group representation and an associated Lie algebra representation, the character <math>\chi_\rho</math> of the Lie algebra representation is related to the character <math>\Chi_\rho</math> of the group representation by the formula

<math>\chi_\rho(X)=\Chi_\rho(e^X)</math>.

Suppose now that <math>\mathfrak g</math> is a complex semisimple Lie algebra with Cartan subalgebra <math>\mathfrak h</math>. The value of the character <math>\chi_\rho</math> of an irreducible representation <math>\rho</math> of <math>\mathfrak g</math> is determined by its values on <math>\mathfrak h</math>. The restriction of the character to <math>\mathfrak h</math> can easily be computed in terms of the weight spaces, as follows:

<math>\chi_\rho(H) = \sum_\lambda m_\lambda e^{\lambda(H)},\quad H\in\mathfrak h</math>,

where the sum is over all weights <math>\lambda</math> of <math>\rho</math> and where <math>m_\lambda</math> is the multiplicity of <math>\lambda</math>.<ref>Template:Harvnb Proposition 10.12</ref>

The (restriction to <math>\mathfrak h</math> of the) character can be computed more explicitly by the Weyl character formula.