Character theory

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This article refers to the use of the term character theory in mathematics, for the media studies definition see Character theory (Media).

In mathematics, more specifically in group theory, the character of a group representation is a function on the group which 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 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.

1 Applications
2 Definitions
3 Properties
- 3.1 Arithmetic properties
4 Character tables
- 4.1 Orthogonality relations
- 4.2 Character table properties
5 Induced characters and Frobenius reciprocity
6 Mackey decomposition
7 Clifford's theorem
8 References

[edit] 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 relying on character theory include the Burnside 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 2 subgroup.

[edit] 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

$\chi_{\rho}(g) = \mathrm{Tr}(\rho(g))\,$

where $Tr$ is the trace.

A character χ_ρ is called irreducible if ρ is an irreducible representation. It is called linear if the dimension of ρ is 1. The kernel of a character χ_ρ is the set:

$\ker \chi_{\rho} := \left \lbrace g \in G \mid \chi_{\rho}(g) = \chi_{\rho}(1) \right \rbrace$

where χ_ρ(1) is the value of χ_ρ on the group identity. If ρ is a representation of G of dimension k and 1 is the identity of G then

$\chi_{\rho}(1) = \operatorname{Tr}(\rho(1)) = \operatorname{Tr} \begin{bmatrix}1 & & 0\\ & \ddots & \\ 0 & & 1\end{bmatrix} = \sum_{i = 1}^k 1 = k = \dim \rho$

Unlike the situation with the character group, the characters of a group do not, in general, form a group themselves.

[edit] Properties

Characters are class functions, that is, they each take a constant value on a given conjugacy class.

Isomorphic representations have the same characters. If char(F)=0, then 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 $\chi\ (g)$ 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 is the field of complex numbers, every such character value is an algebraic integer.

If F is the field of complex numbers, and $χ$ is irreducible, then $[G:C_G(x)]\frac{\chi(x)}{\chi(1)}$ is an algebraic integer for each x in G.

If F is algebraically closed and char(F) does not divide |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.

[edit] Arithmetic properties

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

$\chi_{\rho \oplus \sigma} = \chi_\rho + \chi_\sigma$

$\chi_{\rho \otimes \sigma} = \chi_\rho \cdot \chi_\sigma$

$\chi_{\rho^*} = \overline {\chi_\rho}$

$\chi_{{\scriptscriptstyle \rm{Alt}^2} \rho}(g) = \frac{1}{2} \left[ \left(\chi_\rho (g) \right)^2 - \chi_\rho (g^2) \right]$

$\chi_{{\scriptscriptstyle \rm{Sym}^2} \rho}(g) = \frac{1}{2} \left[ \left(\chi_\rho (g) \right)^2 + \chi_\rho (g^2) \right]$

where $\rho \oplus \sigma$ is the direct sum, $\rho \otimes \sigma$ is the tensor product, $\rho^*\$ denotes the conjugate transpose of ρ, and Alt² is the alternating product Alt² (ρ) = $\rho \wedge \rho$ and Sym² is the symmetric square, which is determined by

$\rho \otimes \rho = \left(\rho \wedge \rho \right) \oplus \textrm{Sym}^2 \rho$ .

[edit] 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 character and the entries in the row are the values of that character on the representatives of 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 degrees of the irreducible characters. Characters of degree 1 are known as linear characters.

Here is the character table of $C_3 = \langle u \rangle$ , the cyclic group with three elements and generator u:

	(1)	(u)	(u²)
1	1	1	1
χ₁	1	ω	ω²
χ₂	1	ω²	ω

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. 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).

[edit] Orthogonality relations

Main article: Schur orthogonality relations

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

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

where $\overline{\beta(g)}$ means the complex conjugate of the value of $β$ on 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:

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

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

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

where the sum is over all of the irreducible characters $χ i$ of G and the symbol $\left | C_G(g) \right |$ 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.

[edit] 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). 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

[edit] Induced characters and Frobenius reciprocity

Th 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 define 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

$\langle \theta^{G}, \chi \rangle = \langle \theta,\chi_H \rangle$

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. Writing G as a disjoint union of right cosets of H, say

$G = Ht_1 \cup \ldots \cup Ht_n,$

and given an element h of H, the value $θ G (h)$ is precisely the sum of those $\theta(t_iht_i^{-1})$ for which the conjugate $t_iht_i^{-1}$ is also in H. 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.

[edit] 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.

$G = \bigcup_{t \in T} HtK$

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

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

where $θ t$ is the class function of $t - 1 H t$ defined by $θ t (t - 1 h t) = θ(h)$ for each 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:

$\langle \theta^{G},\psi^{G} \rangle$

$= \langle \left(\theta^{G}\right)_{K},\psi \rangle = \sum_{ t \in T} \langle \left([\theta^{t}]_{t^{-1}Ht \cap K}\right)^{K}, \psi \rangle$

$= \sum_{t \in T} \langle \left(\theta^{t} \right)_{t^{-1}Ht \cap K}, \psi_{t^{-1}Ht \cap K} \rangle$ ,

(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 $t^{-1}Ht \cap K$ . If θ and ψ are both trivial characters, then the inner product simplifies to |T|.

[edit] Clifford's theorem

Clifford's theorem, proved by A. H. Clifford in 1935, yields information about the restriction of a complex irreducible character of a finite group G to a normal subgroup N. If μ is a complex character of N, then for a fixed element g of G, another character, $μ (g)$ , of N may be constructed by setting

μ (g) (n) = μ(g n g - 1)

for all n in N. The character $μ (g)$ is irreducible if and only if μ is. Clifford's theorem states that if χ is a complex irreducible character of G, and μ is an irreducible character of N with

$\langle \chi_N,\mu \rangle \neq 0,$ then

$\chi_N = e\left(\sum_{i=1}^{t} \mu^{(g_i)}\right),$

where e and t are positive integers, and each $g i$ is an element of G. The integers e and t both divide the index [G:N] . The integer t is the index of a subgroup of G, containing N, known as the inertial subgroup of μ. This is

$\{ g \in G: \mu^{(g)} = \mu \}$

and is often denoted by

I G (μ).

The elements $g i$ may be taken to be representatives of all the right cosets of the subgroup $I G (μ)$ in G.

In fact, the integer e divides the index

[I G (μ): N],

though the proof of this fact requires some use of Schur's theory of projective representations.

The proof of Clifford's theorem is best explained in terms of modules ( and the module-theoretic version works for irreducible modular representations). Let F be a field, V be an irreducible F[G]-module, V_N be its restriction to N and U be an irreducible F[N]-submodule of V_N. For each g in G, U.g is an irreducible F[N]-submodule of V_N, and $\sum_{g \in G} U.g$ is an F[G]-submodule of $V$ , so must be all of V by irreducibility. Now $V N$ is expressed as a sum of irreducible submodules, and this expression may be refined to a direct sum. The proof of the character-theoretic statement of the theorem may now be completed in the case $F = \mathbb{C}$ . Let χ be the character of G afforded by V and μ be the character of N afforded by U. For each g in G, the $\mathbb{C}N$ -submodule U.g affords the character $μ (g)$ and $\langle \chi_N,\mu^{(g)}\rangle = \langle \chi_N^{(g)},\mu^{(g)}\rangle = \langle \chi_N,\mu \rangle$ . The respective equalities follow because χ is a class-function of G and N is a normal subgroup. The integer e appearing in the statement of the theorem is this common multiplicity.

A corollary of Clifford's theorem, which is often exploited, is that the irreducible character χ appearing in the theorem is induced from an irreducible character of the inertial subgroup $I G (μ)$ . If, for example, the irreducible character χ is primitive (that is, χ is not induced from any proper subgroup of G), then $G = I G (μ)$ and χ_N = eμ. A case where this property of primitive characters is used particularly frequently is when N is Abelian and χ is faithful (that is, its kernel contains just the identity element). In that case, μ is linear, N is represented by scalar matrices in any representation affording character χ and N is thus contained in the center of G (that is, the subgroup of G consisting of those elements which themselves commute with every element of G). For example, if G is the symmetric group S₄, then G has a faithful complex irreducible character χ of degree 3. There is an Abelian normal subgroup N of order 4 (a Klein 4-subgroup) which is not contained in the center of G. Hence χ is induced from a character of a proper subgroup of G containing N. The only possibility is that χ is induced from a linear character of a Sylow 2-subgroup of G.

Clifford's theorem has led to a branch of representation theory in its own right, now known as Clifford theory. This is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound. For more general finite groups, Clifford theory often allows representation-theoretic questions to be reduced to questions about groups which are close (in a sense which can be made precise) to being simple.

[edit] References

Fulton, William; and Harris, Joe (1991). Representation Theory, A First Course. Springer, New York. ISBN 0-387-97495-4. See chapter 2.
Isaacs, I.M. (1994). Character Theory of Finite Groups. Dover. ISBN 0-486-68014-2. Corrected reprint of the 1976 original, published by Academic Press.
James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X.
Jean-Pierre, Serre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 0-387-90190-6.
http://planetmath.org/encyclopedia/Character.html
Character Tables for chemically important point groups - Lists most of the point groups and gives their character tables in notation used in Chemistry.

Categories: Representation theory of groups

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Character theory

From Wikipedia, the free encyclopedia

Contents

[edit] Applications

[edit] Definitions

[edit] Properties

[edit] Arithmetic properties

[edit] Character tables

[edit] Orthogonality relations

[edit] Character table properties

[edit] Induced characters and Frobenius reciprocity

[edit] Mackey decomposition

[edit] Clifford's theorem

[edit] References

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