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REPRESENTATIONS OF FINITE GROUPS

1. Definition and Introduction

1.1. Definitions. Let V be a vector space over the field C of complex numbers and let GL(V ) be the group of isomorphisms of V onto itself. An element a of GL(V ) is a linear mapping of V into V which has a linear inverse a−1. When V has dimension n and has a finite basis (ei)ni=1, each map a : V → V is defined by a square matrix (aij) of order n. The coefficients aij are complex numbers. They are obtained by expressing the image a(ej) in terms of the basis (ei):

X

a(ej) =

aijei.

i

Remark 1.1. Saying that a is an isomorphism is equivalent to saying that the determinant det(a) = det(aij) of a is not zero. The group GL(V ) is thus identified with the group of invertible square matrices of order n.

Suppose G is a finite group with identity element 1. A representation of a finite group
G on a finite-dimensional complex vector space V is a homomorphism ρ : G → GL(V ) of G to the group of automorphisms of V . We say such a map gives V the structure of a G-module. The dimension V is called the degree of ρ.

Remark 1.2. When there is little ambiguity of the map ρ, we sometimes call V itself a representation of G. In this vein we often write gv˙ or gv for ρ(g)v.

Remark 1.3. Since ρ is a homomorphism, we have equality

ρ(st) = ρ(s)ρ(t) for s, t ∈ G.

In particular, we have

  • ρ(1) = 1, , ρ(s−1) = ρ(s)−1
  • .

A map ϕ between two representations V and W of G is a vector space map ϕ : V → W such that

ϕ

V −−−−→ W

y
y

  • g
  • g

(1.1)

V −−−−→ W

ϕ

commutes for all g ∈ G. We will call this a G-linear map when we want to distinguish it from an arbitrary linear map between the vector spaces V and W. We can then define Ker ϕ, Im ϕ, and Coker ϕ, which are also G-modules. (Check this.)

∗

These are some notes I wrote for the students in the class Representations of Finite Groups SS2013.
The materials are from the following two books.

J. P. Serre Linear representations of finite groups GTM42

W. Fulton; J. Haris Representation theory Part 1, GTM129

1
2

1.2. Basic examples.

1.2.1. Characters. A representation of degree 1 of a group G is a homomorphism ρ : G → C×, where C× denotes the multiplicative group of nonzero complex numbers. Since

each element of G has finite order, the values ρ(g) of ρ are roots of unity. In particular, |ρ(g)| = 1. Degree 1 representations are called characters.
If we take ρ(g) = 1 for all g ∈ G, we obtain a representation of G which is called the trivial representation or the unit representation.

1.2.2. Regular representation. Let |G| be the order of G, and let V be a vector space of dimension |G| with basis (eg)g∈G indexed by the elements g of G. For g ∈ G, let ρ(g) be the linear map of V into V which sends eg to egh. This defines a linear representation, which is called the regular representation of G. Its degree is equal to the order of G. Note that eg = ρg(e1). Hence the images of e1 form a basis of V . Conversely, let W be a representation of G containing a vector w such that ρg(w) for g ∈ G form a basis of W, then W is isomorphic to the regular representation. Define τ : V → W by τ(eg) = ρg(w), then τ gives us such an isomorphism.

1.2.3. Permutation representations. Suppose that G acts on a finite set X. This means that, for each g ∈ G, there is a permutation x → gx of X, satisfying the identities

1x = x; g(hx) = (gh)x if g, h ∈ G, x ∈ X.

Let V be a vector space having a basis (ex)x∈X indexed by the elements of X. For g ∈ G, let ρg be the linear map of V into V which sends ex to egx. The linear representation of G thus obtained is called the permutation representation associated with X.

1.3. Subrepresentations and irreducible representations.

1.3.1. Some linear algebra. Let V be a vector space, and let W and W0 be two subspaces of V . The space V is said to be the direct sum of W and W0 if each x ∈ V can be written uniquely in the form x = w + w0, with w ∈ W and w0 ∈ W0. This amounts to saying that the intersection W ∩ W0 = 0 and that dim V = dim W + dim W0. We then write V = W ⊕ W0 and say that W0 is a complement of W in V . The mapping p which sends each x ∈ V to its component of w ∈ W is called the projection of V onto W associated with the decomposition V = W ⊕ W0. The image of p is W and p(x) = x for x ∈ W. Conversely, if p is a linear map of V into itself satisfying these two properties, one checks that V is the direct sum of W and the kernel W0 of p.

1.3.2. Subrepresentations. A subrepresentation of a representation V is a vector subspace W of V which is invariant under G. A representation V is called irreducible if there is no proper nonzero invariant subspace W of V .

Example 1.4. Take V for example the regular representation of G, and let W be the

P

subspace of dimension 1 of V generated by the element x = g∈G eg. We have ρgx = x for all g ∈ G. Therefore, W is a subrepresentation of V , isomorphic to the unit representation. Later we will determine all the subrepresentations of the regular representation.

Theorem 1.5. Let ρ : G → GL(V ) be a linear representation of G in V and let W be a vector subspace of V stable under G. Then there exists a complement W0 of W in V which stable under G.

3

Proof. We give two proofs of this theorem.
(1) Recall a Hermitian product on V is a map ( , ) : V × V → C such that

• (w, z) = (z, w) • (w + w0, z) = (w, z) + (w0, z)

• (aw, bz) = a¯b(w, z)

Let H0 be any Hermitian product on V , we define

X

  • H(x, y) =
  • H0(gx, gy).

g∈G

It is easy to check that H is also a Hermitian product on V and H(gx, gy) = H(x, y). Let W0 be the orthogonal complement of W in V . Then W0 is stable under G and we have our theorem.
(2) Let W0 be any complement of W in V and let p be the corresponding projection of
V onto W. Form the average p0 of the conjugates of p by the elements of G:

X

1

p0 =

  • ρg ◦ p ◦ ρ−g 1
  • .

|G|

g∈G

Since p maps V into W and ρg preserves W we see that p0 maps V into W. Note that ρ−g 1x ∈ W for x ∈ W, so p0x = x. Thus p0 is a projection of V onto W, corresponding to some complement W0 of W. It is easy to check that

ρg ◦ p0 ◦ ρ−g 1 = p0.

Thus ρg ◦ p0 = p0 ◦ ρg. If x ∈ W0 and g ∈ G, we have p0x = 0, so p0gx = xp0x = 0. That shows that W0 is stable under G and the theorem follows.

ꢀ

1.3.3. Irreducible representations. Let V be a representation of G. By definition, V is irreducible if V is not the direct sum of two representations except for the trivial decomposition V = 0 ⊕ V . A representation of degree 1 is evidently irreducible.

Theorem 1.6. Every representation is a direct sum of irreducible representations.

Proof. Let V be a linear representation of G. We proceed by induction on dim(V ). If dim(V ) = 0, there is nothing to prove. Suppose that dim(V ) ≥ 1. If V is irreducible, there is nothing to prove. Otherwise, because of last theorem, V can be decomposed into a direct sum V 0 ⊕ V 00 with dim(V 0) < dim(V ) and dim(V 00) < dim(V ). By the induction hypothesis, V 0 and V 00 are direct sums of irreducible representations, and so the same is true for V .

ꢀ

Remark 1.7. Let V be a representation of G, and let V = W1⊕· · ·⊕Wk be a decomposition of V into a direct sum of irreducible representations. We can ask if this decomposition is unique. The case where all the ρg are equal to 1 shows that this is not true in general. Nevertheless, we will show later that the number of Wi isomorphic to a given irreducible representation does not depend on the chosen decomposition.

4

1.4. Schur’s Lemma.

Theorem 1.8. If V and W are irreducible representations of G and ϕ : V → W is a G-module homomorphism, then
(1) Either ϕ is an isomorphism or ϕ = 0. (2) If V = W, then ϕ = λ · I for some λ ∈ C, I the identity map.

Proof. The first claim follows from the the fact that Ker ϕ and Im ϕ are invariant subspaces. For the second, since C is an algebraic closed field, ϕ must have an eigenvalue λ, i.e., for some λ ∈ C, ϕ − λI has a nonzero kernel. By (1), we must have ϕ − λI = 0 and the claim follows.

ꢀ

Proposition 1.9. For any representation V of a finite group G, there is a decomposition

V = V1a ⊕ · · · ⊕ Vka

1

k

,

where the Vi are distinct irreducible representations. The decomposition of V into a direct sum of the k factors is unique, as are the Vi that occur and their multiplicities ai.

Proof. It follows from Schur’s lemma that if W is another representation of G, with a decomposition W = ⊕jWb and ϕ : V → W is a map of representations, then ϕ must map

j

j

  • the factors Via into the factors Wjb for which Vi
  • W . If we apply this to the identity

j

ji

∼

=map of V to V , the stated uniqueness follows.

ꢀ

1.5. Construct new representations from old ones.

1.5.1. General construction. If V and W are representations, the direct sum V ⊕ W and the tensor product are also representations, via

g(v ⊕ w) = gv ⊕ gw, g(v ⊗ w) = gv ⊗ gw.

For a representation V , the nth tensor power V ⊗n is again a representation of G by this

V

rule, and the exterior power n(V ) and symmetric power Symmn(V ) are subrepresentations of it.
The dual V ∗ = Hom(V, C) of V is also a representation of G via

ρ∗(g) = tρ(g−1) : V ∗ → V ∗.

Note under this definition, we have

hρ∗(g)(v∗), ρ(g)(v)i = hv∗, vi,

where h, i is the natural pairing between V and V ∗.

1.5.2. Symmetric square and alternation square. Suppose that the representations V1 and

V2 are identical to the same representation V , so that V1 ⊗ V2 = V ⊗ V . If (ei) is a basis of V , let θ be the automorphism of V ⊗ V such that

θ(ei ⊗ ej) = ej ⊗ ei, for all (i, j).
It follows that θ(x ⊗ y) = y ⊗ x for all x, y ∈ V , and θ is independent of the chosen basis (ei). Moreover, θ2 = 1. The space V ⊗ V then decomposes into a direct sum

2

^

V ⊗ V = Symm2(V ) ⊕ (V ),

5

V

where Symm2(V ) is the set of elements z ∈ V ⊗ V such that θ(z) = z and 2(V ) is the set of elements of z ∈ V ⊗ V such that θ(z) = −z. The elements (ei ⊗ ej + ej ⊗ ei)i≤j for a

V

basis of Symm2(V ), and the elements (ei ⊗ ej − ej ⊗ ei)i<j for a basis of 2(V ). We have

2

^

dim Symm2(V ) = n(n + 1)/2 dim (V ) = n(n − 1)/2 if dim(V ) = n.

1.6. Further remarks.

1.6.1. Complete reducibility. We saw that any representation of a finite group is a direct sum of irreducible representations. This property is called complete reducibility, or semisimplicity. We may show that, for continuous representations, any compact group has this property. Integration over the group with respect to an invariant measure on the group plays the role of averaging in the proof of the theorem. The additive group R, on the other hand, does not have this property. Let V = R2, the representation of

  • ꢀ
  • ꢁ

1 a 0 1

a →

leaves the x axis fixed, but there is no complementary subspace.
The complete reducibility may also fail if the vector space V is over a field of finite characteristic.

1.6.2. Representations of abelian groups. If V is a representation of the finite group G, abelian or not, each g ∈ G gives a map ρ(g) : V → V . But this map in general is not a G-module homomorphism: for general h ∈ G, we will have

g(h(v)) = h(g(v)).
Indeed, ρ(g) : V → V will be a G-linear for every ρ if and only if g is in the center Z(G) of G. In particular, if G is abelian, and V is an irreducible representation, then by Schur’s lemma every element g ∈ G acts on V by a scalar multiple of the identity. Every subspace of V is thus invariant, so that V must be one dimensional. The irreducible representations of an abelian group G are thus simply elements of the dual group, that is, homomorphisms

ρ : G → C×.

1.6.3. Representations of S3. We consider the simplest nonabelian group G = S3. To begin with, we have two one-dimensional representations: the trivial representation, which we will denote U, and the alternating representation U0, defined by

gv = sgn(g)v,

for g ∈ G, v ∈ V . Next, since G comes to us as a permutation group, we have a natural permutation representation, in which G acts on C3 by permuting the coordinates.

Explicitly, if (e1, e2, e3) is the standard basis, then g(ei) = eg(i), or equivalently,

g(z1, z2, z2) = (zg−1(1), zg−1(2), zg−1(3)).

This representation is not irreducible. The line spanned by the sum (1, 1, 1) of the basis vectors is invariant, with complementary subspace

V = {(z1, z2, z3) ∈ C3 : z1 + z2 + z3 = 0}.

6

This two-dimensional representation V is easily seen to be irreducible, we call it the standard representation of S3.
Now we describing an arbitrary representation of S3. (Later we will develop the so called character theory, which is a wonderful tool for describing arbitrary representation of a group.) We have seen that the representation theory of a finite abelian group is virtually trivial, we start our analysis of an arbitrary representation W of S3 by looking at the action of the abelian subgroup U3 = Z/3 ⊂ S3 on W. Let τ be any generator of

U3, the space W is spanned by eigenvectors vi for the action of τ, whose eigenvalues are of course all powers of a cube root of unity ω = e2πi/3. Thus

W = ⊕Vi

where Vi = Cvi and τvi = ωαi vi.
Next, we ask how the remaining elements of S3 act on W in terms of this decomposition.
Let σ be any transposition, so that τ and σ together generate S3, with the relation στσ = τ2. Let v be an eigenvector of τ with eigenvalue ωi, we have

τ(σ(v)) = σ(τ2(v)) = σ(ω2i · v) = ω2i · σ(v).
Therefore, σ(v) is again an eigenvector for τ with eigenvalue ω2i.

Lemma 1.10. Let σ = (12), τ = (123). Then the standard representation of S3 has a

basis α = (ω, 1, ω2), β = (1, ω, ω2), with τα = ωα, τβ = ω2β, σα = β, σβ = α.

Suppose we start with an eigenvector v for τ. If the eigenvalue of v is ωi = 1, then σ(v) is an eigenvector with eigenvalue ω2i = ωi, and so is independent of v. Now v and σ(v) together span a two-dimensional subspace V 0 of W invariant under S3. In fact, V 0 is isomorphic to the standard representation, which follows from the lemma. If, on the other hand, the eigenvalue of v is 1, then σ(v) may or may not be independent of v. If it is not, then v spans a one-dimensional subrepresentation of W, isomorphic to the trivial representation if σ(v) = v and to the alternating representation if σ(v) = −v. If σ(v) and v are independent, then v + σ(v) and v − σ(v) span one-dimensional representations of W isomorphic to the trivial representation and alternating representation respectively. Thus, we have the following proposition.

Proposition 1.11. For an arbitrary representation W of S3, we can write

W = U⊕a ⊕ (U0)⊕b ⊕ V ⊕c

Moreover, c is the number of independent eigenvectors for τ with eigenvalue ω, a+c is the

.

multiplicity of 1 as an eigenvalue of σ, and b+c is the multoplicity of −1 as an eigenvalue of σ.

2. Character theory

2.1. The character of a representation. Let V be a vector space of dimension n having a basis (ei). Let A be a linear map of V into itself with matrix (aij)n×n. By the trace of A we mean the scalar

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  • Permutations Applied to Construction Company Management

    Permutations Applied to Construction Company Management

    Permutations Applied To Construction Company Management PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sat, 21 Sep 2013 01:41:34 UTC Contents Articles Permutation 1 Group theory 12 Permutation group 20 References Article Sources and Contributors 22 Image Sources, Licenses and Contributors 23 Article Licenses License 24 Permutation 1 Permutation In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). For example, an anagram of a word is a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×1, which is commonly denoted as "n factorial" and written "n!". Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S The 6 permutations of 3 balls occurs exactly once as image value).
  • Regular Representations of Finite Groups Via Hypergraphs

    Regular Representations of Finite Groups Via Hypergraphs

    Regular representations of finite groups via hypergraphs Robert Jajcay Indiana State University [email protected] April 14, 2005 Theorem 1 (Cayley) Every (finite) group (G, ·) is isomorphic to a permutation group GL ≤ SymG GL = {σa | ∀a ∈ G σa(b) = a · b} GL is called the left regular representation of G. Example: = 1 2 2 D3 { D3, r, r , s, sr, sr } 1 u 2 u 3 4 u u 5 6 u u 1 1, 2, 2 3, 4, 5, 2 6 D3 7→ s 7→ sr 7→ sr 7→ r 7→ r 7→ σr = (1, 5, 6)(2, 3, 4) 1 Definition 1 Let Γ = (V, E) be a graph. The (full) automorphism group of Γ, AutΓ, is the subgroup of SymV of all permutations σ that preserve the structure of Γ, i.e., σ(u)σ(v) iff uv Example: Γ 1 @ u @ @ @ 2 @ @ @ u @ @ @ @ @ @ 3 @ 4 @ ¨¨ HH @ ¨ u u H ¨¨ HH @ ¨¨ HH@ ¨ ¨ H@ 5 ¨ H@H 6 u u D3 < AutΓ 2 Question 1 Is there a graph Γ such that AutΓ is exactly (D3)L? Definition 2 A graph Γ is a graphical regular representation for a group G if AutΓ = GL. Example: Kn is the GRR for Symn Question 2 Which finite groups allow for a GRR? Definition 3 Given a (finite) group G and a set X of elements of G, X ⊂ G, closed under taking inverses, X = X−1, and not contain- ing 1G, the Cayley graph Γ = C(G, X) is the graph (G, E) where E = {{g, g · x} | g ∈ G, x ∈ X}.
  • ON REPRESENTATION THEORY of QUANTUM Slq(2) GROUPS at ROOTS of UNITY

    ON REPRESENTATION THEORY of QUANTUM Slq(2) GROUPS at ROOTS of UNITY

    QUANTUM GROUPS AND QUANTUM SPACES BANACH CENTER PUBLICATIONS, VOLUME 40 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1997 ON REPRESENTATION THEORY OF QUANTUM SLq(2) GROUPS AT ROOTS OF UNITY PIOTRKONDRATOWICZ Department of Mathematical Methods in Physics, University of Warsaw Ho˙za74, 00-682 Warszawa, Poland E-mail: [email protected] PIOTRPODLES´ Department of Mathematical Methods in Physics, University of Warsaw Ho˙za74, 00-682 Warszawa, Poland E-mail: [email protected] Abstract. Irreducible representations of quantum groups SLq(2) (in Woronowicz’ approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the case of q being an odd root of unity. Here we find the irreducible representations for all roots of unity (also of an even degree), as well as describe ”the diagonal part” of the tensor product of any two irreducible representations. An example of a not completely reducible representation is given. Non-existence of Haar functional is proved. The corresponding representations of universal enveloping algebras of Jimbo and Lusztig are provided. We also recall the case of general q. Our computations are done in explicit way. 0. Introduction. The quantum SL(2) group is by definition a quantum group (A; 4) that has the same representation theory as SL(2), i.e. all nonequivalent irreducible rep- s 1 resentations are u , s = 0; 2 ; 1;::: such that t+s s t s L r dim u = 2s + 1; u > u ≈ u ; r=jt−sj; step=1 1 1 2 s; t = 0; 2 ; 1;::: and matrix elements of u generate A as an algebra with unity I.
  • The Group Determinant Determines the Group

    The Group Determinant Determines the Group

    proceedings of the american mathematical society Volume 112, Number 3, July 1991 THE GROUP DETERMINANT DETERMINES THE GROUP EDWARD FORMANEK AND DAVID SIBLEY (Communicated by Warren J. Wong) Abstract. Let G = {gx, ... , gn} be a finite group of order n , let K be a field whose characteristic is prime to n , and let {x \g 6 G} be independent commuting variables over K . The group determinant of G is the determinant of the n x n matrix (x _ i ). We show that two groups with the same group determinant are isomorphic. 1. Introduction Let G be a finite group of order n , let {xg\g G G} be n independent vari- ables over a field K, and let K(x ) be the field of rational functions over K in these variables. The group determinant, 3¡G = 3¡G(x ) G K(x ), is the determi- nant of the nxn matrix (x -i ), where G = {gx, ... , gn} . Equivalently, it is the determinant of the endomorphism of the group algebra K(x )G induced by left multiplication by the "generic" element S?G = Zxg. The group determi- nant of G is a homogeneous polynomial of degree n with integer coefficients. The representation theory of finite groups was developed by Frobenius in order to solve the problem of factoring the group determinant as an element of C[x ], the polynomial ring over C, the field of complex numbers. In modern language, his solution is as follows: The group algebra CG is isomorphic as a C-algebra to a direct product of matrix algebras over C.