Representations of Finite Groups I (Math 240A)

93 Pages·2016·0.555 MB·English

by  Robert Boltje

Representations of Finite Groups I
(Math 240A)
(Robert Boltje, UCSC, Spring 2016)
Contents
1 Representations and Characters 1
2 Orthogonality Relations 13
3 Algebraic Integers 26
4 Burnside’s paqb-Theorem 33
5 The Group Algebra and its Modules 38
6 The Tensor Product 49
7 Induction 57
8 Frobenius Groups 69
9 Elementary Clifford Theory 74
10 Artin’s and Brauer’s Induction Thoerems 81
1 Representations and Characters
Throughout this section, F denotes a field and G a finite group.
1.1 Definition A (matrix) representation of G over F of degree n ∈ N is
a group homomorphism ∆: G → GL (F). The representation ∆ is called
n
faithful if ∆ is injective. Two representations ∆: G → GL (F) and Γ: G →
n
GL (F) are called equivalent if n = m and if there exists an invertible matrix
m
S ∈ GL (F) such that Γ(g) = S∆(g)S−1 for all g ∈ G. In this case we often
n
write Γ = S∆S−1 for short.
1.2 Remark In the literature, sometimes a representation of G over F is
defined as a pair (V,ρ) where V is a finite-dimensional F-vector space and
ρ: G → Aut (V)(= GL(V)) is a group homomorphism into the group of
F
F-linear automorphisms of V. These two concepts can be translated into
each other. Choosing an F-basis of V we obtain a group isomorphism
∼
Aut (V) → GL (F) (where n = dim V) and composing ρ with this iso-
F n F
∼
morphism we obtain a representation ∆: G → Aut (V) → GL (F). If we
F n
choose another basis then we obtain equivalent representations. Conversely,
if ∆: G → GL (F) is a representation, we choose any n-dimensional F-
n
vector space V and a basis of V (e.g. V = Fn with the canonical basis)
∼
and obtain a homomorphism ρ: G → GL (F) → Aut (V). Representations
n F
(V,ρ) and (W,σ) are called equivalent if there exists an F-linear isomorphism
ϕ: V → W such that σ(g) = ϕ ◦ ρ(g) ◦ ϕ−1 for all g ∈ G. The above con-
structions induce mutually inverse bijections between the set of equivalence
classes of representations (V,ρ) and the set of equivalence classes of matrix
representations ∆ of G over F.
If one takes V := Fn, the space of column vectors together with the
standard basis (e ,...,e ) then the matrix representation ∆ corresponds to
1 n
(Fn,ρ) with ρ(g)v = ∆(g)v.
1.3 Examples (a)Thetrivialrepresentationisthe homomorphism∆: G →
GL (F), g (cid:55)→ 1.
1
(b) For n (cid:62) 2 let D = (cid:104)σ,τ | σn = 1,τ2 = 1,τστ = σ−1(cid:105) be the dihedral
2n
group of order 2n. We define a faithful representation ∆ of D over R by
2n
(cid:32) (cid:33) (cid:32) (cid:33)
cosφ −sinφ −1 0
∆(σ) := and ∆(τ) := ,
sinφ cosφ 0 1
1
where φ = 2π/n. Note that ∆(σ) is the counterclockwise rotation about
2π/n and τ is the reflection about the vertical axis in R2.
(c) The map ∆: Sym(n) → GL (F), which maps σ ∈ Sym(n) to the
n
permutation matrix of σ is a faithful representation of Sym(n) of degree
n over any field. It is called the natural representation of Sym(n). (The
permutation matrix of σ has in its i-th column the canonical basis vector
e , i.e., ∆(σ) maps e to e .)
σ(i) i σ(i)
(d)If∆: G → GL (F)isarepresentationandH (cid:54) Gthentherestriction
n
of ∆ to H,
ResGH(∆): H i (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)G ∆ (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)GLn(F)
is a representation of H over F of the same degree. Here, i: H → G denotes
the inclusion map. More generally, if f: H → G is an arbitrary group
homomorphism between two arbitrary finite groups then also
f ∆
Resf(∆): H (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)G (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)GLn(F)
is a representation of H over F of degree n.
(e) If G = {g ,...,g } and if we define the homomorphism f: G →
1 n
Sym(n) by (f(g))(i) = j, when gg = g , then
i j
f ∆
G (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)Sym(n) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)GLn(F),
with ∆ as in (c), is a representation, called the regular representation of G. If
we rearrange the ordering of the elements g ,...,g we obtain an equivalent
1 n
representation.
(f) If ∆ : G → GL (F) and ∆ : G → GL (F) are representations of G
1 n1 2 n2
over F then also their direct sum
(cid:32) (cid:33)
∆ (g) 0
∆ ⊕∆ : G → GL (F), g (cid:55)→ 1
1 2 n1+n2 0 ∆ (g)
2
is a representation of G over F. Since
(cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33)
∆ (g) 0 0 I ∆ (g) 0 0 I
2 = n2 1 n1 ,,
0 ∆ (g) I 0 0 ∆ (g) I 0
1 n1 2 n2
we see that ∆ ⊕∆ and ∆ ⊕∆ are equivalent.
1 2 2 1
(g) Let E/F be a Galois extension of degree n with Galois group G.
Then G (cid:54) Aut (E) and we obtain a representation (E,i) in the sense of
F
Remark 1.2, with i: G → Aut (E) the inclusion.
F
2
1.4 Definition Let ∆: G → GL (F) be a representation.
n
(a) We say that ∆ is decomposable if it is equivalent to ∆ ⊕∆ for some
1 2
representations∆ and∆ ofGoverF. Otherwiseitiscalledindecomposable.
1 2
(b)Wesaythat∆isreducibleifitisequivalenttoablockuppertriangular
representation, i.e., to a representation of the form
(cid:32) (cid:33)
A(g) B(g)
g (cid:55)→
0 C(g)
where A(g) ∈ Mat (F), B(g) ∈ Mat (F), C(g) ∈ Mat (F), and where
n1 n1×n2 n2
n ,n ∈ N are independent of g ∈ G. Otherwise, it is called irreducible.
1 2
1.5 Remark Let ∆: G → GL (F) be a representation.
n
(a)If∆isdecomposablethen∆isalsoreducible. Thus, if∆isirreducible
it is also indecomposable.
(b) A subspace U of Fn is called invariant under ∆ (or ∆-stable) if
∆(g)u ∈ U for all g ∈ G and u ∈ U. More generally, if (V,ρ) is a rep-
resentation corresponding to ∆, a subspace U of V is called invariant under
∆ (or ∆-stable) if (ρ(g))(u) ∈ U for all g ∈ G and u ∈ U. Note that ∆ is
decomposable if and only if V has a decomposition V = U ⊕ W into two
non-zero ∆-invariant subspaces U and V. And ∆ is reducible if V has a
∆-invariant subspace U with {0} =(cid:54) U (cid:54)= V.
In the following remark we recall a few results from linear algebra.
1.6 Remark Let V be a finite-dimensional C-vector space.
(a) A map
(−,−): V ×V → C
iscalledahermitianscalarproductonV ifthefollowingholdsforallx,x(cid:48),y,y(cid:48) ∈
V and λ ∈ C:
(x+x(cid:48),y) = (x,y)+(x(cid:48),y), (x,y +y(cid:48)) = (x,y)+(x,y(cid:48)),
¯
(λx,y) = λ(x,y), (x,λy) = λ(x,y),
(y,x) = (x,y), (x,x) ∈ R ,
(cid:62)0
(x,x) = 0 ⇐⇒ x = 0.
3
(b) Let (−,−) be a hermitian scalar product on V. And let f: V → V
be an automorphism of V. Recall that f is called unitary with respect to
(−,−) if (f(x),f(y)) = (x,y) for all x,y ∈ V. The following results hold:
(i) The vector space V has an orthonormal basis with respect to (−,−),
i.e., a basis (v ,...,v ) such that (v ,v ) = δ for i,j ∈ {1,...,n}.
1 n i j i,j
(ii) An automorphism f of V is unitary if and only if the representing
matrix A of f with respect to an orthonormal basis is unitary (i.e., AA∗ = 1,
where A∗ = At).
(c) The complex vector space Cn has the standard hermitian scalar prod-
uct
(x,y) = x y¯ +x y¯ +···+x y¯ .
1 1 2 2 n n
The standard basis (e ,...,e ) is an orthonormal basis with respect to this
1 n
product. An automorphism f: Cn → Cn is unitary with respect to (−,−) if
and only if the unique matrix A ∈ GL (C) with the property f(x) = Ax, for
n
all x ∈ Cn, is a unitary matrix, since A is the representing matrix of f with
respect to (e ,...,e ).
1 n
(d) The set of unitary n × n-matrices is a subgroup U(n) of GL (C).
n
Every unitary matrix is diagonalizable, i.e., conjugate to a diagonal matrix.
Its eigenvalues are complex numbers of absolute value 1.
1.7 Proposition Every complex representation ∆: G → GL (C) is equiv-
n
alent to a unitary representation, i.e., to a representation Γ: G → GL (C)
n
with Γ(g) ∈ U(n) for all g ∈ G.
Proof For x,y ∈ Cn define
(cid:88)
(cid:104)x,y(cid:105) := (∆(g)x,∆(g)y),
g∈G
where (−,−) is the standard hermitian scalar product on Cn. It is easy to
check that < −,− > is again a hermitian scalar product on Cn and we have
(cid:88) (cid:88)
(cid:104)∆(g)x,∆(g)y(cid:105) = (∆(h)∆(g)x,∆(h)∆(g)y) = (∆(k)x,∆(k)y) = (cid:104)x,y(cid:105),
h∈G k∈G
for all x,y ∈ Cn and g ∈ G. Thus, ∆(g) is unitary with respect to (cid:104)−,−(cid:105), for
every g ∈ G. Let S ∈ GL (C) be a matrix such that its columns (v ,...,v )
n 1 n
form an orthonormal basis of Cn with respect to (cid:104)−,−(cid:105). Then the represent-
ing matrix S−1∆(g)S of ∆(g) with respect to (v ,...,v ) is unitary. Thus,
1 n
the representation Γ = S−1∆S is unitary.
4
1.8 Corollary Let ∆: G → GL (C) be a representation and let g ∈ G be
n
an element of order k. Furthermore, set ζ := e2πi/k ∈ C, a primitive k-th
root of unity. Then there exists S ∈ GL (C) (which can depend on g) such
n
that
ζi1 
S∆(g)S−1 = diag(ζi1,...,ζin) =  ...  ,
 
ζin
the diagonal matrix with diagonal entries ζi1,...,ζin, where the exponents
i ,...,i are elements in {0,1,...,k −1}.
1 n
Proof By Proposition 1.7, the matrix ∆(g) is conjugate to a unitary matrix.
Moreover, by Remark 1.6(d), every unitary matrix is conjugate to a diagonal
matrix. Thus, there exists S ∈ GL (C) such that S∆(g)S−1 is a diagonal
n
matrix, say with diagonal entries d ,...,d . Finally, since gk = 1, we have
1 n
(S∆(g)S−1)k = S∆(g)kS−1 = S∆(gk)S−1 = I . Thisimpliesthattheentries
n
d must satisfy dk = 1 for i = 1,...,n. Now the statement of the corollary
i i
follows immediately.
Notethatthepreviouscorollaryonlysaysthateachmatrix∆(g), g ∈ G,
is individually diagonalizable. This does not mean that ∆ as a repre-
sentation is diagonalizable, i.e., ∆ is equivalent to a representation with
diagonal matrices as values. Clearly, ∆ as a representation is diagonal-
izable if and only if it is equivalent to a direct sum of representations of
degree 1.
1.9 Theorem Assume that |G| = |G|·1 is invertible in F. Let ∆: G →
F
GL (F) be a representation and let U ⊆ Fn be an F-subspace that is ∆-
n
invariant. Then there exists a ∆-invariant F-subspace V ⊆ Fn such that
Fn = U ⊕V.
Proof Let u ,...,u be an F-basis of U and extend it to an F-basis
1 r
u ,...,u of Fn. Let A ∈ Mat (F) be defined by Au = u for i = 1,...,r
1 n n i i
and Au = 0 for i = r+1,...,n. We set
i
1
A(cid:48) := (cid:88) ∆(g−1)A∆(g).
|G|
g∈G
Then we have
(i) A(cid:48)u = u for all u ∈ U.
5
(ii) A(cid:48)x ∈ U for all x ∈ Fn.
(iii) A(cid:48)A(cid:48)x = A(cid:48)x for all x ∈ Fn.
(iv) A(cid:48)∆(h) = ∆(h)A(cid:48) for all h ∈ G.
In fact, to see (i), note that
1 1 1
A(cid:48)u = (cid:88) ∆(g−1)A∆(g)u = (cid:88) ∆(g−1)∆(g)u = (cid:88) u = u,
|G| |G| |G|
g∈G g∈G g∈G
since ∆(g)u ∈ U for all g ∈ G and u ∈ U. To see (ii), note that A∆(g)x ∈ U
for all x ∈ Fn. Now (iii) follows immediately from (i) and (ii). And (iv)
follows from the computation
1 1
A(cid:48)∆(h) = (cid:88) ∆(g−1)A∆(gh) = (cid:88) ∆(hg˜−1)A∆(g˜) = ∆(h)A(cid:48),
|G| |G|
g∈G g˜∈G
where we made the substitution g˜ = gh.
Now set V := kerA(cid:48). We first show that V is ∆-invariant. For v ∈ V and
g ∈ G we obtain from (iv) that A(cid:48)∆(g)v = ∆(g)A(cid:48)v = 0, which means that
∆(g)v ∈ kerA(cid:48) = V. Next we show that Fn = U +V. In fact, let x ∈ Fn.
Then using (ii) and (iii) we obtain x = A(cid:48)x+(x−A(cid:48)x) ∈ U +V. Finally, we
show that U ∩V = {0}. So let x ∈ U ∩V. Then 0 = A(cid:48)x = x by (i). This
completes the proof.
1.10 Corollary Assume that |G| is invertible in F and let ∆: G → GL (F)
n
be a representation. Then one has:
(a) The representation ∆ is equivalent to a direct sum of irreducible
representations.
(b) The representation ∆ is indecomposable if and only if it is irreducible.
Proof (a) We prove this by induction on n. If n = 1, the statement holds,
since every representation of degree 1 is irreducible by definition. Assume
now that n > 1. If ∆ is irreducible then we are done. So assume that ∆ is
not irreducible. Then there exists a ∆-invariant F-subspace {0} (cid:54)= U (cid:54)= Fn.
By Theorem 1.9, there exists a ∆-invariant F-subspace V of Fn such that
Fn = U ⊕ V. Therefore, ∆ is equivalent to ∆ ⊕ ∆ for representations of
1 2
degree n ,n < n. By induction the claim follows.
1 2
(b) If ∆ is irreducible then it is also indecomposable for trivial reasons.
Now assume that ∆ is not irreducible. Then there exists a ∆-invariant sub-
space {0} (cid:54)= U (cid:54)= Fn. By Theorem 1.9, U has a ∆-stable complement V in
Fn. This implies that ∆ is not indecomposable.
6
We recall the following theorem from Linear Algebra without proof.
1.11 Theorem Let A ,...,A ∈ Mat (F). Then the following are equiva-
1 k n
lent:
(i) The matrices A ,...,A are simultaneously diagonalizable over F,
1 k
i.e., there exists S ∈ GL (F) such SA S−1 is a diagonal matrix, for all
n i
i = 1,...,k.
(ii) The matrices A ,...,A are individually diagonalizable over F and
1 k
they commute, i.e., A A = A A for all i,j ∈ {1,...,k}.
i j j i
1.12 Corollary (a) Let ∆: G → GL (C) be a representation satisfying
n
∆(g)∆(h) = ∆(h)∆(g) for all g,h ∈ G. Then ∆ is equivalent to a direct sum
of representations of degree 1.
(b) Assume that G is abelian. Then every representation ∆: G →
GL (C) is equivalent to a direct sum of representations of degree 1. In par-
n
ticular, every irreducible representation of G has degree 1. In other words,
the irreducible representations of G are just the homomorphisms from G to
the multiplicative group C× of C.
1.13 Definition Let ∆: G → GL (F) be a representation. The character
n
of ∆ is defined as the function
∆ tr
χ := χ∆: G (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)GLn(F) (cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)(cid:113)F .
The characters of representations of G over F are called the F-characters of
G. If F = C, we just call them characters, or sometimes complex characters
of G. If F = C and ∆ is irreducible then χ is called a (complex) irreducible
character.
In the following remark we derive a few first properties of characters.
1.14 Remark Assume that ∆: G → GL (F) is a representation and let
n
χ = χ : G → F denote its character.
∆
(a) Recall that for two matrices A,B ∈ Mat (F) one has tr(AB) =
n
tr(BA). In particular, if S ∈ GL (F) one obtains tr(SAS−1) = tr(AS−1S) =
n
tr(A). This implies that if ∆ and Γ are equivalent representations then
χ = χ . Moreover, by the same property of the trace map we obtain, for
∆ Γ
any elements g,x ∈ G, that
χ(xgx−1) = tr(∆(xgx−1) = tr(∆(x)∆(g)∆(x)−1) = tr(∆(g)) = χ(g).
7
This means that the function χ: G → F is constant on conjugacy classes of
G. Any such function is called a class function or a central function on G.
The class functions f: G → F form an F-vector space CF(G,F) with the
usual addition and scalar multiplication. The dimension of CF(G,F) is equal
to the number k(G) of conjugacy classes of G. In fact, the characteristic class
functions which have value one on one particular conjugacy class and value
zero on all others form an F-basis of CF(G,F).
(b) Let ∆ : G → GL (F) and ∆ : G → GL (F) be two representa-
1 n1 2 n2
tions. Then, for every g ∈ G, we have
(cid:32) (cid:33)
∆ (g) 0
χ (g) = tr 1 = tr(∆ (g))+tr(∆ (g)) = χ (g)+χ (g).
∆1⊕∆2 0 ∆ (g) 1 2 ∆1 ∆2
2
Thus, χ = χ +χ in the vector space CF(G,F).
∆1⊕∆2 ∆1 ∆2
(c) The F-characters and representations of degree 1 are precisely the
group homomorphisms in Hom(G,F×). They form an abelian group under
the product defined by (φ · ψ)(g) = φ(g)ψ(g), for φ,ψ ∈ Hom(G,F×) and
g ∈ G. Recall that we have an isomorphism of abelian groups
Hom(G/G(cid:48),F×) → Hom(G,F×), f (cid:55)→ f ◦ν,
where G(cid:48) denotes the commutator subgroup of G, generated by all commuta-
tors [x,y] := xyx−1y−1, x,y ∈ G, and where ν: G → G/G(cid:48), g (cid:55)→ gG(cid:48), denotes
the natural epimorphism.
(d) Assume from now on that F ⊆ C. Note that
χ(1) = n.
Thus, the character of ∆ ‘knows’ the degree of the representation ∆. We call
χ(1) the degree of the character χ.
Let g ∈ G and assume that gk = 1. Then, by Corollary 1.8 we have
ζi1 
∆(g) = S ... S−1,
 
ζin
for some S ∈ GL (C), with ζ := e2πi/k and i ,...,i ∈ Z. Therefore,
n 1 n
χ(g) = ζi1 +···+ζin.
8