1. Representation Theory of Finite Groups Sample
Spencer Leonardis
Math 194 Spring 2016
University of California, Santa Cruz
August 19, 2016
Definition 3.9. Two representations Γ(1)
: G → AutF (V ) and Γ(2)
: G → AutF (V ) are
called equivalent (Γ(1)
∼ Γ(2)
) if there is an F-linear isomorphism T : V → W such that
Γ(2)
g = T ◦ Γ(1)
g ◦ T−1
for every g ∈ G.
This means that T commutes with ∆(1)
and ∆(2)
. The scenario can be expressed visually by
the following commutative diagram:
V V
W W
T
Γ
(1)
g
T
Γ
(2)
g
Notice that we can start from the upper left-hand corner and follow the arrows in two
separate directions to arrive at the bottom right corner. This is what we mean when we say
a diagram commutes.
Remark 3.10. For representations ∆(1)
: G → GLn(F) and ∆(2)
: G → GLm(F), we say
that ∆(1)
∼ ∆(2)
if there is an invertible matrix A (which corresponds to T by props 2.19
and 3.4) such that ∆
(2)
g = A∆
(1)
g A−1
for every g ∈ G.
Definition 3.11. Let Γ(1)
and Γ(2)
by the representations in definition 3.9 (but not neces-
sarily equivalent). Then the direct sum of two representations is defined
Γ(1)
⊕ Γ(2)
g (v1, v2) = (Γ(1)
g (v1), Γ(2)
g (v2)).
If ∆(1)
and ∆(2)
are the representations in remark 3.10, then the direct sum is defined by the
map
∆(1)
⊕ ∆(2)
: G → Gn+m(F), g →
∆(1)
0
0 ∆(2) .
1

2. Definition 3.12. Let ∆ : G → GLn(F) be a representation. Then a subspace U ⊆ Fn
is
called ∆–invariant (or G-invariant) if ∆gu ∈ U for every g ∈ G and u ∈ U.
Definition 3.13. Let ∆ : G → GLn(F)
∼
−→ AutF (V ) be a representation.
(i). We call ∆ decomposable if ∆ ∼ ∆(1)
⊕ ∆(2)
for some representations ∆(1)
and ∆(1)
.
Equivalently, ∆ is decomposable if V splits into a direct sum U ⊕W of two non-trivial
∆–invariant subspaces. Otherwise ∆ is termed indecomposable.
(ii). We call Γ reducible if ∆ is equivalent to a representation of the form g →
A(g) B(g)
0 C(g)
A(g) ∈ Matn(F), B(g) ∈ Matn×m(F), C(g) ∈ Matn(F). Equivalently, a representa-
tion ∆ is reducible if there exists a non-trivial ∆–invariant subspace U ⊂ V . Otherwise
∆ is called irreducible.
Remark 3.14. Alternatively, one could say that a representation ∆ of a group G is irre-
ducible if no subrepresentation Γ ⊂ ∆ is closed under the action {∆(g) : g ∈ G}. One
special type of representation decomposes into a direct sum of irreducible representations–
it’s called a unitary representation. If a group G has a unitary representation ∆, then G can
be completely examined by looking at each of the individual representations in the direct
sum.
Example 3.15. Define a representation φ : D4 → GL2(C) of the dihedral group D4 =
r, s|ord(r) = 4, ord(s) = 2, srs = r−1
by
rk
→
ik
0
0 (−i)k and srk
→
0 (−i)k
ik
0
.
Observe that φt
rk = φsrk . Then φ is an irreducible representation.
Theorem 3.16. Assume |G| is invertible in F. Let ∆ : G → GLn(F) be a representation
and U ⊆ Fn
a ∆–invariant F–subspace. Then there is a ∆–invariant F-subspace such that
Fn
= U ⊕ V .
proof. Let B = (u1, . . . , ur) be an F-basis of U. Since B is linearly independent, it can be
extended to a basis B = (u1, . . . , un) of Fn
by 2.9.i. Now let A ∈ Matn(F) be defined by
Aui :=
ui : ui ∈ B,
0 : ui ∈ B B.
Now set
A := ∆g−1 A∆g =
1
|G|
|G| ∆g−1 A∆g =
1
|G| g∈G
∆g−1 A∆g.
Next we prove four individual statements (a)–(d):
2

3. (a). A u = u for every u ∈ U:
Observe that
A u =
1
|G| g∈G
∆g−1 A ∆gu
∈U
=
1
|G| g∈G
∆g−1 ∆gu =
1
|G| g∈G
(∆g)−1
∆gu
=
1
|G| g∈G
u = u.
(b). A x ∈ U for every x ∈ Fn
:
There are two cases, either ∆gx ∈ U or ∆gx ∈ Fn
:
(i). A ∆gx
∈U
= ∆gx ∈ U,
(ii). A ∆gx
∈Fn
= 0 ∈ U.
In either case A∆gx ∈ U, so it follows that A x = 1
|G|
g∈G
∆g−1 A∆gx ∈ U since by
∆–invariance, ∆g−1 u ∈ U for every u ∈ U.
(c). A A = A x for every x ∈ Fn
:
This is clear from parts (a) and (b).
(d). A ∆h = ∆hA for every h ∈ G:
We have
A ∆h =
1
|G| g∈G
∆g−1 A∆gh = A ∆h =
1
|G| g∈G
∆hk−1 A∆k
=
1
|G| g∈G
∆h∆k−1 A∆k = ∆hA .
by substituting k = gh.
Now set V := ker(A ) = {x ∈ Fn
: A x = 0}. For every v ∈ V and g ∈ G, it follows from
(d) that A ∆gv = ∆gA v = 0, so ∆gv ∈ V , implying V is ∆–invariant. Let x ∈ Fn
. Then
by part (b), A x ∈ U and by (c), A (x − A x) = A x − A x = 0, so x − A x ∈ V . Thus
x = A x + (x − A x) ∈ U + V . Hence Fn
= U + V . Now if x ∈ U ∩ V =⇒ x = A x = 0 =⇒
U ∩ V = {0}. Therefore Fn
= U ⊕ V , completing the proof.
References
[1] Cooperstein, Bruce N. Advanced Linear Algebra. CRC press. University of Cal-
ifornia, Santa Cruz. (2015), no. 2, 52–54, 69–70, 72, 73, 75, 81–84.
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4. [2] Boltje, Robert. Algebra I (Math 111A). University of California, Santa Cruz.
(2014), 12–17, 58–59.
[3] Boltje, Robert, Representations of Finite Groups I (Math 240A). University of
California, Santa Cruz. (2016), 1–23.
[4] Steinberg, Benjamin. Representation Theory of Finite Groups. Carleton Univer-
sity. (2009), 1–35.
[5] Lent, C. T. Representation Theory. Berlin: Springer, University of California,
Berkeley. (1986), 1–10.
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