Proposition 3.12. Let ρ : G ’’ GL (V ) be a representation, and v = {v1 , . . . , vn } a basis
of V . Suppose that relative to v,
[ρg ]v = [rij (g)] (g ∈ G).
Then relative to the dual basis v— = {v1 , . . . , vn }, we have
— —
[ρ— ]v— = [rji (g ’1 )] (g ∈ G),
g
or equivalently,
[ρ— ]v— = [ρg’1 ]T .
g
Proof. If we write
[ρ— ]v— = [tij (g)],
g
then by de¬nition,
n
ρ— vs =
— —
trs (g)vr .
g
r=1
Now for each i = 1, . . . , n,
n
(ρ— vj )(vi )
— —
= trj (g)vr (vi ),
g
r=1
28 3. REPRESENTATIONS OF FINITE GROUPS
which gives
—
vj (ρg’1 vi ) = tij (g),
and hence
n
—
rki (g ’1 )vi ) = rji (g ’1 ).
tij (g) = vj (
k=1
Another perspective on the above is provided by the next result, whose proof is left as an
exercise.
Proposition 3.13. The klinear isomorphism
Hom (V, W ) ∼ W — V —
=

is a Gisomorphism where the right hand side carries the tensor product representation σ — ρ— .
Using these ideas together with Proposition 3.11 we obtain the following useful result
Proposition 3.14. For k of characteristic 0, the Ghomomorphism
µ : Hom (V, W ) ’’ Hom (V, W )
of Proposition 3.11 has image equal to the set of Ghomomorphisms V ’’ W , Hom (V, W )G
which is also Gisomorphic to (W — V — )G .
Now let ρ : G ’’ GL (V ) be a representation of G and let H G. We can restrict ρ to
H and obtain a representation ρH : H ’’ GL (V ) of H, usually denoted ρ “G or ResG ρ; the
H
H
G or ResG V .
Hmodule V is also denoted V “H H
Similarly, if G K, then we can form the induced representation ρ ‘K : K ’’ GL (V ‘K )
G G
as follows. Take KR to be the Gset consisting of the underlying set of K with the Gaction
g · x = xg ’1 .
De¬ne
V ‘K = IndK V = Map(KR , V )G = {f : K ’’ V : f (x) = ρg f (xg) ∀x ∈ K}.
G G
Then K acts linearly on V ‘K by
G
(k · f )(x) = f (kx),
and so we obtain a linear representation of K. The induced representation is often denoted
ρ ‘K or IndK ρ. The dimension of V ‘K is dim V ‘K = K/G dim V . Later we will meet
G
G G G
Reciprocity Laws relating these induction and restriction operations.
4. Permutation representations
Let G be a ¬nite group and X a ¬nite Gset, i.e., a ¬nite set X equipped with an action
of G on X, written gx. A ¬nite dimensional Grepresentation ρ : G ’’ GLC (V ) over k is a
permutation representation on X if there is an injective Gmap j : X ’’ V and im j = j(X) ⊆ V
is a kbasis for V . Notice that a permutation representation really depends on the injection
j. We frequently have situations where X ⊆ V and j is the inclusion of the subset X. The
condition that j be a Gmap amounts to the requirement that
ρg (j(x)) = j(gx) (g ∈ G, x ∈ X).
Definition 3.15. A homomorphism from a permutation representation j1 : X1 ’’ V1 to a
second j2 : X2 ’’ V2 is a Glinear transformation ¦ : V1 ’’ V2 such that
¦(j1 (x)) ∈ im j2 (x ∈ X1 ).
A Ghomomorphism of permutation representations which is a Gisomorphism is called a G
isomorphism of permutation representations.
4. PERMUTATION REPRESENTATIONS 29
Notice that by the injectivity of j2 , this implies the existence of a unique Gmap • : X1 ’’
X2 for which
j2 (•(x)) = ¦(j1 (x)) (x ∈ X1 ).
Equivalently, we could specify the Gmap • : X1 ’’ X2 and then ¦ : V1 ’’ V2 would be the
unique linear extension of • restricted to im j2 (see Proposition 1.1). In the case where ¦ is a
Gisomorphism, it is easily veri¬ed that • : X1 ’’ X2 is a Gequivalence.
To show that such permutations representations exist in abundance, we proceed as follows.
Let X be a ¬nite set equipped with a Gaction. Let k[X] = Map(X, k), the set of all functions
X ’’ k. This is a ¬nite dimensional kvector space with addition and scalar multiplication
de¬ned by
(f1 + f2 )(x) = f1 (x) + f2 (x), (tf )(x) = t(f (x)),
for f1 , f2 , f ∈ Map(X, k), t ∈ k and x ∈ X. There is an action of G on Map(X, k) given by
(g · f )(x) = f (g ’1 x).
If Y is a second ¬nite Gset, and • : X ’’ Y a Gmap, then we de¬ne the induced function
•— : k[X] ’’ k[Y ] by
(•— f )(y) = f (x) = f (x).
x∈•’1 {y} •(x)=y
Theorem 3.16. Let G be a ¬nite group.
a) For a ¬nite Gset X, k[X] is a ¬nite dimensional permutation representation of di
mension dim k[X] = X.
b) For a Gmap • : X ’’ Y , the induced function •— : k[X] ’’ k[Y ] is a Glinear
transformation.
Proof.
a) For each x ∈ X we have a function δx : X ’’ k given by
1 if y = x,
δx (y) =
0 otherwise.
The map j : X ’’ k[X] given by
j(x) = δx
is easily seen to be an injection. It is also a Gmap, since
δx (g ’1 y) = 1,
δgx (y) = 1 ⇐’
and hence
j(gx)(y) = δgx (y) = δx (g ’1 y) = (g · δx )(y).
Given a function f : X ’’ k, consider
f’ f (x)δx ∈ k[X].
x∈X
Then for y ∈ X,
f (y) ’ f (x)δx (y) = f (y) ’ f (y) = 0,
x∈X
hence f ’ x∈X f (x)δx is the constant function taking the value 0 on X. So the functions δx
(x ∈ X) span k[X]. They are also linearly independent, since if the 0 valued constant function
is expressed in the form
tx δx
x∈X
30 3. REPRESENTATIONS OF FINITE GROUPS
for some tx ∈ k, then for each y ∈ X,
0= tx δx (y) = ty ,
x∈X
hence all the coe¬cients tx must be 0.
b) The klinearity of •— is easily checked. To show it is a Gmap, for g ∈ G,
(g · •— f )(y) = (•— f )(g ’1 y)
= f (x)
x∈•’1 {g ’1 y}
f (g ’1 x),
=
x∈•’1 {y}
since we have
•’1 {g ’1 y} = {x ∈ X : g•(x) = y} = {x ∈ X : •(gx) = y} = {g ’1 x : x ∈ X, x ∈ •’1 {y}}.
Since by de¬nition
(g · f )(x) = f (g ’1 x),
we have
(g · •— f ) = •— (g · f ).
Given a permutation representation k[X], we will often identify X with a subset of k[X],
using the injection j. If • : X ’’ Y is a Gmap, notice that
•— ( tx δ x ) = tx δ•(x) .
x∈X x∈X
We will sometimes write x instead of δx , and a typical element of k[X] as x∈X tx x rather
than x∈X tx δx , where tx ∈ k. Another useful notational device is to list the elements of X
as x1 , x2 , . . . , xn and then identify n = {1, 2, . . . , n} with X via the correspondence k ←’ xk .
Then we can identify k[n] ∼ kn with k[X] using the correspondence
=
n
(t1 , t2 , . . . , tn ) ←’ tk xk .
k=1
5. Properties of permutation representations
Let X be a ¬nite Gset. The result shows how to reduce an arbitrary permutation repre
sentation to a direct sum of those induced from transitive Gsets.
Proposition 3.17. Let X = X1 X2 where X1 , X2 ⊆ X are closed under the action of G.
Then there is a Gisomorphism
k[X] ∼ k[X1 ] • k[X2 ].
=
Proof. Let j1 : X1 ’’ X and j2 : X2 ’’ X be the inclusion maps, which are Gmaps. By
Theorem 3.16(b), there are Glinear transformations j1 — : k[X1 ] ’’ k[X] and j2 — : k[X2 ] ’’
k[X]. For
f= tx x ∈ k[X],
x∈X
we have the ˜restrictions™
f1 = tx x, f2 = tx x.
x∈X1 x∈X2
We de¬ne our linear map k[X] ∼ k[X1 ] • k[X2 ] by
=
f ’’ (f1 , f2 ).
5. PROPERTIES OF PERMUTATION REPRESENTATIONS 31
It is easily seen that this is a linear transformation, and moreover has an inverse given by
(h1 , h2 ) ’’ j1 — h1 + j2 — h2 .
Finally, this is a Gmap since the latter is the sum of two Gmaps, hence its inverse is.
Let X1 and X2 be Gsets; then X = X1 — X2 can be made into a Gset with action given by
g · (x1 , x2 ) = (gx1 , gx2 ).
Proposition 3.18. Let X1 and X2 be Gsets. Then there is a Gisomorphism
k[X1 ] — k[X2 ] ∼ k[X1 — X2 ].
=
Proof. The function F : k[X1 ] — k[X2 ] ’’ k[X1 — X2 ] de¬ned by
F( sx x, ty y) = sx ty (x, y)
x∈X1 y∈X2 x∈X1 y∈X2
is kbilinear. Hence by the universal property of the tensor product (Section 4, UPTP), there
is a unique linear transformation F : k[X1 ] — k[X2 ] ’’ k[X1 — X2 ] for which
F (x — y) = (x, y) (x ∈ X1 , y ∈ X2 ).
This is easily seen to to be an isomorphism and also Glinear.
Definition 3.19. Let G be a ¬nite group. The regular representation over k is the G
representation k[G], of dimension dim k[G] = G.
Proposition 3.20. The regular representation of a ¬nite group G over a ¬eld k is a ring
(in fact a kalgebra). Moreover, this ring is commutative if and only if G is abelian.
Proof. Let a = g∈G ag g and b = g∈G bg g where ag , bg ∈ G. Then we de¬ne the product
of a and b by
ab = ah bh’1 g g.
g∈G h∈G
Note that for g, h ∈ G in k[G] we have
(1g)(1h) = gh.
For commutativity, each such product (1g)(1h) must agree with (1h)(1g), which happens if and
only if G is abelian. The rest of the details are left as an exercise.
The ring k[G] is called the group algebra or group ring of G over k. The next result is
left as an exercise for those who know about modules. It provides a link between the study
of modules over k[G] and Grepresentations, and so the group ring construction provides an
important source of noncommutative rings and their modules.
Proposition 3.21. Let V be a k vector space. Then if V carries a Grepresentation, it
admits the structure of a k[G] module de¬ned by
( ag g)v = ag gv.
g∈G g∈G
Conversely, if V is a k[G]module, then it admits a Grepresentation with action de¬ned by
g · v = (1g)v.
32 3. REPRESENTATIONS OF FINITE GROUPS
6. Calculating in permutation representations
In this section, we determine how the permutation representation k[X] looks in terms of the
basis consisting of elements x (x ∈ X). We know that g ∈ G acts by sending x to gx. Hence, if
we label the rows and columns of a matrix by the elements of X, the X — X matrix [g] of g
with respect to this basis has xy entry
1 if x = gy,
(6.1) [g]xy = δx,gy =
0 else,
where δa,b denotes the Kronecker δ function which is 0 except for when a = b and it then takes