Lecture notes for Introduction to Representation Theory

and the action g(f)(x) = f(xg) ⊕g G.
Remark 4.29. In fact, Ind G H V is naturally isomorphic to Hom H (k[G], V ).
Let us check that Ind G
H V
is indeed a representation:
g(f)(hx) = f(hxg) = δ V (h)f(xg) = δ V (h)g(f)(x), and g(g (f))(x) = g (f)(xg) = f(xgg ) =
(gg )(f)(x) for any g, g , x G and h H.
Remark 4.30. Notice that if we choose a representative x ε from every right H-coset ε of G, then
any f Ind G is uniquely determined by {f(x ε )}.
H V
Because of this,
dim(Ind G |G|
H V ) = dim V · .
| H|
Problem 4.31. Check that if K → H → G are groups and V a representation of K then IndH G IndH
is isomorphic to Ind G K V
K V .
Exercise. Let K → G be finite groups, and ν : K ⊃ C ⊕ be a homomorphism. Let C α be the
corresponding 1-dimensional representation of K. Let
e α =
1
ν(g)
−1
g C[K]
|K| gK
be the idempotent corresponding to ν. Show that the G-representation Ind G
morphic to C[G]e α (with G acting by left multiplication).
K C α
is naturally iso-
4.9 The Mackey formula
Let us now compute the character ν of Ind G
H V .
x ε .
In each right coset ε H\G, choose a representative
Theorem 4.32. (The Mackey formula) One has
ν(g) =
−1
εH\G:x gx H
ν V (x ε gx − ε 1 ).
Remark. If the characteristic of the ground field k is relatively prime to | H | , then this formula
can be written as
1
ν(g) =
ν V (xgx −1 ).
|H|
xG:xgx −1 H
Proof. For a right H-coset ε of G, let us define
V = {f Ind G ε H V | f(g) = 0 ⊕g ⇒ ε}.
Then one has
and so
Ind G =
H V V ε ,
ε
ν(g) = ν ε (g),
ε
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