Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

2.2. Groups and representations 31
Example 2.2.1.2. Here are some actions: g ∈ GL(V ) acts on
(1) V ∗ by α ↦→ α ◦ g −1 .
(2) End(V ) by f ↦→ g ◦ f.
(3) A second action on End(V ) is by f ↦→ g ◦ f ◦ g −1 .
(4) The vector space of homogeneous polynomials of degree d on V for
each d by P ↦→ g · P,where g · P(v) = P(g −1 v). Note that this
agrees with (1) when d = 1.
(5) Let V = C2 � �
a b
so the standard action of ∈ GL(V ) on C
c d
2 is,
� � � �� �
x a b x
↦→ . Then GL2 also acts on C
y c d y
3 by
⎛ ⎞
x
⎛
a
⎝y⎠
↦→ ⎝
z
2 ac c2 2ab ad + bc 2cd
b2 bd d2 ⎞⎛
⎞
x
⎠⎝y⎠
.
z
The geometry of this action is explained by Exercise 2.6.23
These examples give group homomorphisms GL(V ) → GL(V ∗ ), GL(V ) →
GL(End(V )) (two different ones) and GL(V ) → GL(S d V ∗ ), where S d V ∗
denotes the vector space of homogeneous polynomials of degree d on V .
Exercise 2.2.1.3: Verify that each of the above examples are indeed actions,
e.g., show (g1g2) · α = g1(g2 · α).
Exercise 2.2.1.4: Let
�
dim
�
V = 2, choose a basis of V so that g ∈ GL(V )
a b
may be written g = . Write out the 4 × 4 matrices for examples
c d
2.2.1.2.(2) and 2.2.1.2.(3).
2.2.2. Modules and submodules. If W is a G-module and there is a
linear subspace U ⊂ W such that g · u ∈ U for all g ∈ G and u ∈ U, then
one says U is a G-submodule of W.
Exercise 2.2.2.1: Using Exercise 2.2.1.4 show that both actions on End(V )
have nontrivial submodules, in the first case (when dimV = 2) one can find
two-dimensional subspaces preserved by GL(V ) and in the second there is a
unique one dimensional subspace and an unique three dimensional subspace
preserved by GL(V ).⊚
A module is irreducible if it contains no nonzero proper submodules.
For example, the action 2.2.1.2.(3) restricted to the trace free linear maps
is irreducible.
If Z ⊂ W is a subset and a group G acts on W, one says Z is invariant
under the action of G if g · z ∈ Z for all z ∈ Z and g ∈ G.