Representation Theory: Proving Invariant Subspaces and Decomposition Properties

[Ted123]

Homework Statement

I think I've done (a) and (b) correctly (please check).

I'm stuck as to how to describe all subspaces of $V$ that are preserved by the operators $\varphi(t)$ and how to prove that $\varphi$ can't be decomposed into a direct sum $\varphi = \varphi |_U \oplus \varphi |_{U'}$ where $U,U'\neq \{0\},V$ are 2 $\varphi$-invariant subpaces of $V\neq U\oplus U'$.

2. Relevant definitions

Let $(\varphi , V)$ be a representation of a group $G$ and $U \subset V$ be a vector subspace of $V$. Then $U$ is called $\varphi$-invariant if for all $u\in U, g\in G$ we have $\varphi(g)(u) \in U$.

$(\varphi , V)$ is called irreducible if the only $\varphi$-invariant subspaces of $V$ are the trivial ones: $\{0\}$ and $V$.

$(\varphi , V)$ is called decomposable if $V=U\oplus U'$ where $U, U'$ are $\varphi$-invariant subspaces of $V$ such that $U,U' \neq \{0\}, V$. Then $\varphi = \varphi |_U \oplus \varphi |_{U'}$ where $\varphi |_U$ is the restriction of $\varphi$ to the subspace $U$ etc.

The Attempt at a Solution

My answers for (a) and (b)

(a) $(\varphi , V)$ is a representation of the group $(\mathbb{R},+)$, for:

$\varphi(s+t)(P) = P(x+s+t)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = p_0 (x+s+t)^2 + p_1(x+s+t) + p_2$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t) ( p_0 (x+s)^2 + p_1(x+s)+p_2)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t)(P(x+s))$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t)(\varphi(s)(P))$

so that $\varphi$ is a homomorphism, hence a representation.

(b) The subspace $U\subset V$ consisting of all complex-valued polynomials $Q(x) = q_0x+q_1$ of degree at most 1 is a $\varphi$-invariant subspace, for:

$\varphi(t)(Q) = \varphi(t)(q_0x+q_1)$
$\;\;\;\;\;\;\;\;\;\;\;\;\; = q_0(x+t) + q_1$
$\;\;\;\;\;\;\;\;\;\;\;\;\; = q_0x + (q_0t+q_1) \in U$
Moreover, $U\neq \{0\}, V$ so that $\varphi$ is not irreducible.

 

[lippyka]

For (c), I think all subspaces of V that are preserved by the operator \varphi(t) are all vector subspaces U \subset V such that for all u\in U, t\in \mathbb{R} we have \varphi(t)(u)\in U. However, I'm not sure how to prove this.For (d), I'm not sure how to prove that \varphi can't be decomposed into a direct sum \varphi = \varphi |_U \oplus \varphi |_{U'} where U,U'\neq \{0\},V are 2 \varphi-invariant subpaces of V\neq U\oplus U'. Any help would be much appreciated!