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Representation theory

  1. Nov 9, 2012 #1
    1. The problem statement, all variables and given/known data

    2a9uzkn.jpg

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

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

    2. Relevant definitions

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

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

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

    3. The attempt at a solution

    My answers for (a) and (b)

    (a) [itex](\varphi , V)[/itex] is a representation of the group [itex](\mathbb{R},+)[/itex], for:

    [itex]\varphi(s+t)(P) = P(x+s+t)[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = p_0 (x+s+t)^2 + p_1(x+s+t) + p_2[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t) ( p_0 (x+s)^2 + p_1(x+s)+p_2)[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t)(P(x+s))[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \varphi(t)(\varphi(s)(P))[/itex]

    so that [itex]\varphi[/itex] is a homomorphism, hence a representation.

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

    [itex]\varphi(t)(Q) = \varphi(t)(q_0x+q_1)[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\; = q_0(x+t) + q_1[/itex]
    [itex]\;\;\;\;\;\;\;\;\;\;\;\;\; = q_0x + (q_0t+q_1) \in U[/itex]
    Moreover, [itex]U\neq \{0\}, V[/itex] so that [itex]\varphi[/itex] is not irreducible.
     
    Last edited: Nov 9, 2012
  2. jcsd
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