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Consider the group G of invertible upper triangular matrices over C. Let ρ: G → Mat2×2(C) be the inclusion map.

(i) Show that (C^2,ρ) is a representation of G.

(ii) Prove that ρ has precisely one sub-representation of degree 1.

(iii) Is this sub-representation isomorphic to a representation we have already seen?

I found part (ii) to be C (complex numbers) equppied with basis of (1 0)^T, as it is G invariant with that basis. Hence, it has degree 1. I'm just unsure exactly for part (iii) how this subrepresentation may be isomorphic to C^2 as they have different dimension? Thanks :) (Apologies for my lack of skills in LaTex)

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    The representation in (ii) satisfies $\rho(g)v=v$ for g\in G$. Have you seen this in another context? Also, do you know how to prove (i)? – 2017-02-08
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    Hiya, yeah sorry i should have said I got part (i), then i am unsure for part (iii) – 2017-02-08

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