Representation theory quick question

Please see attached, only answer part d, thanks

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  • Thank you! Is there any way to show that these are indeed the only subrepresentations?

  • Yes. The one-dimensional subrep is clearly irreducible and the 2-dim is as well, since rho(x) and rho(y) do not have any common eigenspace. So any other subrep must decompose as a sum of irred. reps, which must be the ones previously found. But the only options then are U, V, and U+V = F_2^3.

  • *any other common eigenspace

  • Hello there, are you available to answer some more questions today, with a $60 price for each? They're related to fields and Galois theory

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