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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
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