$Representation theory quick question$

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$Representation theory quick question$

Please see attached, only answer part d, thanks

Algebra

Jbuck

152

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The 1-dimensional subrepresentation is given by $U = \mathbb{F}_2u$, where $u = (1,1,1)$. To see, note that $\rho(x)u = \rho(y)u = u$, so the action of any element in $G$ is trivial on $U$.

The 2-dimensional subrepresentation is given by $V = \mathbb{F}_2 v_1 \oplus \mathbb{F}_2 v_2$, where $v_1 = (1, 0, 1)$ and $v_2 = (1, 0, 0)$. To see, note that for $x$ we have $\rho(x)v_1 = v_1 \in V$ and $\rho(x)v_2 = v_1 + v_2 \in V$, while for $y$ we have $\rho(y)v_1 = v_2 \in V$ and $\rho(y)v_2 = v_1 + v_2 \in V$.

Matchmaticians Alumnus

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Jbuck

0

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

0

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.
Matchmaticians Alumnus

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*any other common eigenspace
Jbuck

0

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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