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I have to prove that there is a representation $L_1:S_3 \to GL_2(F)$ that do this : $$ L_1((12)) =\begin{pmatrix}0&1\\1&0\end{pmatrix} ,\\ L_1(1,2,3)=\begin{pmatrix}0&-1\\1&-1\end{pmatrix} $$ I look for a basis , i found in the internet this $e1,e2,e3=-(e1+e2)$ . then I wrote the basis like that :span $\{ (1 0 -1) , (0 1 -1) \}$ thin with this basis i got it that way : $$ (1,2)( V1-V3)= V2-V3 ,\\ (1,2)( V2-V3)= V1-V3 $$ thats mean: $L(1,2)= \begin{pmatrix}0&1\\1&0\end{pmatrix}$ its right $$ (1,2,3)( V1-V3)= V2-V1 \\ (1,2,3)( V2-V3)= V3-V1 $$ thats mean : $L(1,2,3)=\begin{pmatrix}1&-1\\-1&0\end{pmatrix}$ and its false :\

I also tried to switch betwen $(1 0 -1) , (0 1 -1)$ and didnt get the correct representation.

can you please help me , were is my fault ? and how I can solve it ?

I'm adding L2 definition: $L_2:S_3 \to GL_2(F)$

$L_2(1,2)= \begin{pmatrix}0&1\\1&0\end{pmatrix}$

$L_2(1,2,3)= \begin{pmatrix}0&1\\-1&-1\end{pmatrix}$

I have to prove that L1 & L2 are irreducible in any field, I know that in field C if there are two isomorphic representations then they are irreducible but how can we prove that in any field other than C these two representations are irreducible?

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1 Answer 1

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It is not at all clear what you mean by "I look for a basis", and it's unclear what you're trying to do with all of your work after that.

In order for $L_1$ to define a representation, it must hold that $L_1:S_3 \to GL_2(F)$ is a homomorphism. Because the elements $\sigma := (1\ \ 2\ \ 3)$ and $\tau = (1\ \ 2)$ generate $S_3$, $L_1$ will extend to a homomorphism over $G$ if and only if $L_1(\sigma), L_2(\tau)$ satisfy the relations corresponding to a presentation of $S_3$. In particular, $S_3$ can be presented as $$ S_3 = \langle \sigma, \tau \mid \sigma^3 = \tau^2 = (\sigma \tau)^2 = e \rangle. $$ Thus, in order to prove that $L_1$ defines a representation, it suffices to verify that the matrices $$ A = \pmatrix{0&-1\\1&-1}, \quad B = \pmatrix{0&1\\1&0} $$ satisfy $A^3 = B^2 = (AB)^2 = I$, where $I \in GL_2(F)$ denotes the identity matrix.

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  • $\begingroup$ so I can conclude from what you said , that if proved that A&B satisfy the relations corresponding to a presentation of S3 , its mean that if I show that " A3=B2=(AB)2=I" , I prove that its a representation . $\endgroup$
    – jimmy
    Apr 1, 2021 at 16:39
  • $\begingroup$ Yes. The only computation you need to prove that it's a representation is showing that $A^3 = B^2 = (AB)^2$. $\endgroup$ Apr 1, 2021 at 16:49
  • $\begingroup$ ok thanks alot for the help i really appriciate it!! $\endgroup$
    – jimmy
    Apr 1, 2021 at 17:05
  • $\begingroup$ I forgot to write $\cdots = I$ at the end there. In any case, you're welcome $\endgroup$ Apr 1, 2021 at 17:06
  • $\begingroup$ @jimmy I don't understand what you mean. If you want to ask about another representation which you're calling $L_2$, I would recommend that you post a new question. $\endgroup$ Apr 1, 2021 at 17:13

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