# how to find basis ' representation

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?

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.
• Yes. The only computation you need to prove that it's a representation is showing that $A^3 = B^2 = (AB)^2$. Apr 1, 2021 at 16:49
• I forgot to write $\cdots = I$ at the end there. In any case, you're welcome Apr 1, 2021 at 17:06
• @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. Apr 1, 2021 at 17:13