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I am reading "Representation Theory of Finite Groups: An Introductory Approach" by Benjamin Steinberg and in Chapter 3, Example 3.1.14 the author gives the following example of a representation:

Let $\rho \colon S_3 \rightarrow GL_{2}(\mathbb{C})$ be specified on the generators $(12)$ and $(123)$ by $\rho_{(12)}= \begin{pmatrix} -1 & -1 \\ 0 & 1 \end{pmatrix}, \rho_{(123)}= \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}$

The author then suggests that the reader checks this is a representation but I am unsure how to do this. I checked that $\rho_{(12)}^2 = I_2$ and $\rho_{(123)}^3=I_2$ but I'm not sure what else I need to check.

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  • $\begingroup$ Do you know about presentations of groups? (Not the same thing as representations!) $\endgroup$ Feb 8, 2022 at 2:47
  • $\begingroup$ No I don't, but If it helps to solve this problem then I can look it up. $\endgroup$
    – 123123
    Feb 8, 2022 at 2:49
  • $\begingroup$ You need to make sure $\rho$ respects the relations in $S_3$. $\endgroup$
    – Randall
    Feb 8, 2022 at 3:01
  • $\begingroup$ Would it be possible for you to go into more detail regarding how I can check that $\rho$ respects all relations in $S_3$? $\endgroup$
    – 123123
    Feb 8, 2022 at 3:06

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What @diracdeltafunk and @Randall are getting at is you first want to write down a presentation for $S_3$ with $(12)$ and $(123)$ as generators. You should check this, but I believe $\langle s,t \; | \; s^2 = 1, t^3 = 1, tst^{-2}s = 1\rangle$ is such a presentation. To verify this you should show that the map $f$ defined on generators by $f(s) = (12)$ and $f(t) = (123)$ is a group isomorphism.

You must then check that $\rho_{(12)}$ and $\rho_{(123)}$ satisfy these relations. So you have checked the first two relations, but the matrices also must satisfy the third relation.

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