# How to check that this map is a Group Representation

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.

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

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.