Complete set of irreducible representations of small groups (1)

Posted: February 10, 2011 in Representations of Finite Groups
Tags: ,

In this post and the next one, we are going to give all non-equivalent irreducible representations of groups of very small orders.  Most of what I’m going to say here has already been discussed in previous posts but now I’m going to put them in one place.

Example 1. Irreducible Representations of Finite Abelian Groups: A finite abelian group G has exactly |G| non-equivalent irreducible representations. I already gave explicit description of these representations, with an example. See the Question after the theorem in this post.

Example 2. Irreducible Representations of S_3: In part 3) in here I showed that S_3 has exactly three non-equivalent irreducible representations. One has degree two and the other two have degree one. Note that S_3 \cong D_{6}, the dihedral group of order 6. Let g_1,g_2 \in S_3 be such that g_1^2=g_2^3=(g_1g_2)^2=1. For example you may choose g_1 = (1 \ \ 2) and g_2=(1 \ \ 2 \ \ 3). Then every element of S_3 is written uniquely as g_1^jg_2^k, where 0 \leq j \leq 1 and 0 \leq k \leq 2.
1) Representations of degree one: this for the general case S_n was done in here.
2) Representation of degree two. In here I gave m representations of degree two for D_{2m}. I showed that all of them are irreducible except those cooresponding to \zeta = \pm 1, \ m \geq 3. For our case we have m=3. So we will have two irreducible representations of degree two. We need one of them only, so I pick the one corresponding to \zeta = \exp(2 \pi i/3) and I call it \rho. Let v = \begin{pmatrix} x \\ y \end{pmatrix} \in \mathbb{C}^2. Then, as we saw in there, \rho is defined on S_3 by

\rho(g_1^jg_2^k)(v)= \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}^j \begin{pmatrix} \zeta & 0 \\ 0 & \zeta^{-1} \end{pmatrix}^k v,

for all0 \leq j \leq 1 and 0 \leq k \leq 2. Thus, explicitly, \rho(g_2^k)(v) = \begin{pmatrix} \zeta^k x \\ \zeta^{-k}y \end{pmatrix} and \rho(g_1g_2^k)(v) = \begin{pmatrix} \zeta^{-k}y \\ \zeta^k x \end{pmatrix}.

In part (2), We’ll give all non-equivalent irreducible representations of non-abelian groups of order 8.

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