[Math] irreducible representations and character table of $D_6$

charactersgroup-theoryrepresentation-theory

Let $$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$ $$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$

I would like to compute its character table and its irreducible representations.

I will explain what I have done so far and I will add some doubts I had while doing this.

MY ATTEMPT

  1. Compute conjugacy classes.
    $$C_1=\{e\}, C_2=\{a,a^5\},C_3=\{a^2,a^4\}$$$$C_4=\{a^3\},C_5=\{b,a^2b,a^4b\},C_6=\{ab,a^3b,a^5b\}$$

  2. Find $1$-dimensional representations. Since $D_6/\{a,a^5\}\cong \mathbb{Z}_2$, we have one more representation apart from $\alpha_1=id$. That is $$\alpha_2: G \longrightarrow \mathbb{C}: a \mapsto 1, b\mapsto -1$$
    Again using irreducible representations from quotient group by normal subgroup, I considered $G/\{\overline{1},\overline{a},\overline{b},\overline{ab}\}\cong \mathbb{Z}_2\times\mathbb{Z}_2$ (since it is abelian). Then from here I obtained
    $$\alpha_3:G\longrightarrow \mathbb{C}: a\mapsto -1, b\mapsto 1$$
    $$\alpha_4:G\longrightarrow \mathbb{C}: a\mapsto -1, b\mapsto -1$$

  3. Find $2$-dimensional representations. I have seen in my notes that for $D_n$ we can define $2$-dimensional representations:
    $$\alpha_5: G\longrightarrow GL_2(\mathbb{C}): a\mapsto \begin{bmatrix}cos(\frac{2\pi}{n}) & -sin(\frac{2\pi}{n})\\ sin(\frac{2\pi}{n}) &cos(\frac{2\pi}{n})\end{bmatrix}, b\mapsto \begin{bmatrix}1 & 0\\ 0 &-1\end{bmatrix}$$
    Hence my $$\alpha_5: G\longrightarrow GL_2(\mathbb{C}): a\mapsto \begin{bmatrix}\frac{1}{2} & -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} &\frac{1}{2}\end{bmatrix}, b\mapsto \begin{bmatrix}1 & 0\\ 0 &-1\end{bmatrix}$$

  4. Build my character table.
    \begin{array}{|c|c|c|c|}
    \hline
    & C_1 & C_2 &C_3 &C_4 &C_5 &C_6 \\ \hline
    \chi_1& 1 & 1 &1 &1 &1&1 \\ \hline
    \chi_2& 1 & 1 &1 &1 &-1 &-1 \\ \hline
    \chi_3& 1 & -1 &1 &-1 &1 &-1 \\ \hline
    \chi_4& 1 & -1 &1 &-1 &-1 &1 \\ \hline
    \chi_5& 2 & 1 &-1 &-2 &0 &0 \\ \hline
    \chi_6& 2 & -1 &-1 &2 &0 &0 \\ \hline
    \end{array}

where I have computed $\chi_6$ by the orthogonality formula $(\chi_6|\chi_j)=\delta_{6,j}$.

QUESTIONS

  1. Is $D_6/\{a^2,a^4\}$ really abelian? I can not see it clearly.

  2. My first question comes when I have to find $2$-dimensional irreducible representations. I have find them because I have seen it in my notes. But how could I get $\alpha_5$ and $\alpha_6$ without knowing the special case of $D_n$. I know that I also could get it from $S_3$ (one of them). But I have again the same problem, if you are looking for $2$-dimensional irreducible representations of $S_3$, how do you find them? (Both).

  3. Now consider $X$ to be the set of the vertices of a regular $6$-gon and consider the action of $D_6$ on the set $X$ by restricting the usual action of $D_6$ on the $6$-gon to the set of vertices $X$. Let $\phi$ be the induced permutation representation (over $\mathbb{C}$) of $D_6$. I would like to write it as a sum of irreducible representations by computing the in-product of the irreducible characters with $\chi_{\phi}$. What should I do? I do not understand this induced permutation representation. Any help?

Best Answer

  1. It has four elements, so yes.

  2. Get the characters by messing around with orthogonality, then come up with a representation that does that.

  3. Label the vertices of your hexagon $a_1$ through $a_6$. Then take the vector space $V$ over $\mathbb{C}$ to be the set of formal $\mathbb{C}$-weighted sums of $a_1$ through $a_6$. Then we construct a representation of $D_6$ by an action on this vector space given by defining, for $\sigma \in D_6$, the action of $\sigma$ to be that which sends each $a_i$ to the $a_j$ that $\sigma$ sends $a_i$ to in the action on the hexagon. Calculate the character of this representation, then proceed as the question tells you to.

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