8.1 Problems for section 1

1. 1.

   Find the matrices of all the elements of $S_3$ in the permutation representation, with respect to the basis $e_1,e_2,e_3$.

2. 2.

   Find another basis such that the matrices all take the form

   and determine the unknown entries for your basis.

Solution

Suppose that $V$ is a representation of $G$ and that $W_1$ and $W_2$ are irreducible subrepresentations. Show that either $W_1=W_2$ or $W_1\cap W_2=\{0\}$.

Solution

Consider $G=S_n$ with its permutation representation action on ${ℂ}^n$ characterized by

$$\pi (g)e_i=e_{g(i)},$$

and let $V\subset {ℂ}^n$ be the subspace

$$\{(a_1,\mathrm{\dots },a_n):\sum _{i=1}^n{a}_i=0\}.$$

Mimic the last part of example 1.18 to show that $V$ is irreducible.

Solution

(Lemma 1.23 from the lectures). Let $({\pi }_V,V)$ and $({\pi }_W,W)$ be two representations of a (finite) group $G$.

1. 1.

   Show that if $T\in {\mathrm{Hom}}_G(V,W)$ is a $G$-homomorphism and an isomorphism of vector spaces, then $T^{-1}$ is also a $G$-homomorphism.

2. 2.

   Assume $dimV=dimW=n$ and identify $V$ and $W$ with ${ℂ}^n$ by choosing bases. Show that $V\cong W$ as representations of $G$ if and only if there exists a $T\in {\mathrm{GL}}_n(ℂ)$ such that

   $$T{\pi }_V(g)T^{-1}={\pi }_W(g)$$

   for all $g\in G$.

Solution

1. 1.

   Show that the symmetry group of the tetrahedron is isomorphic to $S_4$.

2. 2.

   Show that the rotational symmetry group of the cube is isomorphic to $S_4$ (hint: consider the action of this group on the four diagonals of the cube).

3. 3.

   Show that the two $3$-dimensional representations of the symmetric group $S_4$ we obtain from (a) and (b) are not isomorphic (hint: what do conjugate matrices have in common?).

Solution

Classify the irreducible representations of $D_n$ when $n\ge 4$ is even. Write out the list explicitly when $n=4$.

Solution

Consider $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$, the quaternion group, which satisfies the relations

$$i^2=j^2=k^2=ijk=-1.$$

For us it is convenient to view $Q_8$ as the group generated by $i,j$ with the relations $i^4=j^4=e$, $i^2=j^2$, and $ij=j{i}^{-1}$ (the first relation actually follows from the other two: $i^4=i{j}^{2}i=j{i}^{-1}ji=j{i}^{-2}j=j{j}^{-2}j=e$).

1. 1.

   Use the technique developed in class for the dihedral groups to determine the irreducible representations of $Q_8$. (Take $⟨i⟩$ as the abelian subgroup). You should get four $1$-dimensional representations and one $2$-dimensional one. Show (by inspection) that the $2$-dimensional representation is faithful, and write down the matrices corresponding to $i$, $j$ and $k$.

2. 2.

   Compare your results with the list of irreducible representations of $D_4$, the dihedral group with $8$ elements. What do you observe?

3. 3.

   (optional) Show that there is no faithful two-dimensional representation of $Q_8$ over the reals.

Solution

Prove Proposition 1.37: Let $(\pi ,V)$ be an irreducible representation of a finite group $G$ and $Z=Z(G)$ be the center of $G$. Then $Z$ acts on $V$ as a character. That is, there exists a homomorphism $\chi :Z\to {ℂ}^{\times }$ such that

$$\pi (z)v=\chi (z)v$$

for all $v\in V$. (Follow the lines of the proof of Theorem 1.33).

Solution

What is the centre of $D_n$? Find the central character of the irreducible two-dimensional representation of $D_n$ coming from its action on the regular $n$-gon.

Solution

Find a two-dimensional irreducible representation of $C_n$ over $ℝ$ (for $n\ge 3$), and prove that it is irreducible. Why does this mean that Schur’s lemma doesn’t hold with real coefficients? Where does the proof from lectures go wrong?

Solution

Show that, if $V$ is a vector space, $W\subset V$ is a subspace, and $\pi :V\to W$ is a projection, then

1. 1.

   $V=W\oplus \ker (\pi )$ (see exercise 1.42), and

2. 2.

   $\mathrm{tr}(\pi )=dim(W)$.

Solution

Do the exercises in section 1.5.3 of the notes; that is, show that every representation of a finite group $G$ over $ℂ$ is unitarizable and use this to give an alternative proof of Maschke’s theorem. Solution

Let $V$ be the permutation representation of $D_5$ on the set of vertices of the regular pentagon. Write $V$ as a direct sum of irreducible subrepresentations.

Hint: first find the eigenvectors for $\rho \mathit{}\mathrm{(}r\mathrm{)}$.

Solution

In the group ring $ℂ[S_3]$, let

$$\alpha =[e]+[(12)]+[(23)]+[(31)]+[(123)]+[(132)]$$

and let

$$\beta =2[e]-[(123)]-[(132)].$$

1. 1.

   Find ${\alpha }^2$, ${\beta }^2$ and $\alpha \beta$. Hint: first consider $\alpha \mathit{}\mathrm{[}g\mathrm{]}$ for any $g\mathrm{\in }{S}_{\mathrm{3}}$.

2. 2.

   For $(x,y,z)\in {ℂ}^3$ (with the permutation representation), compute $\alpha (x,y,z)$ and $\beta (x,y,z)$. What do you notice (compare example 1.18)?

Solution

Suppose that $G$ is a group and $V=ℂ[G]$, and that $\chi :G\to {ℂ}^{\times }$ is a one-dimensional character. Show that

$$v_{\chi }=\sum _{g\in G}{\chi }^{-1}(g)[g]$$

spans a one-dimensional subrepresentation on which $G$ acts via $\chi$.

Solution

Decompose the group ring $ℂ[S_3]$ as a direct sum of irreducible representations of $S_3$. That is, find explicit irreducible subrepresentations of $ℂ[S_3]$ such that it is the direct sum of those subrepresentations.

Solution

Verify the sum of squares formula for $D_n$ (do both the odd and even cases).

Solution

(optional) Find a group $G$ and representation $V$ of $G$ having no irreducible subrepresentation.

Solution