9 Problems for Michaelmas

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.

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 $W_1\cap W_2=\{0\}$.

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 2.18 to show that $V$ is irreducible.

(Lemma 2.23 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 $g\in G$.

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?).

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

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.

Prove Proposition 2.37: 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 2.33).

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.

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?

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 2.42), and

2. 2.

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

Do the exercises in section 2.5.3 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.

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 $\rho (r)$.

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 $\alpha [g]$ for any $g\in S_3$.

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 2.18)?

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 $\chi$.

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.

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

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

Find the character tables of the following groups (you shouldn’t need to use orthogonality for these). Note that you will have to find the conjugacy classes!

1. 1.

   $C_4$

2. 2.

   $C_3\times C_3$

3. 3.

   $D_4$

4. 4.

   $D_5$

5. 5.

   $Q_8$

6. 6.

   $D_n$ .

Let $G$ be a finite group acting on a finite set $X$. Let $\chi$ be the character of the permutation representation. Prove that

$$\chi (g)=|\{x\in X:gx=x\}|.$$

Find the character of the regular representation.

The center $Z(ℂ[G])$ of the group ring $ℂ[G]$ is the set of elements

$$\sum _{g\in G}{a}_g[g]\in ℂ[G]$$

which commute with all other elements of the group ring (it is enough to check that they commute with all elements $[g]$ for $g\in G$).

Show that ${\sum }_{g\in G}{a}_g[g]$ is in $Z(ℂ[G])$ if and only if the function $g\mapsto a_g$ is a class function.

Let $(\rho ,V)$ be a representation of $G$ with character $\chi$ and dimension $d$. Show that

$$|\chi (g)|\le d$$

for all $g\in G$ with equality if and only if $\rho (g)$ is a scalar matrix. Deduce that

$$\ker (\rho )=\{g\in G:\chi (g)=d\}.$$

1. 1.

   Find the character table of $A_4$.

2. 2.

   For each irreducible representation of $S_4$, decompose its restriction to $A_4$ into irreducibles. ⁷⁷ 7 When we say decompose $V$ into irreducible subrepresentations we mean find actual subrepresentations of $V$ such that $V$ is their direct sum. When I say decompose $V$ into irreducible representations, or just irreducibles, I just mean find irreducible representations such that $V$ is isomorphic to their direct sum — you don’t have to say how they live inside $V$.

Do exercise 3.25: if $V$ is the permutation representation attached to $S_4$ acting on the set of edges of the tetrahedron, find the three irreducible subrepresentations of $V$. ⁷

Let $W\subset V$ be finite-dimensional vector spaces and let $\pi :V\to W$ be a projection. Show that

$$\mathrm{tr}(\pi )=dimW.$$

Let $G$ act on a set $X$, and let $(\pi ,V)$ be the associate permutation representation with character ${\chi }_{\pi }$.

1. 1.

   Show that $dim{V}^G$ is the number of orbits of $G$ acting on $X$.

2. 2.

   By considering $⟨\mathrm{𝟙},{\chi }_{\pi }⟩$, prove Burnside’s lemma: the number of orbits on $X$ is the average number of fixed points of elements of $G$.

Consider the representation $\rho$ of $S_3$ on $V={ℂ}^2$ given by

with $\omega =e^{2\pi i/3}$.

1. 1.

   Write down the matrices of $(\rho \otimes \rho )(12)$ and $(\rho \otimes \rho )(123)$ with respect to the basis

   $$e_1\otimes e_1,e_1\otimes e_2,e_2\otimes e_1,e_2\otimes e_2$$

   of $V\otimes V$ (where $e_1$ and $e_2$ are the standard basis of $V$).

2. 2.

   Write the character of $V\otimes V$ as a sum of irreducible characters.

3. 3.

   For each of the irreducible characters of $V\otimes V$ used in the previous part, find a subrepresentation of $V\otimes V$ with that character.

4. 4.

   Find a $G$-isomorphism $ϵ\otimes V\to V$. Find a $G$-isomorphism $V^{\ast }\to V$.

5. 5.

   Let $V^{\otimes n}=V\otimes \mathrm{\dots }\otimes V$ with $n$ factors. Decompose $V$ into irreducible representations. ⁷

Exercise 3.40. If $V={ℂ}^2$, prove that $e_1\otimes e_2+e_2\otimes e_1\in V\otimes V$ cannot be written in the form $v\otimes w$.

Let $V$ be an irreducible five-dimensional representation of $S_5$. Decompose ${\mathrm{Sym}}^{2}V$ and ${\mathrm{\Lambda }}^{2}V$ into irreducible representations.

Let $V={ℂ}^4$ be the permutation representation of $S_4$, and let $W$ be the two-dimensional irreducible representation of $S_4$.

1. 1.

   Show that ${\mathrm{Sym}}^{2}V$ has a unique subrepresentation isomorphic to $W$.

2. 2.

   Use the projection operator to find that subrepresentation.

Using the character table of $S_5$, find the character table of $A_5$. Using the character table, show that $A_5$ is simple (that is, it has no nontrivial proper normal subgroups). Hint: every element of $A_5$ is conjugate to its inverse. Why does this imply the character values are all real?

A group $G$ of order $168$ has conjugacy classes $C_1$, $C_2$, $C_3$, $C_4$, $C_{7A}$ and $C_{7B}$ where each conjugacy class is labelled by the order of any element in that class (so, for example, any element of $C_{7A}$ or $C_{7B}$ has order $7$). The following shows one of the rows of the character table of $G$.

(9.1)

1. 1.

   Show that, if $x$ is an element of $C_{7A}$ or $C_{7B}$, then $x$ is conjugate to $x^2$.

2. 2.

   Find the character table of $G$. Assume the result of the first part if you were not able to prove it.

(challenge) Show that, if $V$ is any faithful representation of $G$ and $W$ is an irreducible representation of $G$, then $W$ is isomorphic to a subrepresentation of $V^{\otimes n}$ for some $n\ge 1$.

Let $\rho$ be an irreducible representation of $G$. Show that $\rho$ is isomorphic to a subrepresentation of a representation induced from an irreducible representation of $H$.

Let $\chi$ be an irreducible character of $H$ and let

$${\mathrm{Ind}}_H^G\chi =d_1{\chi }_1+\mathrm{\dots }+d_r{\chi }_r$$

be decomposition of its induction into irreducible characters of $G$, with ${\chi }_i$ pairwise distinct and $d_i\in {\mathbb{Z}}_{\ge 0}$.

Show that

$$\sum _{i=1}^r{d}_i^2\le [G:H].$$

Let $H$ be a subgroup of $G$. Show that:

1. 1.

   If $W_1,W_2$ are representations of $H$, then

   $${\mathrm{Ind}}_H^G(W_1\oplus W_2)\cong {\mathrm{Ind}}_H^G{W}_1\oplus {\mathrm{Ind}}_H^G{W}_2.$$

2. 2.

   If $K\subset G$ is a subgroup containing $H$, and $W$ is a representation of $H$, then

   $${\mathrm{Ind}}_H^{G}W\cong {\mathrm{Ind}}_K^G({\mathrm{Ind}}_H^{K}W).$$

3. 3.

   If $V$ is a representation of $G$, then

   $${\mathrm{Ind}}_H^G{\mathrm{Res}}_H^{G}V\cong V\otimes {\mathrm{Ind}}_H^G\mathrm{𝟙}.$$

Note that all of these may be proved either from the definition (using the ’induction recognition’ corollary) or using characters and Frobenius reciprocity.

Let $H\subset G$ be groups and let $\chi$ be a character of $H$. Let $\dot{\chi }(g)=\chi (g)$ if $g\in H$ and $0$ otherwise.

1. 1.

   Show that the formula for the induced character may be rewritten

   $$({\mathrm{Ind}}_H^G\chi )(g)=\frac{1}{|H|}\sum _{x\in G}\dot{\chi }(x^{-1}gx).$$

2. 2.

   If $g_{1}H,\mathrm{\dots },g_{r}H$ are the left cosets of $H$ in $G$, show that

   $$({\mathrm{Ind}}_H^G\chi )(g)=\sum _{i=1}^r\dot{\chi }(g_i^{-1}g{g}_i).$$

For each irreducible representation of $S_4$, decompose its induction to $S_5$ (where $S_4$ is regarded as the subgroup of elements of $S_5$ that fix $5\in \{1,\mathrm{\dots },5\}$.)

Let $\chi$ be the irreducible degree 3 character of $H=A_5$ such that

$$\chi ((12345))=\frac{1+\sqrt{5}}{2}.$$

Let $G=A_6$, with $H$ as the subgroup fixing 6. Compute the character

$${\mathrm{Ind}}_H^G(\chi ).$$

Let $H=⟨(12\mathrm{\dots }p)⟩\subset S_p=G$, for $p$ a prime, and $\chi$ be a nontrivial character of $H$.

1. 1.

   Find

   $${\mathrm{Ind}}_H^G\chi .$$

2. 2.

   By considering $⟨{\mathrm{Ind}}_H^G\chi ,{\mathrm{Ind}}_H^G\chi ⟩$, show that

   $$(p-1)!\equiv -1modp.$$

You may use that, if $\zeta$ is a nontrivial $n$th root of unity, then

$$1+\zeta +{\zeta }^2+\mathrm{\dots }+{\zeta }^{n-1}=0.$$

Let

$$G=⟨a,x:a^7=x^3=e,xa{x}^{-1}=a^2⟩.$$

You may assume that $G$ has 21 elements given by

$$\{a^i{x}^j:i=0,1,\mathrm{\dots },6,j=0,1,2\}.$$

1. 1.

   Show that $H=⟨a⟩$ is a normal subgroup of $G$.

2. 2.

   By considering representations lifted from $G/H$ and induced from $H$, find the character table of $G$.

Fill in the missing proofs from section 4.6. (This is more of a mega-problem!)