Elements of representation theory

Prove that the map $\rho : \mathbb {Z} \rightarrow \mathbf{G L}_{2}(\mathbb {C})$ given by

is a reducible two-dimensional complex representation of the group $\mathbb {Z}$ and is not equivalent to the direct sum of two one-dimensional representations.

Prove that the map $\rho :\langle a\rangle_{p} \rightarrow \mathbf{G L}_{2}\left(\mathbb {F}_{p}\right)$ ($p$ a prime number) given by

is a reducible two-dimensional representation of the cyclic group $\langle a\rangle_{p}$ and is not equivalent to the direct sum of two one-dimensional representations.

Let $A \in \mathbf{G L}_{n}(\mathbb {C})$. Prove that the map $\rho_{A}: \mathbb {Z} \rightarrow \mathbf{G} \mathbf{L}_{n}(\mathbb {C})$ given by $\rho_{A}(n)=A^{n}$ is a representation of the group $\mathbb {Z}$, and that the representations $\rho_{A}$ and $\rho_{B}$ are equivalent if and only if the Jordan normal forms of the matrices $A$ and $B$ coincide (up to the order of the blocks).

Is the map $L$ defined by the formulas below a linear representation of the group $\mathbb {R}$ on the space $C(\mathbb {R})$ of continuous functions on the real line:

Which of the subspaces of $C(\mathbb {R})$ are invariant under the linear representation $L$ from problem 69.4, (a):

Find the subspaces of the space of polynomials invariant under the representation $L$ from problem 69.4a.

Write in matrix form (in some basis) the restriction of the linear representation $L$ from problem 69.5 to the subspace of polynomials of degree $\leqslant 2$.

Write in matrix form (in some basis) the restriction of the linear representation $L$ from problem 69.5 to the linear span of the functions $\sin x$ and $\cos x$.

Prove that each of the following formulas defines a linear representation of the group $\mathbf{G} \mathbf{L}_{n}(F)$ on the space $\mathbf{M}_{n}(F)$:

Prove that the linear representation $\Lambda$ (see problem 69.9, (a)) is completely reducible, and that its invariant subspaces coincide with the left ideals of the algebra $\mathbf{M}_{n}(K)$.

Prove that if char $F$ does not divide $n$, then the linear representation Ad (see problem 69.9, (b)) is completely reducible and its nontrivial invariant subspaces are the space of matrices with zero trace and the space of scalar matrices.

Prove that if char $F \neq 2$, then the linear representation $\Phi$ (see problem 69.9, (c)) is completely reducible and its nontrivial invariant subspaces are the spaces of symmetric and skew-symmetric matrices.

Let $V$ be a two-dimensional space over a field $F$. Show that there exist representations $\rho_{1}$ and $\rho_{2}$ of the group $\mathbf{S}_{3}$ on $V$, for which, in some basis of the space $V$, the relations

= 0 1 1 0 , _1 1 2 3 = 0 -1 1 -1 _2 1 2 = 0 1 1 0 , _2 1 2 3 = 0 1 -1 -1 .

hold.

Prove that these representations are isomorphic if and only if $\operatorname {char} F \neq 3$.

Let $V$ be a two-dimensional vector space over a field $F$. Show that there exist two representations $\rho_{1}, \rho_{2}$ of the group

on $V$, for which, in some basis of the space $V$, the relations

, _1(b)= 0 1 1 0 _2(a)= 0 1 -1 0 , _2(b)= 1 0 0 -1

hold.

Are these representations equivalent?

Let $\rho_{1}$ and $\rho_{2}$ be the representations of the groups $\mathbf{S}_{3}$ and $\mathbf{D}_{4}$ from problems 69.13 and 69.14. Are these representations irreducible?

Let $V$ be a vector space over a field $F$ with basis $\left(e_{1}, \ldots , e_{n}\right)$. Define a map $\psi : \mathbf{S}_{n} \rightarrow \mathbf{G L}(V)$ by setting

where $\sigma \in \mathbf{S}_{n}, i=1, \ldots , n$. Prove that:

Let $P_{n, m}$ be the subspace of homogeneous polynomials of degree $m$ in the algebra $F\left[x_{1}, \ldots , x_{n}\right]$, with $\operatorname {char} F=0$. Define a map $\Theta : \mathbf{G} \mathbf{L}_{n}(F) \rightarrow \mathbf{G L}\left(P_{n, m}\right)$ by setting, for $f \in P_{n, m}$ and $A=\left(a_{i j}\right) \in \mathbf{G L}_{n}(F)$:

Prove that $\Theta$ is an irreducible representation of the group $\mathbf{G} \mathbf{L}_{n}(F)$ on the space $P_{n, m}$.

Let there be given an $n$-dimensional space $V$ over a field $F$ of characteristic zero. Define a map $\theta : \mathbf{G L}(V) \rightarrow \mathbf{G L}\left(\Lambda^{m} V\right)$ by setting

where $x_{1}, \ldots , x_{m} \in V$ and $f \in \mathbf{G} \mathbf{L}(V)$. Prove that $\theta$ is an irreducible representation of the group $\mathbf{G L}(V)$.

Prove that:

Let there be given a representation $\Phi : G \rightarrow \mathbf{G L}(V)$ over a field $F$ and a homomorphism $\xi : G \rightarrow F^{*}$. Consider the map $\Phi_{\xi }: G \rightarrow \mathbf{G L}(V)$ given by the rule $\Phi_{\xi }(g)=\xi (g) \Phi (g), g \in G$. Prove that $\Phi_{\xi }$ is a representation of the group $G$. It is irreducible if and only if the representation $\Phi$ is irreducible.

Let $\Phi$ be a complex representation of a finite group $G$. Prove that every operator $\Phi_{g}, g \in G$, is diagonalizable.

Let $\rho : G \rightarrow \mathbf{G L}(V)$ be a finite-dimensional representation of a group $G$ over a field $F$. Prove that there exists a basis of $V$ in which, for every $g \in G$, the matrix $\rho (g)$ has the block upper-triangular form

where $\rho_{i}$ are irreducible representations of the group $G$.

Let $\rho : G \rightarrow \mathbf{G L}(V)$ be a finite-dimensional representation of a group $G$, and suppose there exists a basis $\left(e_{1}, \ldots , e_{n}\right)$ of $V$ in which, for every $g \in G$, the matrix $\rho (g)$ has the block upper-triangular form from problem 69.22, where the size $d_{i}$ of the square matrix $\rho_{i}(g)$ does not depend on $g$. Prove that:

Let $\rho : G \rightarrow \mathbf{G L}(V)$ be a representation of a group $G$.

Prove that:

Let $\rho : G \rightarrow \mathbf{G L}(V)$ be a representation of a group $G$ and $H$ a subgroup of $G$, $[G: H]=k<\infty$. Prove that if a subspace $U$ is invariant under the restriction of the representation $\rho$ to the subgroup $H$, then the dimension of the minimal subspace containing $U$ that is invariant under the representation $\rho$ does not exceed $k \cdot \operatorname {dim} U$.

Let $V$ be a vector space over the field $\mathbb {C}$ with basis $\left(e_{1}, \ldots , e_{n}\right)$. Define on $V$ a representation $\Phi$ of the cyclic group $\langle a\rangle_{n}$ by setting $\Phi (a)\left(e_{i}\right)=e_{i+1}$ for $i<n$ and $\Phi (a)\left(e_{n}\right)=e_{1}$. For $n=2 m$, find the dimension of the minimal invariant subspace containing the vectors:

Prove that any set of pairwise commuting operators on a finite-dimensional complex vector space $V$ has a common eigenvector.

Prove that every irreducible representation of an abelian group on a finite-dimensional vector space over the field $\mathbb {C}$ is one-dimensional.

Let $G=\langle a\rangle_{p} \times \langle b\rangle_{p}$, where $p$ is a prime number, and let $K$ be a field of characteristic $p$. Suppose $V$ is a vector space over $K$ with basis $x_{0}, x_{1}, \ldots , x_{n}, y_{1}, \ldots , y_{n}$. Define a map $\rho : G \rightarrow \mathbf{G L}(V)$ by setting

Prove that $\rho$ extends to a representation of the group $G$. Check that this representation is indecomposable.

Prove that the irreducible complex representations of the group $\mathbf{U}_{p^{\infty }}$ are in bijective correspondence with sequences $\left(a_{n}\right)$ of natural numbers such that

for all $n$.

Prove that the irreducible complex representations of the group $\mathbb {Q} / \mathbb {Z}$ are in bijective correspondence with sequences of natural numbers $\left(a_{n}\right)$ such that

if $n$ divides $m$.

Let $\mathcal{A}$ and $\mathcal{B}$ be two commuting operators on a finite-dimensional vector space $V$ over $\mathbb {C}$, with $\mathcal{A}^{m}=\mathcal{B}^{n}=\mathcal{E}$ for some natural numbers $m$ and $n$. Prove that the space $V$ decomposes as a direct sum of one-dimensional subspaces invariant under both $\mathcal{A}$ and $\mathcal{B}$.

List all irreducible complex representations of the groups:

Let $V$ be a vector space over a field $F$, $\mathcal{A} \in \mathbf{G L}(V)$, and $\mathcal{A}^{n}=\mathcal{E}$.

Decompose into a direct sum of one-dimensional representations the regular representation of the group:

Let $H=\langle a\rangle_{3}$ be a cyclic subgroup of a group $G$, $\Phi$ the regular representation of $G$, and $\Psi$ its restriction to $H$. Find the multiplicity of each irreducible representation of the group $H$ in the decomposition of the representation $\Psi$ into a sum of irreducible ones:

Find all nonisomorphic one-dimensional real representations of the group $\langle a\rangle_{n}$.

Prove that an irreducible real representation of a cyclic group has dimension at most two.

Let $\rho_{k}:\langle a\rangle_{n} \rightarrow \mathbf{G L}_{2}(\mathbb {R})$ be the representation for which

Prove that:

Find the number of inequivalent irreducible real representations of:

Find the number of inequivalent two-dimensional complex representations of the groups:

Let $G$ be an abelian group of order $n$. Prove that the number of inequivalent $k$-dimensional complex representations of the group $G$ equals the coefficient of $t^{k}$ in the series $(1-t)^{-n}$. Find this coefficient.

Prove that the kernel of a one-dimensional representation of a group $G$ contains the commutator subgroup of that group.

Let $\rho$ be a representation of a group $G$ on a space $V$, and suppose $V$ has a basis in which all the operators $\rho (g)(g \in G)$ are diagonal. Prove that $\operatorname {Ker} \rho \supseteq G^{\prime }$.

Prove that all irreducible complex representations of a finite group are one-dimensional if and only if it is commutative.

Find all nonisomorphic one-dimensional complex representations of the groups $\mathbf{S}_{3}$ and $\mathbf{A}_{4}$.

Find all one-dimensional complex representations of the groups $\mathbf{S}_{n}$ and $\mathbf{D}_{n}$.

Construct an irreducible two-dimensional complex representation of the group $\mathbf{S}_{3}$.

Using a homomorphism of the group $\mathbf{S}_{4}$ onto the group $\mathbf{S}_{3}$, construct an irreducible two-dimensional complex representation of the group $\mathbf{S}_{4}$.

Using the isomorphism between permutation groups and the corresponding groups of motions of the cube and the tetrahedron (see 57.13), construct:

Prove that if $\varepsilon$ is an $n$-th root of 1, then the map

extends to a representation $\rho_{\varepsilon }$ of the group $\mathbf{D}_{n}$. Is it irreducible when $\varepsilon \neq \pm 1$?

Let $\rho_{\varepsilon }$ and $\rho_{\varepsilon^{\prime }}$ be the irreducible two-dimensional complex representations of the group $\mathbf{D}_{n}$ from problem 70.20. Prove that $\rho_{\varepsilon }$ and $\rho_{\varepsilon^{\prime }}$ are isomorphic if and only if $\varepsilon^{\prime }=\varepsilon^{ \pm 1}$.

Let $\rho$ be an irreducible complex representation of the group $\mathbf{D}_{n}$. Prove that $\rho$ is isomorphic to $\rho_{\varepsilon }$ for some $\varepsilon$.

Let $\rho$ be the natural two-dimensional real representation of $\mathbf{D}_{n}$ as the transformations forming a regular $n$-gon. Find $\varepsilon$ such that $\rho$ is isomorphic to $\rho_{\varepsilon }$.

Using the realization of the quaternions as complex matrices of order 2 (see problem 58.11, (c)), construct a two-dimensional complex representation of the group $\mathbf{Q}_{8}$.

Suppose the group $G$ has a faithful reducible two-dimensional representation.

Prove that:

Prove that a faithful two-dimensional complex representation of a finite noncommutative group is irreducible.

Let $G$ be a finite group, $\rho$ its finite-dimensional complex representation, and suppose that, in some basis, the matrices of all the operators $\rho (g)(g \in G)$ are upper triangular. Prove that $\operatorname {Ker}(\rho ) \supseteq G^{\prime }$.

Prove that if, in problems 69.22, 69.23, the base field is the field of complex numbers and the group $G$ is finite, then the representation $\rho$ is equivalent to the direct sum of the representations $\rho_{1}, \ldots , \rho_{m}$.

Prove that if, in problem 69.22, the base field is the field of complex numbers and the group $G$ is finite, then there exists a nonsingular matrix $C$ such that, for all $g \in G$,

Let $G$ be a finite group of order $n$, $\rho$ its regular representation. Prove that

Prove that for any nonidentity element of a finite group there exists an irreducible complex representation that maps it to a nonidentity operator.

Let $\mathcal{A}, \mathcal{B}$ be linear operators on a finite-dimensional vector space $V$ over a field $F$ of characteristic 0, with $\mathcal{A}^{3}=\mathcal{B}^{2}=\mathcal{E}$, $\mathcal{A B}=\mathcal{B} \mathcal{A}^{2}$.

Prove that for every subspace $U$ invariant under $\mathcal{A}$ and $\mathcal{B}$, there exists a subspace $W$ invariant under $\mathcal{A}, \mathcal{B}$ such that $V=U \oplus W$.

Find all inequivalent two-dimensional complex representations of the groups:

Find the number and dimensions of the irreducible complex representations of the groups:

How many direct summands are there in the decomposition into irreducible components of the regular representation of the following groups:

Using representation theory, prove that a group of order 24 cannot equal its own commutator subgroup.

Can the irreducible complex representations of a finite group consist exactly of:

Prove that the group $\mathbf{G L}_{2}(\mathbb {C})$ has no subgroup isomorphic to $\mathbf{S}_{4}$.

Prove the existence of a two-dimensional invariant subspace in any eight-dimensional complex representation of the group $\mathbf{S}_{4}$.

Prove the existence of a one-dimensional invariant subspace in any five-dimensional representation of the group $\mathbf{A}_{4}$.

Prove that the number of irreducible representations of a group $G$ is strictly greater than the number of irreducible representations of any of its quotient groups by a nontrivial normal subgroup.

For which finite groups does the regular representation over the field $\mathbb {C}$ contain only finitely many subrepresentations?

Prove that any irreducible representation of a finite $p$-group over a field of characteristic $p$ is trivial.

Let $G$ be a finite $p$-group and $\rho$ its representation on a finite-dimensional space $V$ over a field of characteristic $p$. Prove that $V$ has a basis such that, for every $g \in G$, the matrix of the operator $\rho (g)$ is upper unitriangular.

Let $H$ be a normal subgroup of a finite group $G$. Prove that the dimension of any irreducible representation of the group $G$ over a field $F$ does not exceed $[G: H] m$, where $m$ is the largest dimension of an irreducible representation of the group $H$ over the field $F$.

Prove that in $\mathbf{G} \mathbf{L}_{n}(\mathbb {C})$ there are only finitely many pairwise nonconjugate subgroups of a fixed finite order.

Let $\rho : G \rightarrow \mathbf{G L}_{3}(\mathbb {R})$ be an irreducible three-dimensional real representation of a finite group $G$, and let the representation $\widetilde{\rho }: G \rightarrow \mathbf{G} \mathbf{L}_{3}(\mathbb {C})$ be obtained as the composition of the map $\rho$ with the standard embedding $\mathbf{G L}_{3}(\mathbb {R}) \rightarrow \mathbf{G L}_{3}(\mathbb {C})$. Prove that the representation $\widetilde{\rho }$ is irreducible.

Prove that every irreducible non-one-dimensional complex representation of a group of order $p^{3}$ is faithful.

Find the number of irreducible complex representations of a noncommutative group of order $p^{3}$ and their dimensions.

The real representation $\Phi$ of the cyclic group $\langle a\rangle$ of order 4, in which

decompose into a direct sum of irreducible ones.

Consider the real three-dimensional representation of the group $G=\langle a\rangle_{2} \times \langle b\rangle_{2}$, where

Decompose $\Phi$ into a direct sum of irreducible representations.

Consider the two-dimensional complex representation $\Phi$ of the group $G=\langle a\rangle_{2} \times \langle b\rangle_{2}$, where

Decompose $\Phi$ into a direct sum of irreducible representations.

Is the algebra of quaternions a real group algebra:

Let $V$ be a vector space over a field $F$ with basis $\left(e_{1}, e_{2}, e_{3}\right)$, $\varphi : F\left[\mathbf{S}_{3}\right] \rightarrow \operatorname {End} V$ the homomorphism where $\varphi (\sigma )\left(e_{i}\right)=e_{\sigma (i)}$ for all $\sigma \in \mathbf{S}_{3}(i=1,2,3)$. Find the dimension of the kernel and the dimension of the image of the homomorphism $\varphi$.

Find a basis for the kernel of the homomorphism $\varphi : \mathbb {C}\left(\langle a\rangle_{n}\right) \rightarrow \mathbb {C}$ given by $\varphi (a)=\varepsilon$, where $\varepsilon$ is an $n$-th root of 1.

Let a group $H$ be isomorphic to a quotient group of a group $G$. Prove that $F[H]$ is isomorphic to a quotient algebra of the algebra $F[G]$.

Let $G=G_{1} \times G_{2}$. Prove that

Let $G$ be a finite group, $R$ the set of maps from $G$ to a field $F$. Define operations on $R$ by setting, for $f_{1}, f_{2} \in R$,

Prove that $R$ is an algebra over the field $F$ and that the map

from $R$ to $F[G]$ is an isomorphism of algebras.

Prove that if a group $G$ contains elements of finite order, then the group algebra $F[G]$ has zero divisors.

Prove that every irreducible $F[G]$-module is isomorphic to a quotient module of the regular $F[G]$-module.