Math Problems Vault

Problem 69.1

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

\rho (n)=\left[\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix}\right], \quad n \in \mathbb {Z},

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.

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Problem 69.2

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

\rho \left(a^{k}\right)=\left[\begin{smallmatrix} 1 & k \cdot 1 \\ 0 & 1 \end{smallmatrix}\right],

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.

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Problem 69.3

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).

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Problem 69.4

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:

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(a)

$(L(t) f)(x)=f(x-t)$ ;

(b)

$(L(t) f)(x)=f(t x)$ ;

(c)

$(L(t) f)(x)=f\left(e^{t} x\right)$ ;

(d)

$(L(t) f)(x)=e^{t} f(x)$ ;

(e)

$(L(t) f)(x)=f(x)+t$ ;

(f)

$(L(t) f)(x)=e^{t} f(x+t)$ ?

Problem 69.5

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

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(a)

the subspace of infinitely differentiable functions;

(b)

the subspace of polynomials;

(c)

the subspace of polynomials of degree $\leqslant n$ ;

(d)

the subspace of even functions;

(e)

the subspace of odd functions;

(f)

the linear span of the functions $\sin x$ and $\cos x$ ;

(g)

the subspace of polynomials in $\cos x$ and $\sin x$ ;

(h)

the linear span of the functions $\cos x, \cos 2 x, \ldots , \cos n x$ ;

(i)

the linear span of the functions $e^{c_{1} t}, e^{c_{2} t}, \ldots , e^{c_{n} t}$ , where $c_{1}, c_{2}, \ldots , c_{n}$ are distinct fixed real numbers?

Problem 69.6

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

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Problem 69.7

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 $\leqslant 2$ .

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Problem 69.8

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 $\cos x$ .

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Problem 69.9

(a)(b)(c)

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

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(a)

$\Lambda (A) \cdot X=A X$ ;

(b)

$\operatorname {Ad}(A) \cdot X=A X A^{-1}$ ;

(c)

$\Phi (A) \cdot X=A X^{t} A$ .

Problem 69.10

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 $\mathbf{M}_{n}(K)$ .

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Problem 69.11

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.

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Problem 69.12

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.

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Problem 69.13

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

\begin{array}{ll} \rho _{1}\left(\left[\begin{smallmatrix} 1 & 2 \end{array}

= 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 $\operatorname {char} F \neq 3$ .

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Problem 69.14

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

\mathbf{D}_{4}= \left\langle a, b \mid a^{4}=b^{2}=(a b)^{2}=1\right\rangle

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

\begin{array}{ll} \rho _{1}(a)=\left[\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{array}

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

hold.

Are these representations equivalent?

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Problem 69.15

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?

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Problem 69.16

(a)(b)(c)

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

\psi _{\sigma }\left(e_{i}\right)=e_{\sigma (i)}

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

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(a)

$\psi$ is a representation of the group $\mathbf{S}_{n}$ ;

(b)

the subspace $W$ of vectors whose coordinates relative to the basis $\left(e_{1}, \ldots , e_{n}\right)$ sum to zero, and the subspace $U$ of vectors with equal coordinates, are invariant under the representation $\psi$ ;

(c)

if $\operatorname {char} F$ does not divide $n$ , then the restriction of the representation $\psi$ to $W$ is an irreducible $(n-1)$ -dimensional representation of the group $\mathbf{S}_{n}$ .

Problem 69.17

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 $A=\left(a_{i j}\right) \in \mathbf{G L}_{n}(F)$ :

\left(\Theta _{A} f\right)\left(x_{1}, \ldots , x_{n}\right)=f\left(\sum _{i=1}^{n} x_{i} a_{i 1}, \ldots , \sum _{i=1}^{n} x_{i} a_{i n}\right)

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

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Problem 69.18

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

\theta (f)\left(x_{1} \wedge \ldots \wedge x_{m}\right)=\left(f x_{1}\right) \wedge \ldots \wedge \left(f x_{m}\right)

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 $\mathbf{G L}(V)$ .

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Problem 69.19

(a)(b)

Prove that:

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(a)

for any representation $\rho$ of a group $G$ there exists a representation $\rho^{\otimes m}$ of the group $G$ on the space

V^{\otimes m}=\underset {m}{\rightarrow } \underbrace{V \otimes \ldots \otimes V}

of $m$ -fold contravariant tensors on the space $V$ such that

\rho ^{\otimes m}(g)\left(v_{1} \otimes \ldots \otimes v_{m}\right)=\left(\rho (g) v_{1}\right) \otimes \ldots \otimes \left(\rho (g) v_{m}\right)

for any $v_{1}, \ldots , v_{m} \in V, g \in G$ ;

(b)

the subspaces of symmetric and skew-symmetric tensors are invariant subspaces for the representation $\rho^{\otimes m}$ . Find the dimensions of these subspaces if $\operatorname {dim} V=n$ .

Problem 69.20

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.

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Problem 69.21

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

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Problem 69.22

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

\rho (g)=\left[\begin{smallmatrix} \rho _{1}(g) & & * \\ & \ddots & \\ 0 & & \rho _{m}(g) \end{smallmatrix}\right]

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

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Problem 69.23

(a)(b)(c)

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:

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(a)

the linear span $V_{i}$ of the vectors $e_{d_{1}+\ldots +d_{i-1}+1}, \ldots , e_{d_{1}+\ldots +d_{i}}$ is a $G$ -invariant subspace $(1 \leqslant i \leqslant m)$ ;

(b)

the map $g \mapsto \rho_{i}(g)$ is a matrix representation of the group $G$ ;

(c)

the linear representation of the group $G$ corresponding to this matrix representation is isomorphic to the representation arising on the quotient space $V_{i} / V_{i-1}$ (with $V_{0}=0$ by definition).

Problem 69.24

(a)(b)(c)

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

Prove that:

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(a)

for every $v \in V$ , the linear span $\langle \rho (g) v \mid g \in G\rangle$ is an invariant subspace for the representation $\rho$ ;

(b)

every vector of $V$ lies in some invariant subspace of dimension $\leqslant \left|G\right|$ .

(c)

the minimal invariant subspace containing a vector $v \in V$ coincides with $\langle \rho (g) v \mid g \in G\rangle$ .

Problem 69.25

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 $k \cdot \operatorname {dim} U$ .

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Problem 69.26

(a)(b)(c)(d)

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:

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(a)

$e_{1}+e_{m+1}$ ;

(b)

$e_{1}+e_{3}+\ldots +e_{2 m-1}$ ;

(c)

$e_{1}-e_{2}+e_{3}-\ldots -e_{2 m}$ ;

(d)

$e_{1}+e_{2}+\ldots +e_{m}$ .

Problem 69.27

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

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Problem 69.28

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

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Problem 69.29

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

\begin{aligned} & \rho (a) x_{i}=\rho (b) x_{i}=x_{i}, \quad 0 \leqslant i \leqslant n, \\ & \rho (a) y_{i}=x_{i}+y_{i}, \quad 1 \leqslant i \leqslant n ; \\ & \rho (b) y_{i}=y_{i}+x_{i-1}, \quad 1 \leqslant i \leqslant n. \end{aligned}

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

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Problem 69.30

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

0 \leqslant a_{n} \leqslant p^{n}-1, \quad a_{n} \equiv a_{n+1} \quad \left(\bmod p^{n}\right)

for all $n$ .

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Problem 69.31

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

0 \leqslant a_{n} \leqslant n-1, \quad a_{n} \equiv a_{m} \quad (\bmod n),

if $n$ divides $m$ .

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