Section III.3.1

Definitions and examples of linear representations

[5/20%]
Show
LaTeX
Problem III.3.1.1

The group SO(2)\mathrm{SO}(2) is given by its natural two-dimensional representation

Φ′(θ)=∥cos⁡θ−sin⁡θsin⁡θcos⁡θ∥ \Phi ^{\prime }(\theta )=\left\Vert \begin{smallmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{smallmatrix}\right\Vert

irreducible over R\mathbb {R}. Verify that

AΦ′(θ)A−1=∥eiθ00e−iθ∥ for A=12∥1ii1∥∈GL⁡(2,C). A \Phi ^{\prime }(\theta ) A^{-1}=\left\Vert \begin{smallmatrix} e^{i \theta } & 0 \\ 0 & e^{-i \theta } \end{smallmatrix}\right\Vert \quad \text{ for } A=\frac{1}{\sqrt{2}}\left\Vert \begin{smallmatrix} 1 & i \\ i & 1 \end{smallmatrix}\right\Vert \in \operatorname {GL}(2, \mathbb {C}).

Hence, Φ′\Phi^{\prime } is a direct sum of two inequivalent (in this case simply distinct) one-dimensional representations.

?
Problem III.3.1.2

Is the GL(n,C)\mathrm{GL}(n, \mathbb {C})-space Mn0(C)M_{n}^{0}(\mathbb {C}) in decomposition (5) irreducible for n=2n=2 and n=3n=3?

?
Problem III.3.1.3

Let Φ\Phi and Ψ\Psi be irreducible complex representations of the cyclic group ⟨a∣an=e⟩\left\langle a \mid a^{n}=e\right\rangle of order nn. Show that

1n∑k=0n−1Φ(ak)Ψ(ak)‾={1, if Φ≈Ψ0, if Φ≉Ψ \frac{1}{n} \sum _{k=0}^{n-1} \Phi \left(a^{k}\right) \overline{\Psi \left(a^{k}\right)}= \begin{cases} 1, \text{ if } \Phi \approx \Psi \\ 0, \text{ if } \Phi \not\approx \Psi \end{cases}
?
Problem III.3.1.4

Relying on Exercise 3, verify the validity of the following statement. Any complex-valued function ff on a finite cyclic group ⟨a∣an=e⟩\langle a \mid a^{n}=e\rangle can be written as an expansion "in elementary harmonics"

f(ak)=∑m=0n−1cmεmk,ε=exp⁡(2πin) f\left(a^{k}\right)=\sum _{m=0}^{n-1} c_{m} \varepsilon ^{m k}, \quad \varepsilon =\exp \left(\frac{2 \pi i}{n}\right)

The "Fourier coefficients" cmc_{m} are computed by the formula

cm=1n∑k=0n−1f(ak)ε−mk c_{m}=\frac{1}{n} \sum _{k=0}^{n-1} f\left(a^{k}\right) \varepsilon ^{-m k}
?
Problem III.3.1.5

From the formula for the number of necklaces (see the beginning of the chapter), derive the elementary consequences:

?
(a)

qp−q≡0( mod p)(q^{p}-q \equiv 0(\bmod p)( Fermat's little theorem, see § 4 Ch. 4);

(b)

∑d∣nφ(d)=n\sum_{d \mid n} \varphi (d)=n.