The representation doesn't have to be a permutation matrix. Consider a primitive $n$-th root of unity $\zeta$ and the matrices $\zeta^k I$ with $I$ the identity.

A complex representation can be one dimensional for cyclic groups. We don't have leave $\mathbb{C}$ at all. Note that real representations don't enjoy this property and you do need cyclic matrices in that case.

@CyclotomicField$:$ I don't get your point. The generators of a cyclic group correspond to an $n$-cycle. Given a $n$-cycle we can consider it's action on the basis elements to get an isomorphism of $\mathbb C^n.$ Isn't the matrix representation of that linear isomorphism a permutation matrix?

@CyclotomicField$:$ Is there any representation of a cyclic group which makes it a pseudoreflection group? I am not familiar with much of representation theory. One dimensional representations are known as characters. Right?

It is a representation but it's not the minimal one when working over the complex numbers. The rotations can be generated by an primitive $n$-th root of unity my favorite being $e^{\frac{2i\pi}n}$. If you want an infinite cyclic group just pick an irrational multiple of $2i\pi$ instead. I actually prefer to call $\mathbb{C}$ the circle numbers for this reason as they encode cyclic properties in a natural way.

Do you mean representation of the following kind $:$ $$\zeta \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \zeta^k \end{pmatrix}$$

That representation works as well but $\zeta I$ would have $\zeta$ along all the diagonal entries. All you need is scalar multiplication without using matrices at all.

It is a representation but it's not the minimal one when working over the complex numbers. The rotations can be generated by an primitive $n$-th root of unity my favorite being $e^{\frac{2i\pi}n}$. If you want an infinite cyclic group just pick an irrational multiple of $2i\pi$ instead. I actually prefer to call $\mathbb{C}$ the circle numbers for this reason as they encode cyclic properties in a natural way. The character is the trace of a complex representation and can be useful when classifying simple groups.

As far as I have understood we need a faithful representation of the cyclic group such that each matrix in the image has the property of a pseudoreflection. That's exactly what I did. Your example is fine as well.

All complex reflections can be written as an identity matrix with a single entry changed to an $n$-th root of unity like you suggested earlier as long as you select the right basis. So circulant matrices and other representations are excluded for dimensional reasons as you observed. The isotypic components are the building blocks for an irreducible representation. Vaguely, think of them as the primes in a prime decomposition and the weights as the powers. It's not that easy but it's those sort of vibes.

You are looking in a wrong place for reflection representations of finite cyclic groups. Start by constructing a 1-dimensional faithful representation and verify that it is a reflection representation.

@MoisheKohan$:$ I think we can do the following $:$ Given a generator $\sigma \in G$ of order $n,$ fix some primitive $n$-th roots of unity $\zeta.$ Then the association $\sigma^k \mapsto \zeta^k$ gives rise to such a one dimensional faithful representation of the cyclic group $G$ of order $n.$ Since $\zeta$ is a pseudoreflection we are done. Is it alright?

Of course. Now, extend this in higher dimensions.

@MoisheKohan$:$ For higher dimensions we could have taken $\sigma^k \mapsto \zeta^k I.$ But I think this argument fails to hold if $G$ is cyclic group of order $1$ i.e. if $G$ is a trivial group because in that case $\zeta = 1$ which does not correspond to a pseudoreflection.

Yes, that would be a wrong thing to use. Given a finite-dimensional linear representation $r: G\to Aut(V)$, what constructions do you know for extending $r$ to a linear representation $G\to Aut(W)$, where $W$ is a finite-dimensional vector space containing $V$?

@MoisheKohan$:$ Can't we extend the action block diagonally as $$g \mapsto \begin{pmatrix} r(g) & 0 \\ 0 & 1 \end{pmatrix}$$

Very good. Now apply this to construct reflection representations of finite cyclic groups.

@MoisheKohan$:$ I think identity matrix is assumed to be a pseudoreflection as it is an element of pseudoreflection groups in any dimension. I don't get your point this time. No matter how we try to construct the representation, identity element should go the identity element and hence the trivial group has only one irreducible representation i.e. the trivial representation and any other representation is the the direct sum of the trivial one i.e. image of the trivial group under any representation of dimension $n$ only contains the $n \times n$ identity matrix.