Group Theory – Why Are These Representations of $\mathbb Z_n$ Not Irreducible?

Question

The Cyclic Groups $(\mathbb Z_n,+)$ have various representations. This answer asserts that the only irreducible representations are

• 1-dimensional, with matrix a real $n$th root of unity;
• 2-dimensional, with matrix $\left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}\right)$ where $n\theta \equiv 0 \pmod{2\pi}$ but $\sin\theta \neq 0$.

But what about other minimal unitary matrix representations?

Take as an example $\mathbb Z_6$, which has other representations. Such as the 6 cyclic permutations of

$$\pmatrix{1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1}
$$

constructed by repeatedly moving the leftmost column to the righthand side of the matrix and shifting all of the other columns left by 1 space.

Or the representation of 6 5×5 matrices constructed from the direct sum of the $\mathbb Z_2$ and $\mathbb Z_3$ representations

$$\left\{\pmatrix{1&0\\0&1},\pmatrix{0&1\\1&0}\right\}
$$
and
$$\left\{\pmatrix{1&0&0\\0&1&0\\0&0&1},\pmatrix{0&0&1\\1&0&0\\0&1&0},\pmatrix{0&1&0\\0&0&1\\1&0&0}\right\}
$$
respectively.

Neither of these representations can be written as a direct sum of other representations of $\mathbb Z_6$. In other words, as far as I can tell, these representations have only trivial subrepresentations; therefore, why should they not be considered irreducible (as per the linked question)?

Answer 1

With $\mathbb{Z}_6$ acting on $\mathbb{R}^6$ by cyclic coordinate shifts, the irreducible subrepresentations are

• The 1D span of $(1,1,1,1,1,1)$
• The 1D span of $(1,-1,1,-1,1,-1)$
• The 2D space $\{(a,b,c,a,b,c):a+b+c=0\}$
• The 2D space $\{(a,-b,c,-a,b,-c):a+b+c=0\}$

With $\mathbb{Z}_6\cong\mathbb{Z}_2\times\mathbb{Z}_3$ acting on $\mathbb{R}^2\oplus\mathbb{R}^3$ by cyclic coordinate shifts in each component, they are

• The 1D span of $(1,1,0,0,0)$
• The 1D span of $(1,-1,0,0,0)$
• The 1D span of $(0,0,1,1,1)$
• The 2D space $\{(0,0,a,b,c):a+b+c=0\}$