What is "cyclic shift unitary" on $M_{n}(\mathbb{C})$?

Question

From page 264 of Brown & Ozawa's $C^*$-algebras and finite-dimensional approximations:

Let $M_{n}(\mathbb{C})$ be the $n\times n$ complex matrices, and what is the "cyclic shift unitary of order $n$" on $M_{n}(\mathbb{C})$?

Maybe it is a very basic concept in functional analysis or matrix theory?

Answer 1

Cyclic shift operator is the matrix $$A=\begin{bmatrix} 0&1 \\ &0&1 \\ & & \ddots \\ &&&&1\\ 1&&&&0 \end{bmatrix}$$ It has order $n$, that is to say, $A^n=I$, and it is a unitary matrix.