# Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$

I would like to know what all the possible finite-dimensional representations of the following relations are.

$$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$

$$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\dagger = 0 \tag{2}$$

$$A_0^\dagger A_1 + A_1^\dagger A_2 + A_2^\dagger A_0 = 0 \tag{3}$$

where $$I$$ is the identity matrix. In other words, what are the matrices (in any dimension) satisfying $$(1)$$, $$(2)$$, $$(3)$$?

A first step is to characterize what the $$3 \times 3$$ matrices satisfying it are. Is there any numerical way to tackle this problem?

Behind this should be the clock/shift algebra (Weyl theorem). Considering the clock and shift operators (of size 3, see https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices) $$X$$, $$Z$$ such that $$ZX=ωXZ$$ with $$ω=e^{2iπ/3}$$. Then $$A_k=ω^{k(k+1)}XZ^k$$, is a solution. A second solution is obtained with $$\omega$$ changed into its conjugate. I expect that all solutions are block diagonal matrices of those two examples.

• I guess the symbol $\dagger$ means conjugate transpose. – YCor May 2 at 10:15
• Yes, sorry I'm a physicist ;) – MarcO May 2 at 10:18
• Actually the dagger symbol is used in mathematics papers as pseudoinverse(Moore-Penrose) also – vidyarthi May 3 at 8:30
• I think the question makes sense: this means classifying finite/low-dimensional representation of the \$C^*-algebra defined by the given presentation. However some motivation/context is maybe missing. – YCor May 3 at 9:12
• I edited to give more context. – MarcO May 3 at 10:10