1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I guess the symbol $\dagger$ means conjugate transpose. $\endgroup$ – YCor May 2 at 10:15
  • $\begingroup$ Yes, sorry I'm a physicist ;) $\endgroup$ – MarcO May 2 at 10:18
  • $\begingroup$ Actually the dagger symbol is used in mathematics papers as pseudoinverse(Moore-Penrose) also $\endgroup$ – vidyarthi May 3 at 8:30
  • $\begingroup$ 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. $\endgroup$ – YCor May 3 at 9:12
  • $\begingroup$ I edited to give more context. $\endgroup$ – MarcO May 3 at 10:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.