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We discuss the existence of an orthogonal basis consisting of decomposable vectors for all symmetry classes of tensors associated with semi-dihedral groups $S{D}_{8n}$. In particular, a necessary and sufficient condition for the existence of such a basis associated with $S{D}_{8n}$ and degree two characters is given.

We consider the problem of reconstructing an $n\times n$ cell matrix $D\left(\overrightarrow{x}\right)$ constructed from a vector $\overrightarrow{x}=(x_1,x_2,\cdots ,x_n)$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectra of cell matrices $D\left(\overrightarrow{x}\right)$ and $D\left(\pi \left(\overrightarrow{x}\right)\right)$ are the same for every permutation $\pi \in S_n$.