Let $\mathcal{ A}$ be a weakly irreducible nonnegative tensor with spectral radius $\rho(\mathcal{ A})$. Let $\mathfrak{ D}$ (respectively, $\mathfrak{ D}^{ (0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $\mathcal{ A}$ corresponding to the eigenvalues with modulus $\rho(\mathcal{ A})$ (respectively, the eigenvalue $\rho(\mathcal{ A})$). It is shown that $\mathfrak{ D}$ is an abelian group containing $\mathfrak{ D}^{ (0)}$ as a subgroup, which acts transitively on the set $\{ e^{ \mathbf{ i} \frac{ 2 \pi j}{ \ell}}\mathcal{ A}:j =0,1, \ldots,\ell-1\}$, where $|\mathfrak{ D}/\mathfrak{ D}^{ (0)}|=\ell$ and $\mathfrak{ D}^{ (0)}$ is the stabilizer of $\mathcal{ A}$. The spectral symmetry of $\mathcal{ A}$ is characterized by the group $\mathfrak{ D}/\mathfrak{ D}^{ (0)}$, and $\mathcal{ A}$ is called spectral $\ell$-symmetric. We obtain the structural information of $\mathcal{ A}$ by analyzing the property of $\mathfrak{ D}$, especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover $\mathcal{ A}$ is symmetric, we prove that $\mathcal{ A}$ is spectral $\ell$-symmetric if and only if it is $(m,\ell)$-colorable. We characterize the spectral $\ell$-symmetry of a tensor by using its generalized traces, and show that for an arbitrarily given integer $m \ge 3$ and each positive integer $\ell$ with $\ell \mid m$, there always exists an $m$-uniform hypergraph $G$ such that $G$ is spectral $\ell$-symmetric.
Categories: math.CO