The multiplicity of eigenvalues of nonnegative weakly irreducible tensors and uniform hypergraphs

Hu and Ye conjectured that for a $k$-th order and $n$-dimensional tensor $\mathcal{A}$ with an eigenvalue $\lambda$ and the corresponding eigenvariety $\mathcal{V}_\lambda(\mathcal{A})$, $$\mathrm{am}(\lambda) \ge \sum_{i=1}^\kappa \mathrm{dim}(V_i)(k-1)^{\mathrm{dim}(V_i)-1},$$ where $\mathrm{am}(\lambda)$ is the algebraic multiplicity of $\lambda$, and $V_1,\ldots,V_\kappa$ are all irreducible components of $\mathcal{V}_\lambda(\mathcal{A})$. In this paper, we prove that if $\mathcal{A}$ is a nonnegative weakly irreducible tensor with spectral radius $\rho$, then $\mathrm{am}(\lambda) \ge |\mathbb{V}_\lambda(\mathcal{A})|$ for all eigenvalues $\lambda$ of $\mathcal{A}$ with modulus $\rho$, where $\mathbb{V}_\lambda(\mathcal{A})$ is the projective eigenvariety of $\mathcal{A}$ associated with $\lambda$. Consequently we confirm Hu-Ye's conjecture for the above eigenvalues $\lambda$ of $\mathcal{A}$ and also the least H-eigenvalue of a weakly irreducible $Z$-tensor. We prove several equality cases in Hu-Ye's conjecture for the eigenvalues of the adjacency tensor or Laplacian tensor of uniform hypergraphs.