Spectral analysis and representations]{Spectral analysis near regular point of reducibility and representations of Coxeter groups

For a tuple of square matrices $A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n$ the determinantal hypersurface is defined as \begin{eqnarray*} &\sigma(A_1,…,A_n)= \ &\Big{[x_1:\cdots :x_n]\in \C{\mathbb P}^{n-1}: det(x_1A_1+\cdots +x_nA_n)=0\Big }. \end{eqnarray*} In this paper we develop a local spectral analysis near a regular point of reducibility of a determinantal hypersurface. We prove a rigidity type theorem for representations of Coxeter groups as an application