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For a tuple of square matrices $A_1,...,A_n$ the determinantal hypersurfaceis 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 aregular point of reducibility of a determinantal hypersurface. We prove arigidity type theorem for representations of Coxeter groups as an application

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