Determinantal Hypersurfaces And Representations Of Coxeter Groups

Let G be a finitely generated group. To every generating set T={g₁,…, g_n} of G and every complex finite dimensional representation ρ of G, we associate the determinantal hypersurface (Formula Presented) is not invertible, where I is the identity operator, and we investigate how the geometry of this hypersurface reflects the properties of ρ. In the classical case when G is a finite Coxeter group of regular type and T is a Coxeter generating set for G we show that D(T, ρ) determines ρ.