Sonlu gruplar için iz formülü üzerine notlar

Let G be a finite group and consider a subgroup Γ of G along with a representation ρ: Γ → GL(V_ρ) of Γ on a finite dimensional C-linear vector space V_ρ. We get the trace formula by computing the trace of the operator Ind_Γ^G(ρ)(f): Ind_Γ^G(V_ρ) → Ind_Γ^G(V_ρ) for a test function f: G → C. Our aim is to present this computation in two different ways, that is, by expressing the spectral side J(ρ,f) and the geometric side I(ρ,f) of the formula. The expression J(ρ,f) is a sum over non-isomorphic irreducible representations of G having the multiplicities of the irreducible representations emerging in the decomposition of the induced representation Ind_Γ^G(ρ) within and the geometric side I(ρ,f) of the trace formula is obtained by calculating the conjugacy classes of the groups Γ and G. We call the generalized formula J(ρ,f) = Tr(Ind_Γ^G(ρ)(f)) = I(ρ,f) the trace formula for the finite group G with respect to the subgroup Γ and ρ: Γ → GL(V_ρ).