# A Coxeter type classification of Dynkin type πΈ_n non-negative posets

We continue the Coxeter spectral analysis of finite connected posets I that are non-negative in the sense that their symmetric Gram matrix G_I:=1/2(C_I + C_I^tr)βπ_n(β) is positive semi-definite of rank n-ππ«π€_Iβ₯ 0, where C_Iβπ_n(β€) is the incidence matrix of I encoding the relation βΌ_I. We extend the results of Fundam. Inform., 139.4(2015), 347β367] and give a complete Coxeter spectral classification of finite connected posets I of Dynkin type πΈ_m. We show that such posets I, with |I|>1, yield exactly βn/2β Coxeter types, one of which describes the positive (i.e., ππ«π€_I=0) ones. We give an exact description and calculate the number of posets of every type. Moreover, we prove that, given a pair of such posets I and J, the incidence matrices C_I and C_J are β€-congruent if and only if π¬π©πππ_I = π¬π©πππ_J, and present deterministic algorithms that calculate a β€-invertible matrix defining such a β€-congruence in a polynomial time.

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