A Coxeter type classification of Dynkin type 𝔸_n non-negative posets

05/31/2022

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by   M. Gąsiorek, et al.

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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)∈𝕄_n(ℚ) is positive semi-definite of rank n-𝐜𝐫𝐤_I≥ 0, where C_I∈𝕄_n(ℤ) is the incidence matrix of I encoding the relation ≼_I. We extend the results of Fundam. Inform., 139.4(2015), 347–367] and give a complete Coxeter spectral classification of finite connected posets I of Dynkin type 𝔸_m. We show that such posets I, with |I|>1, yield exactly ⌊n/2⌋ Coxeter types, one of which describes the positive (i.e., 𝐜𝐫𝐤_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 ℤ-congruent if and only if 𝐬𝐩𝐞𝐜𝐜_I = 𝐬𝐩𝐞𝐜𝐜_J, and present deterministic algorithms that calculate a ℤ-invertible matrix defining such a ℤ-congruence in a polynomial time.