Structure of non-negative posets of Dynkin type 𝔸_n

05/30/2022

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by   Marcin Gąsiorek, et al.

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We study, in terms of directed graphs, partially ordered sets (posets) I=({1,…, n}, ≼_I) that are non-negative in the sense that their symmetric Gram matrix G_I:=1/2(C_I + C_I^tr)∈𝕄_|I|(ℚ) is positive semi-definite, where C_I∈𝕄_n(ℤ) is the incidence matrix of I encoding the relation ≼_I. We give a complete, up to isomorphism, structural description of connected posets I of Dynkin type Dyn_I=𝔸_n in terms of their Hasse digraphs ℋ(I) that uniquely determine I. One of the main results of the paper is the proof that the matrix G_I is of rank n or n-1, i.e., every non-negative poset I with Dyn_I=𝔸_n is either positive or principal. Moreover, we depict explicit shapes of Hasse digraphs ℋ(I) of all non-negative posets I with Dyn_I=𝔸_n. We show that ℋ(I) is isomorphic to an oriented path or cycle with at least two sinks. By giving explicit formulae for the number of all possible orientations of the path and cycle graphs, up to the isomorphism of unlabeled digraphs, we devise formulae for the number of non-negative posets of Dynkin type 𝔸_n.