Structure of non-negative posets of Dynkin type \mathbbA_n

2022/05/30 by Gąsiorek, Marcin
#05C30 #05C50 #06A07 #06A11 #15A63 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #G.2.1 #G.2.2

paper · doi:10.48550/arxiv.2205.15032

Abstract

A poset I=(\1,…, n\, ≤_I) is called non-negative if the symmetric Gram matrix G_I:=(1)/(2)(C_I + C_I^tr)∈\mathbbM_n(ℝ) is positive semi-definite, where C_I∈\mathbbM_n(ℤ) is the (0,1)-matrix encoding the relation ≤_I. Every such a connected poset I, up to the ℤ-congruence of the G_I matrix, is determined by a unique simply-laced Dynkin diagram Dyn_I∈\\mathbbA_m, \mathbbD_m,𝔼₆,𝔼₇,𝔼₈\. We show that Dyn_I=\mathbbA_n implies that the matrix G_I is of rank n or n-1. Moreover, we depict explicit shapes of Hasse digraphs H(I) of all such posets~I and devise formulae for their number.

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