On algorithmic Coxeter spectral analysis of positive posets

摘要

Following a general framework of Coxeter spectral analysis of signed graphs Δ and finite posets I introduced by Simson (SIAM J. Discrete Math. 27:827–854, 2013) we present efficient numerical algorithms for the Coxeter spectral study of finite posets I=({1,…,n},⪯I) that are positive in the sense that the symmetric Gram matrix GI:=12(CI+CItr)∈Mn(Q) is positive definite, where CI∈Mn(Z) is the incidence matrix of I encoding the relation ⪯I. In the framework of scientific computing we present a complete Coxeter spectral classification of finite positive posets I of size n=|I|<20. It extends one of the main results obtained in Ga̧siorek et al. (Eur. J. Comb. 48:127–142, 2015) for posets of size n ≤ 10. We also show that the connectivity of such posets I is determined by the complex Coxeter spectrum speccI⊆C; equivalently, by the Coxeter polynomial coxI(t)∈Z[t] od I.One of the main results of the paper is a new technique to compute in a polynomial time a Z-invertible matrix BI∈Mn(Z) such that BItr·CI·BI=GˇDI, where GˇDI∈Mn(Z) is a non-symmetric Gram matrix of a simply-laced Dynkin diagram DI∈{An,Dn,E6,E7,E8} associated with a finite positive poset I.