A Coxeter type classification of Dynkin type \mathbbA_n non-negative posets

2022/05/31 by M. Gąsiorek, Gąsiorek, M.
Mathematics · Chemistry · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.2205.15813

Abstract

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)∈\mathbbM_m(ℚ) is positive semi-definite of rank n≥ 0, where C_I∈\mathbbM_m(ℤ) is the incidence matrix of I encoding the relation \preceq_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 \mathbbA_n. We show that such posets I, with |I|>1, yield exactly \lfloor(m)/(2)\rfloor Coxeter types, one of which describes the positive (i.e., with n=m) 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.

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