45 research outputs found
A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots
Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈J(ℚ) is positive semidefinite, where CJ∈J(ℤ) is the incidence matrix of J. Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number cJ. In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number čJ, and the defect homomorphism ∂J:ℤJ→ℤ. In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity. In case GJ is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets J with the Coxeter spectral properties of a simply laced Euclidean diagram DJ∈{̃n,̃6,̃7,̃8} associated with J. Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ, čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix A∈n(ℤ), there is B∈n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix
Structure of non-negative posets of Dynkin type
We study, in terms of directed graphs, partially ordered sets (posets)
that are non-negative in the sense that their
symmetric Gram matrix is positive semi-definite, where
is the incidence matrix of encoding the
relation . We give a complete, up to isomorphism, structural
description of connected posets of Dynkin type
in terms of their Hasse digraphs
that uniquely determine . One of the main results of the paper is the proof
that the matrix is of rank or , i.e., every non-negative poset
with is either positive or principal.
Moreover, we depict explicit shapes of Hasse digraphs of all
non-negative posets with . We show that
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
.Comment: 17 pages; Example 2.4 corrected, typos corrected, references adde
Algorithmic computation of principal posets using Maple and Python
We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets I that are principal, i.e., the rational symmetric Gram matrix GI : = 1/2[CI+CItr] ∈ MI(Q) of I is positive semi-definite of corank one, where CI ∈ MI(Z) is the incidence matrix of I. With any such a connected poset I, we associate a simply laced Euclidean diagram DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈}, the Coxeter matrix CoxI := −CI ⋅ C−trI, its complex Coxeter spectrum speccI, and a reduced Coxeter number cI. One of our aims is to show that the spectrum speccI of any such a poset I determines the incidence matrix CI (hence the poset I) uniquely, up to a Z-congruence. By computer calculations, we find a complete list of principal one-peak posets I (i.e., I has a unique maximal element) of cardinality ≤ 15, together with speccI, cI, the incidence defect ∂I : ZI → Z, and the Coxeter-Euclidean type DI. In case when DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈} and n := |I| is relatively small, we show that given such a principal poset I, the incidence matrix CI is Z-congruent with the non-symmetric Gram matrix GˇDI of DI, speccI = speccDI and cˇI = cˇDI. Moreover, given a pair of principal posets I and J, with |I| = |J| ≤ 15, the matrices CI and CJ are Z-congruent if and only if speccI = speccJ
Four Variations on Graded Posets
We explore the enumeration of some natural classes of graded posets,
including all graded posets, (2+2)- and (3+1)-avoiding graded posets,
(2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain
enumerative and structural theorems for all of them. Along the way, we discuss
a situation when we can switch between enumeration of labeled and unlabeled
objects with ease, generalize a result of Postnikov and Stanley from the theory
of hyperplane arrangements, answer a question posed by Stanley, and see an old
result of Klarner in a new light.Comment: 28 page