A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Abstract

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 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶^tr_𝐽]∈𝕄_𝐽(ℚ) is positive semidefinite, where 𝐶_𝐽∈𝕄_𝐽(ℤ) is the incidence matrix of 𝐽. Following the idea of Drozd mentioned earlier, we associate to 𝐽 its Coxeter matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶^−tr_𝐽, its Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽, a Coxeter polynomial cox_𝐽(𝑡)∈ℤ[𝑡], and a Coxeter number𝐜_𝐽. In case 𝐺_𝐽 is positive semi-definite, we also associate to 𝐽 a reduced Coxeter number 𝐜̌_𝐽, and the defect homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ. In this case, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of the unit circle and consists of roots of unity. In case 𝐺_𝐽 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∈{𝔻^∼_𝑛,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} associated with 𝐽. Our aim of the Coxeter spectral analysis of such posets 𝐽 is to answer the question when the Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜̌_𝐽) of 𝐽 determines its incidence matrix 𝐶_𝐽 (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∈𝕄_𝑛(ℤ), there is 𝐵∈𝕄_𝑛(ℤ) such that 𝐴^tr=𝐵^tr⋅𝐴⋅𝐵 and 𝐵²=𝐸 is the identity matrix.

1. Introduction

In the present paper, we continue our Coxeter spectral study of finite posets, started in [1], in a close connection with the Coxeter spectral technique introduced in [2–4] for acyclic edge-bipartite graphs or signed graphs in the sense of [5]. We also follow some of the techniques of representation theory, graph combinatorics, and the spectral graph theory; see [6–31].

Here, we use the terminology and notation introduced in [1, 4, 26–28]. We denote by ℕ⊆ℤ⊆ℚ the set of nonnegative integers, the ring of integers, and the rational number field. Given 𝑚≥1, we view ℤ^𝑚 as a free abelian group and denote by 𝑒₁,…,𝑒_𝑚 the standard ℤ-basis of ℤ^𝑚. Given an index set 𝐽, we denote by ℤ^𝐽 the abelian group of all vectors 𝑣=(𝑣_𝑗)_𝑗∈𝐽, with integer coordinates 𝑣_𝑗∈ℤ, by 𝕄_𝐽(ℤ) the ℤ-algebra of all square 𝐽 by 𝐽 integral matrices, and by 𝐸∈𝕄_𝐽(ℤ) the identity matrix. In particular, 𝕄_𝑚(ℤ), with 𝑚≥1, is the ℤ-algebra of all square 𝑚 by 𝑚 matrices. The group Gl(𝑚,ℤ)∶=𝐴∈𝕄_𝑚(ℤ),det𝐴∈{−1,1}⊆𝕄_𝑚(ℤ) (1) is called the (integral) general linear group. We say that two square rational matrices 𝐴,𝐴^′∈𝕄_𝑚(ℚ) are ℤ-equivalent, or ℤ-congruent, (and denote 𝐴∼_ℤ𝐴^′) if there is a matrix 𝐵∈Gl(𝑚,ℤ) such that 𝐴^′=𝐵^tr⋅𝐴⋅𝐵. By a poset 𝐽≡(𝐽,⪯) we mean a finite partially ordered set 𝐽 with respect to a partial order relation ⪯. Following [26], a poset 𝐽 is called a one-peak poset if 𝐽 has a unique maximal element ∗. A finite poset 𝐽 is uniquely determined by its incidence matrix  𝐶_𝐽∈𝕄_𝐽(ℤ), that is, the square 𝐽×𝐽 matrix, as follows:𝐶_𝐽=𝑐_{𝑖𝑗𝑖,𝑗∈𝐽}∈𝕄_𝐽(ℤ),with𝑐_𝑖𝑗=1,for𝑖⪯𝑗,0,for𝑖⪯^╱𝑗. (2) Following an idea of Drozd [32] (developed in [27]), we have introduced in [1, 28] the Tits matrix  𝐶_𝐽∈𝕄_𝐽(ℤ) of 𝐽 to be the integral matrix 𝐶_𝐽=𝑐̂_{𝑖𝑗𝑖,𝑗∈𝐽}∈𝕄_𝐽(ℤ),with𝑐̂_𝑖𝑗1,𝑖=𝑗or𝑗⪯𝑖,𝑖,𝑗∉max𝐽,0,𝑖,𝑗incomparable,or𝑖⪯𝑗and𝑖,𝑗∉max𝐽,−1,if𝑖≺𝑗and𝑗∈max𝐽, (3) where max𝐽 is the set of all maximal elements of 𝐽. Usually, we equip the elements of 𝐽 with a numbering; that is, 𝐽 is viewed as 𝐽={𝑎₁,…,𝑎_𝑚}, 𝑚=|𝐽|≥1. Throughout, we fix such a numbering and make the identifications 𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) and ℤ^𝑚≡ℤ^𝐽. The incidence matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) and the Tits matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) depend on the numbering of 𝑎₁,…,𝑎_𝑚. Namely, if 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} is obtained from 𝐽={𝑎₁,…,𝑎_𝑚} by a permutation 𝜎∈𝐒_𝑚 and 𝜎∈Gl(𝑚,ℤ) is the permutation matrix of 𝜎, then 𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎,𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎. (4) Note that any poset 𝐽 admits an upper-triangular numbering  𝐽={𝑎₁,…,𝑎_𝑚}; that is, 𝑎_𝑖⪯𝑎_𝑗 implies 𝑖≤𝑗. In this case, 𝐶_𝐽∈𝕄_𝑚(ℤ) is an upper-triangular matrix with 1 on the main diagonal, and, hence, det𝐶_𝐽=1, and det𝐶_𝐽′=1, for any numbering 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚}.

Fix a numbering 𝑎₁,…,𝑎_𝑚 of elements of 𝐽. Following [1, 28], by the Euler matrix of the poset 𝐽 we mean the inverse 𝐶_𝐽∶=𝐶_𝐽⁻¹∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (5) of 𝐶_𝐽. Following [3, 4], we call 𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr𝑃−2⋅𝐸,_𝐽(𝑡)=det𝑡⋅𝐸−𝐴𝑑_𝐽∈ℤ[𝑡] (6) the symmetric adjacency matrix and the characteristic polynomial of the poset 𝐽. The set 𝐬𝐩𝐞𝐜_𝐽 of all 𝑚=|𝐽| real roots of 𝑃_𝐽(𝑡) is defined to be the (real) spectrum of the poset 𝐽.

We denote by 𝑞_𝐽,𝑞_𝐽,𝑞_𝐽∶ℤ^𝐽≡ℤ^𝑚→ℤ the incidence quadratic form, the Tits quadratic form, and the Euler quadratic form of 𝐽 defined by the formulae 𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=_𝑗∈𝐽𝑥²_𝑗+_𝑖≺𝑗𝑥_𝑖𝑥_𝑗,𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=_𝑗∈𝐽𝑥²_𝑗+_{𝑖≺𝑗∈̌‌𝐽}𝑥_𝑖𝑥_𝑗−_{𝑝∈max𝐽𝑖≺𝑝}𝑥_𝑖𝑥_𝑝,𝑞_𝐽(𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr=𝑥⋅𝐶_𝐽⁻¹⋅𝑥^tr, (7) respectively, where 𝐽^̌‌=𝐽⧵max𝐽, max𝐽 is the set of all maximal elements in 𝐽, and 𝐶_𝐽∈𝕄_𝐽(ℤ) is the Tits matrix of 𝐽; see (27) and [1, 28] for a definition. The matrices 𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr,𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr,𝐺_𝐽1∶=2𝐶_𝐽+𝐶_𝐽^tr∈𝕄_𝐽(ℚ), (8) with rational coefficients, are called the symmetric incidence Gram matrix, the symmetric Tits-Gram matrix, and the symmetric Euler-Gram matrix of 𝐽. The matrices 𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr−2⋅𝐸,𝐴𝑑_𝐽∶=𝐶_𝐽+𝐶_𝐽^tr−2⋅𝐸=𝐶_𝐽⁻¹+𝐶_𝐽^−tr−2⋅𝐸, (9) with integer coefficients, are called the Tits adjacency matrix, and the Euler adjacency matrix of 𝐽. The polynomials 𝑃_𝐽(𝑡)∶=det𝑡⋅𝐸−𝐴𝑑_𝐽=det𝑡⋅𝐸−𝐴𝑑_𝐽,𝑃_𝐽(𝑡)∶=det𝑡⋅𝐸−𝐴𝑑_𝐽 (10) are called the characteristic polynomial of 𝐽 and the Euler-characteristic polynomial of 𝐽, respectively.

Example 1. (a) If 𝐼 is the poset (11) then 𝑃_𝐼(𝑡)=𝑃_𝐼(𝑡)=𝑡⁴−5𝑡²−4𝑡; that is, the characteristic polynomial 𝑃_𝐼(𝑡) of 𝐼 coincides with the Euler-characteristic polynomial 𝑃_𝐼(𝑡) of 𝐼.
(b) If 𝐽 is the poset (12) of the Dynkin type 𝔼₆, then the characteristic polynomial 𝑃_𝐽(𝑡) of 𝐽 does not coincide with the Euler-characteristic polynomial 𝑃_𝐽(𝑡) of 𝐽, because 𝑃_𝐽(𝑡)=𝑡⁶−13𝑡⁴−26𝑡³−15𝑡²+2𝑡+3,𝑃_𝐽(𝑡)=𝑡⁶−5𝑡⁴+5𝑡²−1. (13)
Following [17, 33], we introduce the following definition.

Definition 2. (a) We define a poset 𝐽 to be positive (resp., nonnegative) if the incidence form 𝑞_𝐽∶ℤ^𝐽→ℤ of 𝐽 is positive (resp., nonnegative); that is, 𝑞_𝐽(𝑣)>0, for any nonzero 𝑣∈ℤ^𝐽 (resp., 𝑞_𝐽(𝑣)≥0, for any 𝑣∈ℤ^𝐽).
(b) We define a poset 𝐽 to be principal if its incidence form 𝑞_𝐽∶ℤ^𝐽→ℤ is principal in the sense of [34, Definition 2.1]; that is, 𝑞_𝐽 is nonnegative, not positive, and the kernel Ker𝑞_𝐽∶=𝑣∈ℤ^𝐽;𝑞_𝐽(𝑣)=0 (14) is an infinite cyclic subgroup of ℤ^𝐽.
Following the main idea of the Coxeter spectral analysis of acyclic edge-bipartite graphs (signed graphs) presented in [3, 4], we study finite posets 𝐽 (with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}) by means of the Coxeter spectrum𝐬𝐩𝐞𝐜𝐜_𝐽⊆ℂ (15) of 𝐽, that is, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| eigenvalues of the Coxeter matrixCox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (16) of 𝐽, or equivalently, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| roots of the Coxeter polynomialcox_𝐽(𝑡)∶=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽∈ℤ[𝑡]; (17) see (31) and [1]. It follows from (4) that the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 of 𝐽 and the spectrum 𝐬𝐩𝐞𝐜_𝐽 of 𝐽 do not depend on the numbering of the elements of the poset 𝐽.

A motivation. We recall from [26, 27] that the problems we study in the paper have a bimodule matrix problem interpretation and have essential applications in reducing some classes of partitioned matrices with coefficients in a field 𝐾 to their canonical forms. For simplicity of its presentation, we illustrate it in case when 𝑞_𝐽(𝑥) is the Tits quadratic form (7) of the poset 𝐽={𝑎₁,…,𝑎_𝑛,∗,+}, with an upper-triangular partial order ⪯ such that 𝐽 has precisely two maximal elements ∗∶=∗_𝑛+1 and +∶=+_𝑛+2. In this case, we have 𝑞_𝐽(𝑥)=_{𝑎𝑖∈𝐽}𝑥²_𝑖+_{𝑎𝑖≺𝑎𝑗,𝑖,𝑗≤𝑛}𝑥_𝑖𝑥_𝑗−_{𝑎𝑖≺∗}𝑥_𝑖𝑥_∗−_𝑎𝑗≺+𝑥_𝑗𝑥₊. (18) Fix a vector 𝑣=(𝑣₁,…,𝑣_𝑛,𝑣_∗,𝑣₊)∈ℕ^𝑛+2≡ℕ^𝐽, and consider the 𝐾-vector space 𝐌𝐚𝐭^𝐽_𝑣 of all partitioned matrices of the form (compare with [27]) (19) with coefficients in 𝐾, where 𝐴_𝑖∗=0 if 𝑎_𝑖≺∗̸ and 𝐴_𝑗+=0 if 𝑎_𝑗≺+̸. Consider the group 𝐆^𝐽_𝑣 generated by the elementary transformations of the following three types:(a)all simultaneous transformations on rows inside each horizontal block;(b)all simultaneous transformations on columns inside each vertical block;(c)all simultaneous transformations on columns from the 𝑖th column block to 𝑗th column block, if the relation 𝑎_𝑖⪯𝑎_𝑗 holds in the poset 𝐽⧵{∗,+} (with natural zero-adjustments).

It follows from [27, Section 2] (see also [16, 26, 32]) that the problem of finding canonical forms of matrices in 𝐌𝐚𝐭^𝐽_𝑣, with respect to the elementary transformations from the set 𝐆^𝐽_𝑣, is controlled by the Tits quadratic form 𝑞_𝐽 in the following sense. For any 𝑣∈ℕ^𝐽, there is only a finite number 𝐆^𝐽_𝑣-canonical forms of matrices in 𝐌𝐚𝐭^𝐽_𝑣 if and only if the form 𝑞_𝐽 is weakly positive; that is, 𝑞_𝐽(𝑣) is positive, for all nonzero vectors 𝑣∈ℕ^𝐽. Moreover, there is one-to-one correspondence between the irreducible 𝐆^𝐽_𝑣-canonical forms in 𝐌𝐚𝐭^𝐽_𝑣 and the vectors 𝑣∈ℕ^𝐽 satisfying the equation 𝑞_𝐽(𝑣)=1. A solution of the problem is given in [27] and [1, Theorem 1.6]. A useful homological interpretation (in terms of the Euler characteristic) of the ℤ-bilinear Tits form 𝑏̂_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr (26) and ℤ-bilinear Euler form 𝑏_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr is given in [1, (1.3)]. The reader is referred to [6–8, 25] for a detailed study and a solution of other important matrix problems of high computational complexity that have many useful applications in representation theory; see [16, 26].

We show in Section 3 that the Coxeter spectral analysis of principal posets 𝐽 essentially uses the Coxeter spectra of the simply laced Euclidean diagrams presented in Figure 1.

Figure 1 

Simply laced Euclidean (extended Dynkin) diagrams.

The nonsymmetric Gram matrix  𝐺^̌‌_Δ of any graph Δ=(Δ₀,Δ₁)∈{𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} of Figure 1, with the set of vertices Δ₀={1,…,𝑛,𝑛+1} and the set of edges Δ₁, is defined to be the matrix 𝐺^̌‌_Δ=1𝑑^Δ₁₂…𝑑^Δ_1𝑛𝑑^Δ_1𝑛+101…𝑑^Δ_2𝑛𝑑^Δ_2𝑛+1⋮⋮⋱⋮⋮00…1𝑑^Δ_𝑛𝑛+100…01∈𝕄_𝑛+1(ℤ), (20) where 𝑑^Δ_𝑖𝑗=−|Δ₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗. We set 𝑑^Δ_𝑖𝑗=0, if Δ₁(𝑖,𝑗) is empty or 𝑗<𝑖.

The Coxeter polynomialcox_Δ(𝑡)∶=det(𝑡⋅𝐸+𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr) of any diagram Δ=(Δ₀,Δ₁)∈{𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} does not depend on the numbering of the vertices in Δ₀ and is presented in (48). If 𝑛≥1 and Δ=𝔸^∼_𝑛, the Coxeter polynomialcox_Δ(𝑡)∶=det(𝑡⋅𝐸+𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr) of Δ depends on the numbering of the vertices in Δ₀ and is one of the polynomials 𝐹_Δ⁽¹⁾(𝑡),𝐹_Δ⁽²⁾(𝑡),…,𝐹^(𝑚𝑛)_Δ(𝑡) presented in [4], where 𝐹_Δ^(𝑗)(𝑡)=𝑡^𝑛+1−𝑡^{𝑛−𝑗+1}−𝑡^𝑗=()+1𝑡−1²⋅𝔳_𝑗(𝑡)⋅𝔳_{𝑛−𝑗+1}(𝑡),𝑚_𝑛=𝑛2,if𝑛iseven,𝑛+12,if𝑛+1iseven, (21) for 𝑗=1,…,𝑚_𝑛, and 𝔳_𝑚(𝑡)=𝑡^𝑚−1+𝑡^𝑚−2+⋯+𝑡²+𝑡+1. In particular, if 𝑛+1 is even and 𝑗=𝑚_𝑛=(𝑛+1)/2, then 𝑡^{𝑛−𝑗+1}=𝑡^𝑗 and 𝐹^(𝑚𝑛)_Δ(𝑡)=𝐹_Δ^((𝑛+1)/2)(𝑡)=𝑡^𝑛+1−2𝑡^(𝑛+1)/2+1. (22)

Following [4, 21], we associate (in Section 2) to any principal poset 𝐽 a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑛,𝑛≥1,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} such that the incidence symmetric Gram matrix 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶_𝐽^tr]∈𝕄_𝐽(ℚ) is ℤ-congruent to the symmetric Gram matrix𝐺_𝐷𝐽1∶=2^̌‌𝐺_𝐷𝐽+𝐺^̌‌tr_𝐷𝐽∈𝕄_𝐷𝐽(ℚ)=𝕄_𝐽(ℚ) (23) of 𝐷𝐽; that is, there is a ℤ-invertible matrix 𝐵 such that 𝐺_𝐷𝐽=𝐵^tr⋅𝐺_𝐽⋅𝐵.

One of the aims of the Coxeter spectral analysis of nonnegative finite posets is to study the question when the Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽 (24) of a poset 𝐽 determines the matrix 𝐶_𝐽 (and, hence, the poset 𝐽) uniquely, up to a ℤ-congruency. Here, we set 𝐜^̌‌_𝐽=𝐜_𝐽, if 𝐽 is positive. In other words, we claim that, for any pair 𝐽, 𝐼 of nonnegative one-peak posets, 𝐂𝐭𝐲𝐩𝐞_𝐽=𝐂𝐭𝐲𝐩𝐞_𝐼 if and only if the incidence matrices 𝐶_𝐽 and 𝐶_𝐼 are ℤ-congruent. We also study the problem related with the results proved by Horn and Sergeichuk [35], if for any ℤ-invertible matrix 𝐴∈𝕄_𝑛(ℤ), there exists 𝐵∈𝕄_𝑛(ℤ) such that 𝐴^tr=𝐵^tr⋅𝐴⋅𝐵 and 𝐵²=𝐸 is the identity matrix; see [17, 18].

The main results of the present paper on nonnegative posets 𝐽 can be summarised as follows:

(1) canonical equivalences between the incidences, Tits, and Euler quadratic form (and corresponding Coxeter transformations and Coxeter spectra) of any poset 𝐽, established in Proposition 5;

(2) a characterization of principal posets given in Section 3. We show that a connected poset 𝐽 is principal if and only if there exists a simply laced Euclidean diagram 𝐷𝐽∈𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈ (25) such that the symmetric Gram matrix 𝐺_𝐽∶=(1/2)[𝐶_𝐽+𝐶_𝐽^tr]∈𝕄_𝐽(ℚ) of 𝐽 is ℤ-congruent to the symmetric Gram matrix 𝐺_𝐷𝐽∶=(1/2)[𝐺^̌‌_𝐷𝐽+𝐺^̌‌tr_𝐷𝐽]∈𝕄_𝐷𝐽(ℚ) of 𝐷𝐽. Moreover, we show in Section 3 that, given a connected principal poset 𝐽, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, 1∈𝐬𝐩𝐞𝐜𝐜_𝐽, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity;

(3) a Coxeter spectral classification result (Corollary 11) asserting that, given a pair 𝐼, 𝐽 of one-peak principal posets with at most 13 elements, the following conditions are equivalent:(3a)𝐷𝐼=𝐷𝐽,(3b)𝐬𝐩𝐞𝐜𝐜_𝐼=𝐬𝐩𝐞𝐜𝐜_𝐽,(3c)𝐜^̌‌_𝐼=𝐜^̌‌_𝐽 and |𝐼|=|𝐽|,(3d) the incidence matrix 𝐶_𝐽∈𝕄_𝐽(ℤ) is ℤ-congruent to the incidence matrix 𝐶_𝐼∈𝕄_𝐼(ℤ); that is, there is a ℤ-invertible matrix 𝐵 such that 𝐶_𝐼=𝐵^tr⋅𝐶_𝐽⋅𝐵.

In Section 3, we study principal posets by means of the defect and the reduced Coxeter number, and in Section 4, we present a framework for the study of nonnegative posets of corank 𝑟≥2 by means of their defect and the reduced Coxeter number. Examples are given in Sections 3–5.

The reader is referred to [1, 16, 17, 26] for a background of poset representation theory and elementary introduction to the poset matrix problems.

2. A Framework for the Coxeter Spectral Analysis of Finite Posets

The quadratic wanderings on finite posets 𝐽 studied in [1] are playing a key role in the representation theory of posets, algebras, and coalgebras, as well as in the algebraic combinatorics of posets; see [6, 9–14, 16, 24–26, 28, 31, 32, 36–39]. Except for the incidence wandering and the Euler wanderings defined by the incidence matrix 𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) (2), with det𝐶_𝐽=1 and a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, as well as the Euler matrix 𝐶_𝐽∶=𝐶_𝐽⁻¹, we study in [1, 26–28] the Tits wandering defined by the Tits matrix  𝐶_𝐽∈𝕄_𝑚(ℤ)≡𝕄_𝐽(ℤ) of 𝐽 (see [28, (3.6)]), that is, the Gram matrix of the Tits  ℤ-bilinear form  𝑏̂_𝐽∶ℤ^𝐽×ℤ^𝐽→ℤ given by 𝑏̂_𝐽𝑥,𝑦=_{𝑎𝑖∈𝐽}𝑥_𝑖𝑦_𝑖+_{𝑎𝑗≺𝑎𝑖∈̌‌𝐽}𝑥_𝑖𝑦_𝑗−_{𝑎𝑝∈max𝐽𝑎𝑖≺𝑎𝑝}𝑥_𝑖𝑦_𝑝=𝑥⋅𝐶_𝐽⋅𝑦^tr, (26) where max𝐽 is the set of all maximal elements in the poset 𝐽 and 𝐽^̌‌∶=𝐽⧵max𝐽. We call 𝑞_𝐽(𝑥)∶=𝑏̂_𝐽(𝑥,𝑥)=𝑥⋅𝐶_𝐽⋅𝑥^tr the Tits quadratic form of 𝐽.

A homological interpretation of the ℤ-bilinear forms 𝑏̂_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr and 𝑏_𝐽(𝑥,𝑦)=𝑥⋅𝐶_𝐽⋅𝑦^tr is given in [1, (1.3)]. For a geometric interpretation of the Tits form 𝑞_𝐼 of a one-peak poset 𝐼, the reader is referred to Drozd [32] and Simson [26].

Note that, given a one-peak poset 𝐼 of the form 𝐼={1,2,…,𝑛,∗=𝑛+1}, with a unique maximal element ∗=𝑛+1, we have 𝐶_𝐼=𝐶_𝑇^tr−𝑢01∈𝕄_𝑛+1(ℤ)1⋮1,with𝑢=, (27) where 𝐶_𝑇∈𝕄_𝑇(ℤ)=𝕄_𝑛(ℤ) is the incidence matrix of the poset 𝑇=𝐼⧵{∗}={1,2,…,𝑛}; see [26]. Note that 𝑞_𝐼(𝑥)=𝑥⋅𝐶_𝐼⋅𝑥^tr.

Now, we show that, in the Coxeter spectral study of finite posets 𝐽, we can use the Coxeter spectral technique introduced in [2, 4], for the edge-bipartite graphs (signed graphs [5]), and developed in [2, 34, 40] for the matrix morsifications of unit quadratic forms.

Following [3, 4, 24], by an edge-bipartite graph (bigraph, in short), we mean a pair Δ=(Δ₀,Δ₁), where Δ₀ is a finite nonempty set of vertices and Δ₁ is a finite set of edges equipped with a bipartition Δ₁=Δ⁻₁∪Δ⁺₁ such that the set Δ₁(𝑖,𝑗)=Δ⁻₁(𝑖,𝑗)∪Δ⁺₁(𝑖,𝑗) of edges connecting the vertices 𝑖 and 𝑗 does not contain edges lying in Δ⁻₁(𝑖,𝑗)∩Δ⁺₁(𝑖,𝑗), for each pair of vertices 𝑖,𝑗∈Δ₀, and either Δ₁(𝑖,𝑗)=Δ⁻₁(𝑖,𝑗) or Δ₁(𝑖,𝑗)=Δ⁺₁(𝑖,𝑗). Note that the edge-bipartite graphs can be viewed as signed multigraphs satisfying a separation property; see [4, 5].

We visualize Δ as a graph in a Euclidean space ℝ^𝑚, 𝑚≥2, with the vertices numbered by the integers 1,…,𝑛; usually, we simply write Δ₀={1,…,𝑛}. An edge in Δ⁻₁(𝑖,𝑗) is visualised as a continuous one •_𝑖---•_𝑗, and an edge in Δ⁺₁(𝑖,𝑗) is visualised as a dotted one •_𝑖---•_𝑗. A bigraph Δ is said to be loop-free if it has no loops.

We view any finite graph Δ=(Δ₀,Δ₁) as an edge-bipartite one by setting Δ⁻₁(𝑖,𝑗)=Δ₁(𝑖,𝑗) and Δ⁺₁(𝑖,𝑗)=∅, for each pair of vertices 𝑖,𝑗∈Δ₀.

To any loop-free edge-bipartite graph Δ=(Δ₀,Δ₁), with a fixed numbering Δ₀={𝑎₁,…,𝑎_𝑚} of its vertices, we associate the upper-triangular nonsymmetric Gram matrix  𝐺^̌‌_Δ=𝐸+[𝑑^Δ_𝑖𝑗]∈𝕄_𝑚(ℤ) of the form (20), with 𝑚∶=𝑛+1, where 𝑑^Δ_𝑖𝑗=−|Δ⁻₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗, 𝑑^Δ_𝑖𝑗=|Δ⁺₁(𝑖,𝑗)|, if there is an edge •_𝑖---•_𝑗 and 𝑖≤𝑗. We set 𝑑^Δ_𝑖𝑗=0, if Δ₁(𝑖,𝑗) is empty or 𝑗<𝑖. Since Δ is loop-free, we have 𝑑^Δ₁₁=⋯=𝑑^Δ_𝑚𝑚=0 and the main diagonal of 𝐺^̌‌_Δ consists of unities.

Following [4], we call Δ=(Δ₀,Δ₁)  positive (resp., nonnegative), if the symmetric Gram matrix 𝐺_Δ∶=(1/2)𝐺^̌‌_Δ+𝐺^̌‌_Δ^tr (28) of Δ is positive definite (resp., positive semidefinite).

Following [4], we associate to any loop-free edge-bipartite graph Δ, with |Δ₀|=𝑛≥2, the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_Δ⊆ℂ defined to be the spectrum of the Coxeter (-Gram) matrix Cox_Δ∶=−𝐺^̌‌_Δ⋅𝐺^̌‌_Δ^−tr∈𝕄_𝑛(ℤ), (29) the Coxeter polynomial cox_Δ(𝑡)∶=det𝑡⋅𝐸−Cox_Δ∈ℤ[𝑡], (30) the Coxeter transformation Φ_Δ∶ℤ^𝑛→ℤ^𝑛, given by 𝑥↦Φ_Δ(𝑥)∶=𝑥⋅Cox_Δ, the Coxeter number 𝐜_Δ (the order of Φ_Δ in the automorphism group of ℤ^𝑛, i.e., the minimal integer 𝑟≥1 such that Φ^𝑟_Δ=𝐸), the ℤ-bilinear Gram form 𝑏_Δ∶ℤ^𝑛×ℤ^𝑛→ℤ of Δ given by 𝑏_Δ(𝑥,𝑦)∶=𝑥⋅𝐺^̌‌_Δ⋅𝑦^tr, and the integral unit quadratic form 𝑞_Δ(𝑥)∶=𝑏_Δ(𝑥,𝑥)=𝑥²₁+⋯+𝑥²_𝑛+_𝑖<𝑗𝑑^Δ_𝑖𝑗𝑥_𝑖𝑥_𝑗=𝑥⋅𝐺_Δ⋅𝑥^tr. (31) Conversely, following Ovsienko [24], to any integral unit form 𝑞(𝑥)=𝑥²₁+⋯+𝑥²_𝑛+_𝑖<𝑗𝑞_𝑖𝑗𝑥_𝑖𝑥_𝑗,with𝑞_𝑖𝑗∈ℤ, (32) we associate the loop-free bigraph 𝐛𝐢𝐠𝐫(𝑞) of 𝑞 as follows (see also [34, 41]):(a)the vertices of 𝐛𝐢𝐠𝐫(𝑞) are the integers 1,…,𝑛,(b)two vertices 𝑖≠𝑗 are joined by −𝑞_𝑖𝑗 continuous edges of the form •_𝑖---•_𝑗 if 𝑞_𝑖𝑗 is negative, and by 𝑞_𝑖𝑗 dotted edges of the form •_𝑖---•_𝑗, if 𝑞_𝑖𝑗 is positive,(c)there is no edge between 𝑖 and 𝑗, if 𝑞_𝑖𝑗=0, or 𝑖=𝑗.

To any poset 𝐽≡(𝐽,⪯), with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚} of its points, we associate the following three edge-bipartite graphs: Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽,Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽,Δ_𝐽∶=𝐛𝐢𝐠𝐫𝑞_𝐽, (33) where 𝐛𝐢𝐠𝐫(𝑞_𝐽), 𝐛𝐢𝐠𝐫(𝑞_𝐽), and 𝐛𝐢𝐠𝐫(𝑞_𝐽) are the bigraphs of the quadratic forms 𝑞_𝐽, 𝑞_𝐽, and 𝑞_𝐽, respectively; see (7). More precisely, the bigraphs (33) are defined as follows.(i)The set of vertices of each of the bigraphs Δ_𝐽, Δ_𝐽, and Δ_𝐽 is the enumerated set 𝐽={𝑎₁,…,𝑎_𝑚}.(ii)There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖≺𝑎_𝑗 or 𝑎_𝑗≺𝑎_𝑖 holds in 𝐽.(iii)There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖 and 𝑎_𝑗 are not maximal in 𝐽 and 𝑎_𝑖≺𝑎_𝑗 or 𝑎_𝑗≺𝑎_𝑖 holds in 𝐽. There is an edge 𝑎_𝑖---𝑎_𝑗 in Δ_𝐽, if 𝑎_𝑖≺𝑎_𝑗 holds and 𝑎_𝑗 is maximal in 𝐽.(iv)Let 𝐶_𝐽=𝐶_𝐽⁻¹=[𝑐_𝑖𝑗]∈𝕄_𝑚(ℤ) be the Euler matrix of 𝐽. There is an edge 𝑎_𝑖---𝑎_𝑗 (resp., 𝑎_𝑖---𝑎_𝑗) in Δ_𝐽, if 𝑐_𝑖𝑗>0 or 𝑐_𝑗𝑖>0 (resp., 𝑐_𝑖𝑗<0 or 𝑐_𝑗𝑖<0).

We call Δ_𝐽, Δ_𝐽, and Δ_𝐽 the incidence bigraph of  Δ, the Tits bigraph of   Δ, and the Euler bigraph of  Δ, respectively, (with respect to the numbering 𝐽={𝑎₁,…,𝑎_𝑚}).

The following simple lemma is of importance.

Lemma 3. Assume that 𝐽 is a finite poset with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, and let Δ_𝐽, Δ_𝐽, Δ_𝐽 be the loop-free edge-bipartite graphs associated with 𝐽 in (33).(a)The symmetric Gram matrices 𝐺_𝐽, 𝐺_𝐽, 𝐺_𝐽 are ℤ-congruent to the symmetric Gram matrices 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽, respectively. The rank of each of the symmetric Gram matrices 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽 does not depend of the numbering 𝐽={𝑎₁,…,𝑎_𝑚} and coincides with the common rank rk𝐺_Δ𝐽Δ=rk𝐺_𝐽=rk𝐺_Δ𝐽.(b)𝑃_𝐽(𝑡)=𝑃_Δ𝐽Δ(𝑡)=𝑃_𝐽(𝑡).(c)The poset 𝐽 is positive (resp., nonnegative) if and only if the bigraph Δ_𝐽  (and Δ_𝐽, Δ_𝐽) is positive (resp., nonnegative).(d)The poset 𝐽 is principal if and only if the bigraph Δ_𝐽  (and Δ_𝐽, Δ_𝐽) is principal.

Proof. For the proof of (a), we recall that the Gram matrices 𝐺_𝐽, 𝐺_𝐽, 𝐺_𝐽, 𝐺_Δ𝐽, 𝐺Δ_𝐽, 𝐺_Δ𝐽 are invariant, up to ℤ-congruency, under permutations of the elements {𝑎₁,…,𝑎_𝑚}. Since 𝐽 admits an upper-triangular numbering 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} and 𝐺_Δ𝐽′=𝐺_𝐽′, then (a) follows. The proof of remaining statements is left to the reader.

Following the terminology used in [2–4, 34], we introduce the following definition.

Definition 4. Let 𝐽 be a finite poset, with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}.(a) We associate with 𝐽 the following three Coxeter matrices:(a1) the (incidence) Coxeter matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ);(a2) the Coxeter-Tits matrix Cox_𝐽∶=−𝐶_𝐽⋅𝐶_𝐽^−tr∈𝕄_𝑚(ℤ);(a3) the Coxeter-Euler matrix Cox_𝐽∶=−𝐶_𝐽⁻¹⋅𝐶_𝐽^tr∈𝕄_𝑚(ℤ). Moreover, we define the following three Coxeter transformations:(a4) the (incidence) Coxeter transformation Φ_𝑚∶ℤ^𝑚→ℤ^𝑚 of 𝐽;(a5) the Coxeter-Tits transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 of 𝐽;(a6) the Coxeter-Euler transformation Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 of 𝐽, by the following formulae: Φ_𝐽(𝑥)=𝑥⋅Cox_𝐽,Φ_𝐽(𝑥)=𝑥⋅Cox_𝐽,andΦ_𝐽(𝑥)=𝑥⋅Cox_𝐽. (34)(b) The integral polynomial cox_𝐽(𝑡)=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽=det𝑡⋅𝐸−Cox_𝐽∈ℤ[𝑡] (35) is called the Coxeter polynomial of the poset 𝐽.(c) The Coxeter spectrum of 𝐽 is the set 𝐬𝐩𝐞𝐜𝐜_𝐽⊆ℂ of all 𝑚=|𝐽| eigenvalues of the matrix Cox_𝐽, or, equivalently, the set 𝐬𝐩𝐞𝐜𝐜_𝐽 of all 𝑚=|𝐽| roots of the Coxeter polynomial cox_𝐽(𝑡).(d) The order 𝐜_𝐽∶=ord(Φ_𝐽) of the Coxeter transformation Φ_𝐽∶ℤ^𝐽→ℤ^𝐽 is called the Coxeter number of the poset 𝐽. In other words, 𝐜_𝐽 is the minimal integer 𝑟≥1 such that Φ^𝑟_𝐽=𝑖𝑑. We set 𝐜_𝐽=∞, if Φ^𝑟_𝐽≠𝑖𝑑, for any 𝑟≥1.(e) Assume that 𝐽 is nonnegative. The Coxeter type of 𝐽 is defined to be the pair 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽) if 𝐽 is positive, and the triple 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽) if 𝐽 is not positive, where 𝐜^̌‌_𝐽 is the reduced Coxeter number of 𝐽 in the sense of Theorems 10 and 18.
The following proposition shows that equality (35) holds.

Proposition 5. Let 𝐽 be a finite poset, with a fixed numbering 𝐽={𝑎₁,…,𝑎_𝑚}, let 𝑞_𝐽,𝑞_𝐽,𝑞_𝐽∶ℤ^𝑚→ℤ be the incidence, Tits, and Euler quadratic form of 𝐽, and let Φ_𝐽,Φ_𝐽,Φ_𝐽∶ℤ^𝑚→ℤ^𝑚 be the corresponding Coxeter transformations.(a) The following equalities hold 𝐶_𝐽=𝐵⋅𝐶_𝐽⋅𝐵^𝑡𝑟 and 𝐶_𝐽^𝑡𝑟=𝐵^′⋅𝐶_𝐽⋅𝐵^′𝑡𝑟, and the following diagrams are commutative (36) where 𝐵^′=𝐶_𝐽^𝑡𝑟, 𝐶𝐵=^̌‌𝐽00𝐸, 𝐽^̌‌=𝐽⧵max𝐽, and ℎ_𝐵,ℎ_𝐵′∶ℤ^𝑚→ℤ^𝑚 are the group isomorphisms defined by the formulae ℎ_𝐵(𝑥)=𝑥⋅𝐵 and ℎ_𝐵′(𝑥)=𝑥⋅𝐵^′, for 𝑥∈ℤ^𝑚.(b)Cox_𝐽=𝐵⋅Cox_𝐽⋅𝐵⁻¹, Cox_𝐽⁻¹=Cox_𝐽^op=𝐵^′⋅Cox_𝐽⋅𝐵^′−1, and Φ_𝐽^op=Φ_𝐽⁻¹.(c) The Coxeter number 𝐜_𝐽=ord(Φ_𝐽) of the poset 𝐽 coincides with the Coxeter number of 𝐽^op. Moreover, 𝐜_𝐽=ord(Φ_𝐽)=ord(Φ_𝐽) and cox_𝐽^op(𝑡)=cox_𝐽(𝑡).(d) Assume that 𝐽 is connected and nonnegative.(d1) If the numbering 𝐽={𝑎₁,…,𝑎_𝑚} is upper-triangular and Δ_𝐽 is the bigraph (33) associated to 𝐽, then Cox_Δ𝐽=Cox_𝐽 and cox_Δ𝐽(𝑡)=cox_𝐽(𝑡).(d2) The Coxeter type 𝐂𝐭𝐲𝐩𝐞_𝐽∶=(𝐬𝐩𝐞𝐜𝐜_𝐽,𝐜_𝐽,𝐜^̌‌_𝐽) of 𝐽 does not depend on the numbering 𝐽={𝑎₁,…,𝑎_𝑚}.(d3) The Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity.(d4) The poset 𝐽 is positive if and only if 1∉𝐬𝐩𝐞𝐜𝐜_𝐽.

Proof. The first equality 𝐶_𝐽=𝐵⋅𝐶_𝐽⋅𝐵^tr is obvious, and the second one 𝐶_𝐼^tr=𝐵^′⋅𝐶_𝐽⋅𝐵^'tr follows by a direct calculation. Hence, (b) follows and, consequently, the diagrams (36) are commutative; see [1, Proposition 3.13]. Hence, the statement (c) follows from the commutativity of the diagrams (36).(d1) We recall from Section 1 that, given two numberings 𝐽={𝑎₁,…,𝑎_𝑚} and 𝐽^′={𝑎^′₁,…,𝑎^′_𝑚} of elements in 𝐽, we have 𝐶_𝐽′=𝜎⁻¹⋅𝐶_𝐽⋅𝜎, where 𝜎∈Gl(𝑚,ℤ) is the permutation matrix of a permutation 𝜎∈𝐒_𝑚. Hence, (d1) easily follows.(d2) It is sufficient to note that the incidence matrix 𝐶_𝐽 is upper triangular. Hence, 𝐶_𝐽=𝐺^̌‌_Δ𝐽 and Cox_Δ𝐽=Cox_𝐽.
To prove (d3) and (d4), we recall from [2] and [3, Proposition 2.6] that the Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐴 of any matrix morsification 𝐴∈𝐌𝐨𝐫_Δ of a nonnegative bigraph Δ is a subset of the unit circle 𝒮¹ and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐴 is a root of unity (see also [41, 42]). Moreover, Δ is positive iff 1∉𝐬𝐩𝐞𝐜𝐜_𝐴. Assume that 𝐽 is connected and nonnegative. Then, the bigraph Δ_𝐽 (33) associated to 𝐽 is nonnegative, 𝐴∶=𝐺^̌‌_Δ𝐽=∇(𝐶_𝐽) is a morsification of Δ_𝐽, and 𝐬𝐩𝐞𝐜𝐜_𝐽=𝐬𝐩𝐞𝐜𝐜_𝐴, because the incidence matrix 𝐶_𝐽 is quasitriangular and [4, Proposition 2.2] applies. This completes the proof.

Corollary 6. For any poset 𝐽, equality (35) holds.

Proof. Apply Proposition 5(b).

The following example shows that the correspondence 𝐽↦Δ_𝐽 defined in (33) does not preserve the Coxeter types of 𝐽 and Δ_𝐽. In particular, it shows that the equality cox_𝐽(𝑡)=cox_Δ𝐽(𝑡) does not hold in general and the Coxeter polynomial cox_Δ𝐽(𝑡) depends on the numbering of 𝐽, whereas the Coxeter polynomial cox_𝐽(𝑡) does not depend on the numbering of 𝐽.

Example 7. Consider the poset 𝐽 such that its Hasse quiver has the form (37) By a permutation of the elements in 𝐽, we get (38) Note that the first numbering is upper-triangular, whereas the second one is not upper-triangular.

3. Principal Posets

We recall that a poset 𝐽 is principal if its incidence unit form 𝑞_𝐽 is principal in the sense of [34, Definition 2.1]; that is, 𝑞_𝐽∶ℤ^𝐽→ℤ is nonnegative and not positive, and the kernel Ker𝑞_𝐽∶={𝑣∈ℤ^𝐽;𝑞_𝐽(𝑣)=0} is an infinite cyclic subgroup of ℤ^𝐽.

We start with the following useful observation.

Lemma 8. Assume that 𝐽 is a connected principal poset.(a) The Coxeter number 𝐜_𝐽 of 𝐽 is infinite.(b) The Coxeter spectrum 𝐬𝐩𝐞𝐜𝐜_𝐽 is a subset of a unit circle 𝒮¹={𝑧∈ℂ;|𝑧|=1}, 1∈𝐬𝐩𝐞𝐜𝐜_𝐽, and any 𝑧∈𝐬𝐩𝐞𝐜𝐜_𝐽 is a root of unity.(c) If Ker𝑞_𝐽=ℤ⋅𝐡, then Ker𝑞_𝐽=ℤ⋅𝐡 and Ker𝑞_𝐽=ℤ⋅𝐡, where(i)𝐡=𝐡⋅𝐵^′, 𝐡=𝐡⋅𝐵, 𝐡=𝐡⋅𝐵^′⋅𝐵⁻¹,(ii)𝐵^′=𝐶_𝐽^tr, 𝐶𝐵=^̌‌𝐽00𝐸, and 𝐽^̌‌=𝐽⧵max𝐽 are as in Proposition 5.

Proof. (a) By Proposition 5(d2), 𝐜_𝐽 is independent of the numbering of 𝐽. Then, without loss of generality, we may suppose that the numbering of 𝐽 is upper-triangular. Then, by Lemma 3(d) and Proposition 5(d1), the Coxeter number 𝐜_𝐽 coincides with the Coxeter number of the principal edge-bipartite graph Δ_𝐽 associated with 𝐽 in (33). Then, (a) is a consequence of [3, Proposition 3.12].
The statements (b) and (c) follow by applying Proposition 5 and the commutative diagram (36).

Proposition 9. Let 𝐽 be a connected poset, 𝑚=|𝐽|≥2, and let 𝐺_𝐽,𝐺_𝐽,𝐺_𝐽,∈𝕄_𝐽(ℚ) be the symmetric incidence Gram matrix of 𝐽, the symmetric Tits-Gram matrix of 𝐽, and the symmetric Euler-Gram matrix of 𝐽, respectively. The following five conditions are equivalent.(a)The poset 𝐽  is principal.(b)The Gram matrix 𝐺_𝐽 is positive indefinite of rank 𝑚−1.(c)The Tits quadratic form 𝑞_𝐽 of 𝐽 is nonnegative and Ker𝑞_𝐽=ℤ⋅𝐡, for some nonzero vector 𝐡∈ℤ^𝐽. (d)The Euler quadratic form 𝑞_𝐽 of𝐽 is nonnegative and  Ker𝑞_𝐽=ℤ⋅𝐡, for some nonzero vector 𝐡∈ℤ^𝐽.(e)If 𝐺 is any of the symmetric Gram matrices 𝐺_𝐽,𝐺_𝐽,𝐺_𝐽,∈𝕄_𝐽(ℚ) of 𝐽, then there exists a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈}  (uniquely determined by 𝐽) such that the matrix 𝐺 is ℤ-congruent to the symmetric Gram matrix 𝐺_𝐷𝐽∈𝕄_𝐷𝐽(ℚ) of the Euclidean diagram 𝐷𝐽; that is, there is a ℤ-invertible matrix 𝐵∈Gl(𝑚,ℤ) such that𝐺_𝐷𝐽=𝐵^tr⋅𝐺⋅𝐵.

Proof. (a)⇔(b) If 𝑚=|𝐽| and 𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚,𝑣↦𝐷𝑞_𝐽(𝑣)=𝜕𝑞_𝐽𝜕𝑥₁(𝑣),…,𝜕𝑞_𝐽𝜕𝑥_𝑚(𝑣), (39) is the gradient group homomorphism of 𝑞_𝐽, then Ker𝑞_𝐽=Ker[𝐷𝑞_𝐽∶ℤ^𝑚→ℤ^𝑚] and the subgroup Ker𝑞_𝐽 of ℤ^𝑚 is of rank 𝑚−rk𝐺_𝐽 and consists of all integral solutions of the system 2⋅𝐺_𝐽⋅𝑥^tr=0 of linear equations with integral coefficients; see [34, Proposition 2.8]. Then, (a)⇔(b) follows.
The equivalences (a)⇔(c)⇔(d) follow from Proposition 5 (a) and the commutativity of the diagram (36).
(e)⇒(a) Assume that there exist a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} and a ℤ-invertible matrix 𝐵∈Gl(𝑚,ℤ) such that 𝐺_𝐷𝐽=𝐵^tr⋅𝐺⋅𝐵. It follows that the quadratic form 𝑞_𝐷𝐽(𝑥)=𝑥⋅𝐺_𝐷𝐽⋅𝑥^tr is ℤ-congruent to 𝑞_𝐽 and 𝑞_𝐽=𝑞_𝐷𝐽∘ℎ_𝐵. Then, (a) is a consequence of [36, Lemma VII.4.2].
(a)⇒(e) Let Δ_𝐽 be the Euler edge-bipartite graph defined in (33) of 𝐽. By (a) and Lemma 3 (d), Δ_𝐽 is principal and the inflation algorithm defined in [4, 21] applies to Δ_𝐽. Consequently, there exists a simply laced Euclidean diagram 𝐷𝐽∈{𝔸^∼_𝑠,𝑠≥3,𝔻^∼_𝑛,𝑛≥4,𝔼^∼₆,𝔼^∼₇,𝔼^∼₈} and a ℤ-invertible matrix 𝐵∈Gl(𝑛,ℤ) defining the congruence Δ_𝐽≈_ℤ𝐷𝐽; that is, the equality 𝐺_𝐷𝐽=𝐵^tr⋅𝐺_Δ𝐽⋅𝐵=𝐵^tr⋅𝐺_𝐽⋅𝐵 holds. Then, in view of Proposition 5, the implication (a)⇒(e) follows from Lemma 3 (d); see also Section 6.

Theorem 10. Let 𝐽 be a finite principal poset, with a numbering {𝑎₁,…,𝑎_𝑚} of elements of 𝐽. Fix a nonzero vector 𝐡_𝐽∈ℤ^𝐽≡ℤ^𝑚 such that Ker𝑞_𝐽=ℤ⋅𝐡_𝐽.(a) There exist a minimal integer 𝐜^̌‌_𝐽≥2  (called the reduced Coxeter number of 𝐽) and a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ(called the incidence defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)⋅𝐡_𝐽,𝜕_𝐽Φ_𝐽(𝑣)=𝜕_𝐽(𝑣),∀𝑣∈ℤ^𝐽,𝜕_𝐽𝐡_𝐽=0. (40)(b) Assume that 𝐜^̌‌_𝐽≥1 and 𝜕_𝐽∶ℤ^𝐽→ℤ are as in (a), and one sets 𝐡_𝐽=𝐡_𝐽⋅𝐵^′, 𝐡_𝐽=𝐡_𝐽⋅𝐵^′⋅𝐵⁻¹, where 𝐵^′,𝐵∈𝕄_𝐽(ℤ) are as in Proposition 5.(b1) There exists a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ(called the Euler defect of 𝐽) such that Φ^{̌‌𝐜𝐽}_𝐽(𝑣)=𝑣+𝜕_𝐽(𝑣)⋅𝐡_𝐽,∀𝑣∈ℤ^𝐽,𝜕_𝐽∘Φ_𝐽=𝜕_𝐽,𝜕_𝐽∘ℎ_𝐵′=𝜕_𝐽,𝜕_𝐽𝐡_𝐽=0. (41) (b2) There exists a group homomorphism 𝜕_𝐽∶ℤ^𝐽→ℤ