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

Authors: Daniel Simson; Katarzyna Zając
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## 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 posetsJ≡(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]∈𝕄J(ℚ) is positive semidefinite, where CJ∈𝕄J(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 numbercJ. 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∈{𝔻̃n,𝔼̃6,𝔼̃7,𝔼̃8} 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∈𝕄n(ℤ), there is B∈𝕄n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix.


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## Body

## 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 ℤm as a free abelian group and denote by e1,…,em the standard ℤ-basis of ℤm. Given an index set J, we denote by ℤJ the abelian group of all vectors v=(vj)j∈J, with integer coordinates vj∈ℤ, by 𝕄J(ℤ) the ℤ-algebra of all square J by J integral matrices, and by E∈𝕄J(ℤ) the identity matrix. In particular, 𝕄m(ℤ), with m≥1, is the ℤ-algebra of all square m by m matrices. The group
(1)Gl(m,ℤ):={A∈𝕄m(ℤ),detA∈{-1,1}}⊆𝕄m(ℤ)
is called the (integral)  general linear group. We say that two square rational matrices A,A′∈𝕄m(ℚ) are ℤ-equivalent, or ℤ-congruent, (and denote A~ℤA′) if there is a matrix B∈Gl(m,ℤ) such that A′=Btr·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  CJ∈𝕄J(ℤ), that is, the square J×J matrix, as follows:(2)CJ=[cij]i,j∈J∈𝕄J(ℤ),withcij={1,fori⪯j,0,fori⋠j.
Following an idea of Drozd [32] (developed in [27]), we have introduced in [1, 28] the  Tits matrix  C^J∈𝕄J(ℤ) of J to be the integral matrix
(3)C^J=[c^ij]i,j∈J∈𝕄J(ℤ),withc^ij{1,i=jorj⪯i,i,j∉maxJ,0,i,jincomparable,ori⪯jandi,j∉maxJ,-1,ifi≺jandj∈maxJ,
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,…,am}, m=|J|≥1. Throughout, we fix such a numbering and make the identifications 𝕄m(ℤ)≡𝕄J(ℤ) and ℤm≡ℤJ. The incidence matrix CJ∈𝕄m(ℤ)≡𝕄J(ℤ) and the Tits matrix C^J∈𝕄m(ℤ)≡𝕄J(ℤ) depend on the numbering of a1,…,am. Namely, if J′={a1′,…,am′} is obtained from J={a1,…,am} by a permutation σ∈Sm and σ^∈Gl(m,ℤ) is the permutation matrix of σ, then
(4)CJ′=σ^-1·CJ·σ^,C^J′=σ^-1·C^J·σ^.
Note that any poset J admits an  upper-triangular numbering  J={a1,…,am}; that is, ai⪯aj implies i≤j. In this case, CJ∈𝕄m(ℤ) is an upper-triangular matrix with 1 on the main diagonal, and, hence, detCJ=1, and detCJ′=1, for any numbering J′={a1′,…,am′}.Fix a numberinga1,…,am of elements of J. Following [1, 28], by the  Euler matrix of the poset J we mean the inverse
(5)C¯J:=CJ-1∈𝕄m(ℤ)≡𝕄J(ℤ)
of CJ. Following [3, 4], we call
(6)AdJ≔CJ+CJtr-2·E,PJ(t)=det(t·E-AdJ)∈ℤ[t]
the symmetric adjacency matrix and the  characteristic polynomial of the poset J. The set specJ of all m=|J| real roots of PJ(t) is defined to be the (real)  spectrum of the poset J.We denote byqJ,q^J,q¯J:ℤJ≡ℤm→ℤ the   incidence quadratic form, the  Tits quadratic form, and the  Euler  quadratic form of J defined by the formulae
(7)qJ(x)=x·CJ·xtr=∑j∈J‍xj2+∑i≺j‍xixj,q^J(x)=x·C^J·xtr=∑j∈J‍xj2+∑i≺j∈Jˇ‍xixj-∑p∈maxJ‍∑i≺p‍xixp,q¯J(x)=x·C¯J·xtr=x·CJ-1·xtr,
respectively, where Jˇ=J∖maxJ, maxJ is the set of all maximal elements in J, and C^J∈𝕄J(ℤ) is the Tits matrix of J; see (27) and [1, 28] for a definition. The matrices
(8)GJ:=12[CJ+CJtr],G^J:=12[C^J+C^Jtr],G¯J:=12[C¯J+C¯Jtr]∈𝕄J(ℚ),
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
(9)A^dJ:=C^J+C^Jtr-2·E,A¯dJ:=C¯J+C¯Jtr-2·E=CJ-1+CJ-tr-2·E,
with integer coefficients, are called the  Tits adjacency matrix, and the  Euler adjacency matrix of J. The polynomials
(10)PJ(t):=det(t·E-AdJ)=det(t·E-A^dJ),P¯J(t):=det(t·E-A¯dJ)
are called the  characteristic polynomial of J and the  Euler-characteristic polynomial of J, respectively.Example 1.
(a)  IfI is the poset
(11)
then PI(t)=P¯I(t)=t4-5t2-4t; that is, the characteristic polynomial PI(t) of I coincides with the Euler-characteristic polynomial P¯I(t) of I.
(b)  IfJ is the poset
(12)
of the Dynkin type 𝔼6, then the characteristic polynomial PJ(t) of J does not coincide with the Euler-characteristic polynomial P¯J(t) of J, because
(13)PJ(t)=t6-13t4-26t3-15t2+2t+3,P¯J(t)=t6-5t4+5t2-1.
Following [17, 33], we introduce the following definition.Definition 2.
(a)  We define a posetJ to be  positive (resp.,  nonnegative) if the incidence form qJ:ℤJ→ℤ of J is positive (resp., nonnegative); that is, qJ(v)>0, for any nonzero v∈ℤJ (resp., qJ(v)≥0, for any v∈ℤJ).
(b)  We define a posetJ to be  principal if its incidence form qJ:ℤJ→ℤ is principal in the sense of [34, Definition 2.1]; that is, qJ is nonnegative, not positive, and the kernel
(14)KerqJ:={v∈ℤJ;qJ(v)=0}
is an infinite cyclic subgroup of ℤJ.
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,…,am}) by means of the  Coxeter spectrum(15)speccJ⊆ℂ
of J, that is, the set speccJ of all m=|J| eigenvalues of the  Coxeter matrix(16)CoxJ:=-CJ·CJ-tr∈𝕄m(ℤ)≡𝕄J(ℤ)
of J, or equivalently, the set speccJ of all m=|J| roots of the  Coxeter polynomial(17)coxJ(t):=det(t·E-CoxJ)=det(t·E-C^oxJ)=det(t·E-C¯oxJ)∈ℤ[t];
see (31) and [1]. It follows from (4) that the Coxeter spectrum speccJ of J and the spectrum specJ 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,…,an,*,+}, with an upper-triangular partial order ⪯ such that J has precisely two maximal elements *:=*n+1 and +:=+n+2. In this case, we have
(18)q^J(x)=∑ai∈J‍xi2+∑ai≺aj,i,j≤n‍xixj-∑ai≺*‍xix*-∑aj≺+‍xjx+.
Fix a vector v=(v1,…,vn,v*,v+)∈ℕn+2≡ℕJ, and consider the K-vector space MatvJ of all partitioned matrices of the form (compare with [27])
(19)
with coefficients in K, where Ai*=0 if ai⊀* and Aj+=0 if aj⊀+. Consider the group GvJ 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 theith column block to jth column block, if the relation ai⪯aj 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 MatvJ, with respect to the elementary transformations from the set GvJ, is controlled by the Tits quadratic form q^J in the following sense. For any v∈ℕJ, there is only a finite number GvJ-canonical forms of matrices in MatvJ if and only if the form q^J is weakly positive; that is, q^J(v) is positive, for all nonzero vectors v∈ℕJ. Moreover, there is one-to-one correspondence between the irreducible GvJ-canonical forms in MatvJ and the vectors v∈ℕJ 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 b^J(x,y)=x·C^J·ytr (26) and ℤ-bilinear Euler form b¯J(x,y)=x·C¯J·ytr 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 Section3 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 nonsymmetricGram matrix  GˇΔ 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
(20)GˇΔ=[1d12Δ…d1nΔd1n+1Δ01…d2nΔd2n+1Δ⋮⋮⋱⋮⋮00…1dnn+1Δ00…01]∈𝕄n+1(ℤ),
where dijΔ=-|Δ1(i,j)|, if there is an edge •i—•j and i≤j. We set dijΔ=0, if Δ1(i,j) is empty or j<i.The Coxeter polynomialcoxΔ(t):=det(t·E+GˇΔ·GˇΔ-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 polynomialcoxΔ(t):=det(t·E+GˇΔ·GˇΔ-tr) of Δ depends on the numbering of the vertices in Δ0 and is one of the polynomials FΔ(1)(t),FΔ(2)(t),…,FΔ(mn)(t) presented in [4], where
(21)FΔ(j)(t)=tn+1-tn-j+1-tj+1=(t-1)2·𝔳j(t)·𝔳n-j+1(t),mn={n2,ifniseven,n+12,ifn+1iseven,
for j=1,…,mn, and 𝔳m(t)=tm-1+tm-2+⋯+t2+t+1. In particular, if n+1 is even and j=mn=(n+1)/2, then tn-j+1=tj and
(22)FΔ(mn)(t)=FΔ((n+1)/2)(t)=tn+1-2t(n+1)/2+1.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 GJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ) is ℤ-congruent to the  symmetric Gram matrix(23)GDJ:=12[GˇDJ+GˇDJtr]∈𝕄DJ(ℚ)=𝕄J(ℚ)
of DJ; that is, there is a ℤ-invertible matrix B such that GDJ=Btr·GJ·B.One of the aims of the Coxeter spectral analysis of nonnegative finite posets is to study the question when the Coxeter type(24)CtypeJ:=(speccJ,cJ,cˇJ)
of a poset J determines the matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. Here, we set cˇJ=cJ, if J is positive. In other words, we claim that, for any pair J, I of nonnegative one-peak posets, CtypeJ=CtypeI if and only if the incidence matrices CJ and CI 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∈𝕄n(ℤ), there exists B∈𝕄n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix; see [17, 18].The main results of the present paper on nonnegative posetsJ 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
(25)DJ∈{𝔻~n,n≥4,𝔼~6,𝔼~7,𝔼~8}
such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ) of J is ℤ-congruent to the symmetric Gram matrix GDJ:=(1/2)[GˇDJ+GˇDJtr]∈𝕄DJ(ℚ) of DJ. Moreover, we show in Section 3 that, given a connected principal poset J, the Coxeter spectrum speccJ is a subset of a unit circle 𝒮1={z∈ℂ;|z|=1},  1∈speccJ, and any z∈speccJ 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)
speccI=speccJ,(3c)
cˇI=cˇJ and |I|=|J|,(3d)
the incidence matrixCJ∈𝕄J(ℤ) is ℤ-congruent to the incidence matrix CI∈𝕄I(ℤ); that is, there is a ℤ-invertible matrix B such that CI=Btr·CJ·B.In Section3, 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 posetsJ 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 CJ∈𝕄m(ℤ)≡𝕄J(ℤ) (2), with detCJ=1 and a fixed numbering J={a1,…,am}, as well as the Euler matrix C¯J:=CJ-1, we study in [1, 26–28] the Tits wandering defined by the  Tits matrix  C^J∈𝕄m(ℤ)≡𝕄J(ℤ) of J (see [28, (3.6)]), that is, the Gram matrix of the  Tits  ℤ-bilinear form  b^J:ℤJ×ℤJ→ℤ given by
(26)b^J(x,y)=∑ai∈Jxiyi+∑aj≺ai∈Jˇxiyj-∑ap∈maxJ∑ai≺apxiyp=x·C^J·ytr,
where maxJ is the set of all maximal elements in the poset J and Jˇ:=J∖maxJ. We call q^J(x):=b^J(x,x)=x·C^J·xtr the  Tits quadratic form of J.A homological interpretation of theℤ-bilinear forms b^J(x,y)=x·C^J·ytr and b¯J(x,y)=x·C¯J·ytr 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 posetI of the form I={1,2,…,n,*=n+1}, with a unique maximal element *=n+1, we have
(27)C^I=[CTtr-u01]∈𝕄n+1(ℤ),withu=[1⋮1],
where CT∈𝕄T(ℤ)=𝕄n(ℤ) is the incidence matrix of the poset T=I∖{*}={1,2,…,n}; see [26]. Note that q^I(x)=x·C^I·xtr.Now, we show that, in the Coxeter spectral study of finite posetsJ, 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=Δ1-∪Δ1+ such that the set Δ1(i,j)=Δ1-(i,j)∪Δ1+(i,j) of edges connecting the vertices i and j does not contain edges lying in Δ1-(i,j)∩Δ1+(i,j), for each pair of vertices i,j∈Δ0, and either Δ1(i,j)=Δ1-(i,j) or Δ1(i,j)=Δ1+(i,j). Note that the edge-bipartite graphs can be viewed as signed multigraphs satisfying a separation property; see [4, 5].We visualizeΔ as a graph in a Euclidean space ℝm, m≥2, with the vertices numbered by the integers 1,…,n; usually, we simply write Δ0={1,…,n}. An edge in Δ1-(i,j) is visualised as a continuous one •i—•j, and an edge in Δ1+(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 graphΔ=(Δ0,Δ1) as an edge-bipartite one by setting Δ1-(i,j)=Δ1(i,j) and Δ1+(i,j)=∅, for each pair of vertices i,j∈Δ0.To any loop-free edge-bipartite graphΔ=(Δ0,Δ1), with a fixed numbering Δ0={a1,…,am} of its vertices, we associate the upper-triangular nonsymmetric  Gram matrix  GˇΔ=E+[dijΔ]∈𝕄m(ℤ) of the form (20), with m:=n+1, where dijΔ=-|Δ1-(i,j)|, if there is an edge •i—•j and i≤j, dijΔ=|Δ1+(i,j)|, if there is an edge •i---•j and i≤j. We set dijΔ=0, if Δ1(i,j) is empty or j<i. Since Δ is loop-free, we have d11Δ=⋯=dmmΔ=0 and the main diagonal of GˇΔ consists of unities.Following [4], we call Δ=(Δ0,Δ1)  positive (resp.,  nonnegative), if the symmetric Gram matrix
(28)GΔ:=(1/2)(GˇΔ+GˇΔtr)
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 spectrumspeccΔ⊆ℂ defined to be the spectrum of the Coxeter (-Gram) matrix
(29)CoxΔ:=-GˇΔ·GˇΔ-tr∈𝕄n(ℤ),
the Coxeter polynomial
(30)coxΔ(t):=det(t·E-CoxΔ)∈ℤ[t],
the Coxeter transformation ΦΔ:ℤn→ℤn, given by x↦ΦΔ(x):=x·CoxΔ, the Coxeter number cΔ (the order of ΦΔ in the automorphism group of ℤn, i.e., the minimal integer r≥1 such that ΦΔr=E), the ℤ-bilinear Gram form bΔ:ℤn×ℤn→ℤ of Δ given by bΔ(x,y):=x·GˇΔ·ytr, and the integral unit quadratic form
(31)qΔ(x):=bΔ(x,x)=x12+⋯+xn2+∑i<j‍dijΔxixj=x·GΔ·xtr.
Conversely, following Ovsienko [24], to any integral unit form
(32)q(x)=x12+⋯+xn2+∑i<j‍qijxixj,withqij∈ℤ,
we associate the loop-free bigraph bigr(q) of q as follows (see also [34, 41]):(a)
the vertices ofbigr(q) are the integers 1,…,n,(b)
two verticesi≠j are joined by -qij continuous edges of the form •i—•j if qij is negative, and by qij dotted edges of the form •i---•j, if qij is positive,(c)
there is no edge betweeni and j, if qij=0, or i=j.To any posetJ≡(J,⪯), with a fixed numbering J={a1,…,am} of its points, we associate the following three edge-bipartite graphs:
(33)ΔJ:=bigr(qJ),Δ^J:=bigr(q^J),Δ¯J:=bigr(q¯J),
where bigr(qJ), bigr(q^J), and bigr(q¯J) are the bigraphs of the quadratic forms qJ, 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 bigraphsΔJ, Δ^J, and Δ¯J is the enumerated set J={a1,…,am}.(ii)
There is an edgeai---aj in ΔJ, if ai≺aj or aj≺ai holds in J.(iii)
There is an edgeai---aj in Δ^J, if ai and aj are not maximal in J and ai≺aj or aj≺ai holds in J. There is an edge ai—aj in Δ^J, if ai≺aj holds and aj is maximal in J.(iv)
LetC¯J=CJ-1=[c¯ij]∈𝕄m(ℤ) be the Euler matrix of J. There is an edge ai---aj (resp., ai—aj) in Δ¯J, if c¯ij>0 or c¯ji>0 (resp., c¯ij<0 or c¯ji<0).We callΔJ, 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,…,am}).The following simple lemma is of importance.Lemma 3.
Assume thatJ is a finite poset with a fixed numbering J={a1,…,am}, and let ΔJ, Δ^J, Δ¯J be the loop-free edge-bipartite graphs associated with J in  (33).(a)
The symmetric Gram matricesGJ, G^J, G¯J are ℤ-congruent to the symmetric Gram matrices GΔJ, GΔ^J, GΔ¯J, respectively. The rank of each of the symmetric Gram matrices GΔJ, GΔ^J, GΔ¯J does not depend of the numbering J={a1,…,am} and coincides with the common rank rkGΔJ=rkGΔ^J=rkGΔ¯J.(b)
PJ(t)=PΔJ(t)=PΔ^J(t).(c)
The posetJ is positive (resp., nonnegative) if and only if the bigraph ΔJ  (and Δ^J, Δ¯J) is positive (resp., nonnegative).(d)
The posetJ is principal if and only if the bigraph ΔJ  (and Δ^J, Δ¯J) is principal.Proof.
For the proof of (a), we recall that the Gram matricesGJ, G^J, G¯J, GΔJ, GΔ^J, GΔ¯J are invariant, up to ℤ-congruency, under permutations of the elements {a1,…,am}. Since J admits an upper-triangular numbering J′={a1′,…,am′} and GΔJ′=GJ′, 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.
LetJ be a finite poset, with a fixed numbering J={a1,…,am}.(a)
We associate withJ the following three Coxeter matrices:
(a1)
the (incidence) Coxeter matrixCoxJ:=-CJ·CJ-tr∈𝕄m(ℤ);(a2)
the Coxeter-Tits matrixC^oxJ:=-C^J·C^J-tr∈𝕄m(ℤ);(a3)
the Coxeter-Euler matrixC¯oxJ:=-CJ-1·CJtr∈𝕄m(ℤ).Moreover, we define the following threeCoxeter transformations:
(a4)
the (incidence) Coxeter transformationΦm:ℤm→ℤm of J;(a5)
the Coxeter-Tits transformationΦ^J:ℤm→ℤm of J;(a6)
the Coxeter-Euler transformationΦ¯J:ℤm→ℤm of J, by the following formulae:
(34)ΦJ(x)=x·CoxJ,Φ^J(x)=x·C^oxJ,andΦ¯J(x)=x·C¯oxJ.(b)
The integral polynomial(35)coxJ(t)=det(t·E-CoxJ)=det(t·E-C^oxJ)=det(t·E-C¯oxJ)∈ℤ[t]is called theCoxeter polynomial of the posetJ.(c)
TheCoxeter spectrum of J is the set speccJ⊆ℂ of all m=|J| eigenvalues of the matrix CoxJ, or, equivalently, the set speccJ of all m=|J| roots of the Coxeter polynomial coxJ(t).(d)
The ordercJ:=ord(ΦJ) of the Coxeter transformation ΦJ:ℤJ→ℤJ is called the  Coxeter number of the poset J. In other words, cJ is the minimal integer r≥1 such that ΦJr=id. We set cJ=∞, if ΦJr≠id, for any r≥1.(e)
Assume thatJ is nonnegative. The  Coxeter type of J is defined to be the pair CtypeJ:=(speccJ,cJ) if J is positive, and the triple CtypeJ:=(speccJ,cJ,cˇJ) if J is not positive, where cˇJ 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.
LetJ be a finite poset, with a fixed numbering J={a1,…,am}, let qJ,q^J,q¯J:ℤm→ℤ be the incidence, Tits, and Euler quadratic form of J, and let ΦJ,Φ^J,Φ¯J:ℤm→ℤm be the corresponding Coxeter transformations.(a)
The following equalities holdC^J=B·C¯J·Btr and CJtr=B′·C¯J·B′tr, and the following diagrams are commutative
(36)whereB′=CJtr, B=[CJˇ00E], Jˇ=J∖maxJ, and hB,hB′:ℤm→ℤm are the group isomorphisms defined by the formulae hB(x)=x·B and hB′(x)=x·B′, for x∈ℤm.(b)
C^oxJ=B·C¯oxJ·B-1, CoxJ-1=CoxJop=B′·C¯oxJ·B′-1, and ΦJop=ΦJ-1.(c)
The Coxeter numbercJ=ord(ΦJ) of the poset J coincides with the Coxeter number of Jop. Moreover, cJ=ord(Φ^J)=ord(Φ¯J) and coxJop(t)=coxJ(t).(d)
Assume thatJ is connected and nonnegative.
(d1)
If the numberingJ={a1,…,am} is upper-triangular and ΔJ is the bigraph (33) associated to J, then CoxΔJ=CoxJ and coxΔJ(t)=coxJ(t).(d2)
The Coxeter typeCtypeJ:=(speccJ,cJ,cˇJ) of J does not depend on the numbering J={a1,…,am}.(d3)
The Coxeter spectrumspeccJ is a subset of a unit circle 𝒮1={z∈ℂ;|z|=1}, and any z∈speccJ is a root of unity.(d4)
The posetJ is positive if and only if 1∉speccJ.Proof.
The first equalityC^J=B·C¯J·Btr is obvious, and the second one CItr=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 Section1 that, given two numberings J={a1,…,am} and J′={a1′,…,am′} of elements in J, we have CJ′=σ^-1·CJ·σ^, where σ^∈Gl(m,ℤ) is the permutation matrix of a permutation σ∈Sm. Hence, (d1) easily follows.(d2)
It is sufficient to note that the incidence matrixCJ is upper triangular. Hence, CJ=GˇΔJ and CoxΔJ=CoxJ.
To prove (d3) and (d4), we recall from [2] and [3, Proposition 2.6] that the Coxeter spectrum speccA of any matrix morsification A∈MorΔ of a nonnegative bigraph Δ is a subset of the unit circle 𝒮1 and any z∈speccA is a root of unity (see also [41, 42]). Moreover, Δ is positive iff 1∉speccA. Assume that J is connected and nonnegative. Then, the bigraph ΔJ (33) associated to J is nonnegative, A:=GˇΔJ=∇(CJ) is a morsification of ΔJ, and speccJ=speccA, because the incidence matrix CJ is quasitriangular and [4, Proposition 2.2] applies. This completes the proof.Corollary 6.
For any posetJ, equality  (35) holds.Proof.
Apply Proposition5(b).The following example shows that the correspondenceJ↦ΔJ defined in (33) does not preserve the Coxeter types of J and ΔJ. In particular, it shows that the equality coxJ(t)=coxΔJ(t) does not hold in general and the Coxeter polynomial coxΔJ(t) depends on the numbering of J, whereas the Coxeter polynomial coxJ(t) does not depend on the numbering of J.Example 7.
Consider the posetJ 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 posetJ is  principal if its incidence unit form qJ is principal in the sense of [34, Definition 2.1]; that is, qJ:ℤJ→ℤ is nonnegative and not positive, and the kernel KerqJ:={v∈ℤJ;qJ(v)=0} is an infinite cyclic subgroup of ℤJ.We start with the following useful observation.Lemma 8.
Assume thatJ is a connected principal poset.(a)
The Coxeter numbercJ of J is infinite.(b)
The Coxeter spectrumspeccJ is a subset of a unit circle 𝒮1={z∈ℂ;|z|=1},  1∈speccJ, and any z∈speccJ is a root of unity.(c)
IfKerqJ=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′=CJtr,  B=[CJˇ00E],  and  Jˇ=J∖maxJare as in Proposition5.Proof.
(a) By Proposition5(d2), cJ 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 cJ coincides with the Coxeter number of the principal edge-bipartite graph ΔJ associated with J in (33). Then, (a) is a consequence of [3, Proposition 3.12].
The statements (b) and (c) follow by applying Proposition5 and the commutative diagram (36).Proposition 9.
LetJ be a connected poset, m=|J|≥2, and let GJ,G^J,G¯J,∈𝕄J(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 posetJis principal.(b)
The Gram matrixGJis positive indefinite of rankm-1.(c)
The Tits quadratic formq^J of Jis nonnegative andKerq^J=ℤ·h^, for some nonzero vectorh^∈ℤJ.(d)
The Euler quadratic formq¯JofJis nonnegative andKerq¯J=ℤ·h¯, for some nonzero vectorh¯∈ℤJ.(e)
IfGis any of the symmetric Gram matricesGJ,G^J,G¯J,∈𝕄J(ℚ)ofJ, then there exists a simply laced Euclidean diagramDJ∈{𝔸~s,s≥3,𝔻~n,n≥4,𝔼~6,𝔼~7,𝔼~8}(uniquely determined byJ) such that the matrixGisℤ-congruent to the symmetric Gram matrixGDJ∈𝕄DJ(ℚ)of the Euclidean diagramDJ; that is, there is aℤ-invertible matrixB∈Gl(m,ℤ)such thatGDJ=Btr·G·B.Proof.
(a)⇔(b) If m=|J| and
(39)DqJ:ℤm→ℤm,v↦DqJ(v)=(∂qJ∂x1(v),…,∂qJ∂xm(v)),
is the gradient group homomorphism of qJ, then KerqJ=Ker[DqJ:ℤm→ℤm] and the subgroup KerqJ of ℤm is of rank m-rkGJ and consists of all integral solutions of the system 2·GJ·xtr=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 GDJ=Btr·G·B. It follows that the quadratic form qDJ(x)=x·GDJ·xtr is ℤ-congruent to qJ and qJ=qDJ∘hB. 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 ΔJ. 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 GDJ=Btr·GΔ¯J·B=Btr·GJ·B holds. Then, in view of Proposition 5, the implication (a)⇒(e) follows from Lemma 3 (d); see also Section 6.Theorem 10.
LetJ be a finite principal poset, with a numbering {a1,…,am} of elements of J. Fix a nonzero vector hJ∈ℤJ≡ℤm such that KerqJ=ℤ·hJ.(a)
There exist a minimal integercˇJ≥2  (called the  reduced Coxeter number of J) and a group homomorphism ∂J:ℤJ→ℤ(called the  incidence defect  of J) such that
(40)ΦJcˇJ(v)=v+∂J(v)·hJ,∂J(ΦJ(v))=∂J(v),∀v∈ℤJ,∂J(hJ)=0.(b)
Assume thatcˇJ≥1 and ∂J:ℤJ→ℤ are as in  (a), and one sets h¯J=hJ·B′, h^J=hJ·B′·B-1, where B′,B∈𝕄J(ℤ) are as in Proposition 5.
(b1)
There exists a group homomorphism∂¯J:ℤJ→ℤ(called the  Euler defect  of J) such that
(41)Φ¯JcˇJ(v)=v+∂¯J(v)·h¯J,∀v∈ℤJ,∂¯J∘ΦJ=∂¯J,∂¯J∘hB′=∂J,∂¯J(h¯J)=0.(b2)
There exists a group homomorphism∂^J:ℤJ→ℤ  (called the  Tits defect of J) such that
(42)Φ^JcˇJ(v)=v+∂^J(v)·h^J,∀v∈ℤJ,∂^J=∂¯J∘hB=∂J∘hB′-1∘hB,∂^J=∂^J∘ΦJ,∂^J(h^J)=0.(c)
The Coxeter numbercJ of J is infinite, and the incidence defect ∂J:ℤJ→ℤ is nonzero.(d)
Givenv∈ℤJ, the order sv:=|𝒪(v)| of the ΦJ-orbit 𝒪(v) is finite if and only if ∂J(v)=0. If sv=|𝒪(v)| is finite, then sv divides cˇJ and there is a unique integer mv such that
(43)mv·h=v+ΦJ(v)+ΦJ2(v)+⋯+ΦJsv-1(v)=v+ΦJ-1(v)+ΦJ-2(v)+⋯+ΦJ-sv+1(v).Proof.
We recall from the proof of Proposition9 that
(44)ℤ·hJ=KerqJ=Ker[DqJ:ℤm→ℤm],
where m=|J|≥2 and DqJ:ℤm→ℤm,v↦DqJ(v)=((∂qJ/∂x1)(v),…,(∂qJ/∂xm)(v)), is the gradient group homomorphism. It follows that ℤm/ℤ·hJ≅ImDqJ≅ℤm-1. Denote by ϕ:ℤm→ℤm-1 the composite quotient epimorphism. Then, the form qJ induces the form q~J:ℤm-1→ℤ such that q~J(ϕ(x))=qJ(x), for all x∈ℤm. Moreover, the Coxeter transformation ΦJ:ℤm→ℤm induces a group automorphism Φ~J:ℤm-1→ℤm-1 such that
(45)Φ~J∘ϕ=ϕ∘ΦJ,q~J(Φ~J(y))=q~J(y),∀y∈ℤm-1.
It follows that q~J is positive definite and there exists a minimal integer cˇJ≥1 such that Φ~JcˇJ is the identity map on ℤm-1. Hence, (a) follows, because the equalities ∂J(hJ)=0 and ∂J(ΦJ(v))=∂J(v), for all v∈ℤJ, are almost obvious; see [34, Theorem 4.7].
In view of Proposition5, the statements (b)–(d) are a consequence of (a) and Lemma 8(a). The reader is referred to [34, Theorem 4.7, Corollary 4.15] for more details.Corollary 11.
(a) IfJ is a principal connected poset with at most 13 elements, then its Coxeter spectrum speccJ is a subset of a unit circle 𝒮1={z∈ℂ;|z|=1},  1∈speccJ, and any z∈speccJ is an mth root of unity, where m≤cˇJ and cˇJ is the  reduced Coxeter number of J.
(b)  IfI and J are one-peak principal posets with at most 13 elements and DI, DJ are the associated Euclidean diagrams, then the following conditions are equivalent:(b1)
DI=DJ,(b2)
speccI=speccJ,(b3)
cˇI=cˇJ and |I|=|J|,(b4)
the incidence matrixCJ∈𝕄J(ℤ) is ℤ-congruent to the incidence matrix CI∈𝕄I(ℤ); that is, there is a ℤ-invertible matrix B such that CI=Btr·CJ·B.Proof.
(a) By Lemma8, speccJ⊆𝒮1 and 1∈speccJ. Assume that DJ is the associated Euclidean diagram of J, as in Proposition 9. By a computer search (using the results of [43] and the inflation algorithm given in [4, 21]), we show that
(46)coxJ(t)=coxDJ(t),cˇJ=cˇDJ,
for any poset J, with at most 13 elements. Hence, in view of [4, Proposition 2.17], we have
(47)coxJ(t)=coxDJ(t)=FDJ(t),
where
(48)FDJ(t)=tn+1+tn-tn-1-tn-2-t3-t2+t+1=(t-1)2𝔳22(t)𝔳n-2(t),forDJ=𝔻~n,t7+t6-2t4-2t3+t+1=(t-1)2𝔳2(t)𝔳32(t),forDJ=𝔼~6,t8+t7-t5-2t4-t3+t+1=(t-1)2𝔳2(t)𝔳3(t)𝔳4(t),forDJ=𝔼~7,t9+t8-t6-t5-t4-t3+t+1=(t-1)2𝔳2(t)𝔳3(t)𝔳5(t),forDJ=𝔼~8,
where 𝔳m(t)=tm-1+tm-2+tm-3+⋯+t2+t+1. For DJ∈{𝔻~4,𝔻~5}, we have
(49)FDJ(t)={t5+t4-2t3-2t2+t+1,forDJ=𝔻~4,t6+t5-t4-2t3-t2+t+1,forDJ=𝔻~5.
Then, (a) follows by applying [38, Lemma XIII.1.3]. Hence, we also easily conclude that the statements (b1)–(b3) are equivalent.
To finish the proof of (b), we note that the equalityCI=Btr·CJ·B in (b4) implies that the matrices CoxI and CoxJ are conjugate, and, hence, we get speccI=speccJ; that is, the implication (b4)⇒(b2) holds. To prove the inverse implication (b2)⇒(b4), we apply the technique used in [18, Section 6]. On this way, given a principal poset J, with at most 13 elements and the associated Euclidean diagram DJ, we construct (by a computer search) a ℤ-invertible matrix BJ such that GˇDJ=BJtr·CJ·BJ (compare with [17, 18, 33, 43]). Hence, (b4) follows, and the proof is complete.IfJ is a principal poset, then the sets
(50)ℛqJ={v∈ℤm;qJ(v)=1},ℛq^J={v∈ℤm;q^J(v)=1},ℛq¯J={v∈ℤm;q¯J(v)=1}
of roots of the unit forms qJ, q^J, and q¯J have the disjoint union decompositions
(51)ℛqJ=∂J-ℛqJ∪∂J+ℛqJ∪∂J0ℛqJ,ℛq^J=∂^J-ℛq^J∪∂^J+ℛq^J∪∂^J0ℛq^J,ℛq¯J=∂¯J-ℛq¯J∪∂¯J+ℛq¯J∪∂¯J0ℛq¯J,
where
(52)∂J-ℛqJ={v∈ℛqJ;∂J(v)<0},∂J+ℛqJ={v∈ℛqJ;∂J(v)>0},∂J0ℛqJ={v∈ℛqJ;∂J(v)=0};∂^J-ℛq^J={v∈ℛq^J;∂^J(v)<0},∂^J+ℛq^J={v∈ℛq^J;∂^J(v)>0},∂^J0ℛq^J={v∈ℛq^J;∂^J(v)=0};∂¯J-ℛq¯J={v∈ℛq¯J;∂¯J(v)<0},∂¯J+ℛq¯J={v∈ℛq¯J;∂¯J(v)>0},∂¯J0ℛq¯J={v∈ℛq¯J;∂¯J(v)=0}.Note that the group isomorphismℤJ→ℤJ, v↦v^:=-v, restricts to the bijections
(53)∂J-ℛqJ→1-1≃∂J+ℛqJ,∂^J-ℛq^J→1-1≃∂^J+ℛq^J,∂¯J-ℛq¯J→1-1≃∂¯J+ℛq¯J.Example 12.
We compute the reduced Coxeter number, the Coxeter polynomial, and the Euler defect of the following principal two-peak poset(54)
Note that J is principal, because
(55)q¯J(x)=x12+x22+x32+x42+x52+x62+x72-(x1+x2)x3-(x1+x5)x4-x1x6+(x1-x2-x5-x6)x7=(x1-12x4-12x5-12x6+12x7)2+(x2-12x3-12x7)2+512(x3-25x5-45x6+15x7)2+34(-13x3+x4-23x5-13x6+13x7)2+35(x5-12x6-12x7)2+14(x6-x7)2.
It follows thatq¯J is nonnegative and Kerq¯J=ℤ·h, where h=(1,1,1,1,1,1,1); q¯J is critical in the sense of Ovsienko [24]; see also [38, 44]. Note that the Euler matrix C¯J=CJ-1 of J and the inverse of the Coxeter-Euler matrix C¯oxJ:=-CJ-1·CJtr have the forms
(56)C¯J=CJ-1=[10-1-10-1101-1000-100100000001-100000010-1000001-10000001],C¯oxJ-1=[-10-1-1-1-1-10-1-1000-11111112101001000010001011100-110-1001].
Moreover, we haveG𝔼~6=Btr·G¯J·B, and the matrix A:=Btr·C¯J·B is a morsification of the Euclidean diagram 𝔼~6 (see [34, 40]), where
(57)B=[-20100000000001-1000010-10001000001000-11000001000000],A=[11-21110-21000002-110-1-10-1001000-100-1100-100001000000-11].
Hence, in view of [2, Proposition 2.8], we get the following:(i)
the Euclidean typeDJ of J is the diagram 𝔼~6, and the Coxeter polynomial of the poset J has the form
(58)coxJ(t)=t7-t5-t2+1=coxA(t);that is,coxJ(t) is the Coxeter polynomial FΔ(2)(t) (21), of the Euclidean diagram Δ=𝔸~6 (with a particular numbering of vertices), and is the Coxeter polynomial coxA(t) of the morsification A∈𝕄7(ℤ) of the diagram 𝔼~6,(ii)
the Coxeter numbercJ is infinite and the reduced Coxeter number cˇJ equals 10,(iii)
the Euler defect has the form∂¯J(x)=3(x1+x2-x3-x7),(iv)
theΦ¯J-orbit of any vector of defect zero in ∂¯J0ℛq^J is of length 2 or of length 5. It is shown in [1, Remark 4.5] and [34, Example 5.14] that they lie on one sand-glass tube 𝒯1,2 of rank 2 and on six sand-glass tubes of rank five.


## 4. Nonnegative Posets of Positive Corank

In the study of nonnegative posets, the following extensions of  [34, Definition 2.2] are of importance.Definition 13.
Assume thatm≥2, r≥0, and q:ℤm→ℤ is a unit quadratic form.(a)
The formq is defined to be  nonnegative of corank  r≥0, if q is nonnegative and the ℚ-rank rkℚGq of the rational Gram matrix Gq∈𝕄m(ℚ) equals rkℚGq=m-r.(b)
The formq is defined to be  nonnegative critical of corank  r≥1, if q is nonnegative of corank r≥1 and each of the nonnegative quadratic forms q(1),…,q(m):ℤm-1→ℤ is of corank at most r-1≥0, where
(59)q(j)(x1,…,xj-1,xj+1,…,xm)=q(x1,…,xj-1,0,xj+1,…,xm).Lemma 14.
Assume thatm≥2, r≥0, and q:ℤm→ℤ is an integral quadratic form.(a)
q is nonnegative of corank r≥0 if and only if q is nonnegative and the subgroup Kerq of the abelian group ℤm is free of rank r.(b)
q is nonnegative of corank r=0 if and only if q is positive, and q is nonnegative of corank one if and only if q is principal.(c)
q is nonnegative critical of corank r≥1 if and only if q is nonnegative and, for any j∈{1,…,m}, the abelian subgroup ℤm,j∩Kerq of ℤm is free of rank at most r-1, where
(60)ℤm,j:=ℤm-j-1×{0}×ℤj-1={v=(v1,…,vm)∈ℤm;vj=0}⊆ℤmis viewed as a subgroup ofℤm.(d)
q is nonnegative critical of corank r=1 if and only if q is P-critical in the sense of  [34, Definition 2.2] and [44].Proof.
The proof of (a) follows by applying the arguments used in the proof of the equivalence (a)⇔(b) in Proposition 9. The statements (b) and (c) follow from (a).
(c) First, we note that the quadratic formsq(1),…,q(m):ℤm-1→ℤ are nonnegative, if q is nonnegative. Then, (c) is a consequence of the group isomorphism
(61)Kerq(j)→≃ℤm,j∩Kerq,w↦w^(j):=(w1,…,wj-1,0,wj+1,…,wm).
Since (d) is a consequence of (c), the proof is complete.Definition 15.
Assume thatJ is a connected poset and qJ,q^J:ℤJ→ℤ are its incidence and Tits quadratic forms  (6), respectively.(a)
J is defined to be  nonnegative of corank  r≥0 if its incidence quadratic form qJ:ℤJ→ℤ (resp., one of the forms q^J and q¯J) is nonnegative and the free abelian subgroup KerqJ of ℤJ is of ℤ-rank r (resp., Kerq^J≅Kerq¯J≅KerqJ is of ℤ-rank r); see (36).(b)
J is defined to be  nonprincipal critical if the incidence quadratic form qJ:ℤJ→ℤ is nonnegative and not positive, J is not principal, and the quadratic form qJ′:ℤJ′→ℤ is principal or positive, for every proper subposet J′ of J.(c)
A one-peak posetI, with maxI={*}, is defined to be  nonprincipal Tits-critical if the Tits quadratic form q^I:ℤI→ℤ is nonnegative and not positive, I is not principal, and the Tits quadratic form q^I′:ℤI′→ℤ is principal or positive, for every proper subposet I′ of I containing the peak *. We call a nonprincipal Tits-critical poset I  exceptional, if the subposet T=I∖{*} is nonprincipal Tits-critical; see [33, 34].(d)
A posetJ is defined to be  P-hypercritical if J is not nonnegative and each of its proper subposet is nonnegative; see [34, Definition 2.2].Remark 16.
Assume thatT is a poset and T*=T∪{*} is its one-peak enlargement.(a)
IfT* is P-hypercritical, then T is NP-critical in the sense of [14], but not conversely.(b)
By [43], many of the NP-critical posets T listed in [14, Table 2] are of corank at most two.(c)
A Coxeter spectral classification of one-peak positive (resp., almost TitsP-critical) posets is given in [17, 18] (resp., in [33]).
We frequently use the following important characterisation.Theorem 17.
Assume thatJ is a connected poset and qJ,q^J:ℤJ→ℤ are the incidence and the Tits quadratic forms of J  (7), respectively.(a)
IfJ is nonnegative of corank two, then J contains at least 6 elements, and |J|=6 if and only if J is the garland
(62)andKerq𝒢3:=ℤ·h(1)⊕ℤ·h(2), where h(1)=(1,1,-1,-1,0,0) and h(2)=(1,1,0,0,-1,-1). The garland 𝒢3 is nonprincipal critical.(b)
The following four conditions are equivalent.
(b1)
The posetJ is nonprincipal critical.(b2)
|J|≥6 and the form qJ:ℤJ→ℤ is nonnegative critical of corank two.(b3)
|J|≥6 and qJ:ℤJ→ℤ is nonnegative, the group KerqJ is of ℤ-rank two, and for any j∈J, the subposet J(j):=J∖{j} of J is principal or positive.(b4)
|J|≥6 and qJ:ℤJ→ℤ is nonnegative, and the group KerqJ has a ℤ-basis h,h′ such that there is no j∈J, with hj=hj′=0.(c)
LetI be a one-peak poset I, with maxI={*}. The following three conditions are equivalent.
(c1)
I is nonprincipal Tits-critical.(c2)
|I|≥7, the Tits form q^I:ℤI→ℤ is nonnegative, the group Kerq^I is of ℤ-rank two, and for any j∈I∖{*}, the one-peak subposet I(j):=I∖{j} of I is principal or positive.(c3)
|I|≥7 and q^I:ℤI→ℤ is nonnegative, and the group Kerq^I has a ℤ-basis h,h′ such that there is no j∈I∖{*}, with hj=hj′=0.(d)
A nonprincipal Tits-critical one-peak posetI, with maxI={*} and |I|=7, is exceptional if and only if I is the one-peak garland
(63)and Kerq^𝒢3*:=ℤ·h^(1)⊕ℤ·h^(2), where h^(1)=(1,1,-1,-1,0,0,0), h^(2)=(1,1,0,0,-1,-1,0).Proof.
(a) It is easy to check that any posetJ with at most 5 elements is either positive or principal. Moreover, if J is nonnegative of corank two and |J|=6, then J is the garland 𝒢3. Since
(64)q𝒢3(x)=x12+x22+x32+x42+x52+x62+(x1+x2)(x3+x4+x5+x6)+(x3+x4)(x5+x6),
the Lagrange’s algorithm yields
(65)q𝒢3(x)=(x1+12x3+12x4+12x5+12x6)2+(x2+12x3+12x4+12x5+12x6)2+12(x3-x4)2+12(x5-x6)2.
It follows that q𝒢3:ℤ6→ℤ is nonnegative and its kernel is a rank-two free abelian group of the form shown in (a). Hence, (a) follows.
(b) We show by a computer search that there is no nonprincipal critical posetJ such that |J|≤5. Then, in view of Lemma 14, the equivalences (b1)⇔(b2)⇔(b3)⇔(b4) easily follow.
(c) We show by a computer search that there is no one-peak nonprincipal Tits-critical posetI such that |I|≤6. Then, in view of Lemma 14, the equivalences (c1)⇔(c2)⇔(c3) easily follow.
(d) Note that(66)q^𝒢3*(x)=x12+x22+x32+x42+x52+x62+x72+(x1+x2)(x3+x4+x5+x6)+(x3+x4)(x5+x6)-(x1+x2+x3+x4+x5+x6)x7,
and the Lagrange's algorithm yields
(67)q^𝒢3*(x)=(x1+12x3+12x4+12x5+12x6-12x7)2+(x2+12x3+12x4+12x5+12x6-12x7)2+12(x3-x4)2+12(x5-x6)2+12x72.
It follows that q^𝒢3*:ℤ7→ℤ is nonnegative and its kernel is a rank-two free abelian group of the form shown in (d). Hence, the one-peak garland 𝒢3* is nonprincipal Tits-critical and exceptional. On the other hand, one shows by a computer search that 𝒢3* is the only one-peak poset that is nonprincipal Tits-critical and exceptional. This finishes the proof.Following [34, Section 4], we will study nonnegative posets J of corank r≥2 by means of the spectrum speccJ, the reduced Coxeter number cˇJ, and the rank r≥2 defects
(68)∂J=(∂J(1),…,∂J(r)),∂¯J=(∂¯J(1),…,∂¯J(r)),∂^J=(∂^J(1),…,∂^J(r)):ℤJ→ℤr
defined in the following extension of Theorem 10.Theorem 18.
LetJ be a finite nonnegative poset of corank r≥2, and let m=|J|≥2. One fixes nonzero vectors hJ(1),…,hJ(r)∈ℤJ such that KerqJ=ℤ·hJ(1)⊕⋯⊕ℤ·hJ(r)≅ℤr, and one sets hJ=(hJ(1),…,hJ(r)).(a)
There exist a minimal integercˇJ≥1 (called the reduced Coxeter number of J) and a group homomorphism ∂J=(∂J(1),…,∂J(r)):ℤJ→ℤr≅KerqJ(called the  incidence defect of J) such that
(69)ΦJcˇJ(v)=v+∂J(v)•hJ=v+∂J(1)(v)·hJ(1)+⋯+∂J(r)(v)·hJ(r),∂J(ΦJ(v))=∂J(v),∀v∈ℤJ,and∂J(h)=0, for all h∈KerqJ, where one sets
(70)∂J(v)=(∂J(1)(v),…,∂J(r)(v)),∂J(v)•hJ:=∂J(1)(v)·hJ(1)+⋯+∂J(r)(v)·hJ(r).(b)
Assume thatcˇJ≥1 and ∂J:ℤJ→ℤr are as in  (a), and one sets
(71)h¯J(1)=hJ(1)·B′,…,h¯J(r)=hJ(r)·B′,h^J(1)=hJ(1)·B′·B-1,…,h^J(r)=hJ(r)·B′·B-1,whereB′,B∈𝕄J(ℤ) are as in Proposition 5.
(b1)
There exists a group homomorphism∂¯J=(∂¯J(1),…,∂¯J(r)):ℤJ→ℤr≅Kerq¯J(called the  Euler defect of J) such that
(72)Φ¯JcˇJ(v)=v+∂¯J(v)•h¯J=v+∂¯J(1)(v)·h¯J(1)+⋯+∂¯J(r)(v)·h¯J(r),∀v∈ℤJ,∂¯J∘Φ¯J=∂¯J,∂¯J=∂¯J∘hB′, and ∂¯J(h)=0,  for all h∈Kerq¯J, where one sets
(73)h¯J=(h¯J(1),…,h¯J(r)),∂¯J(v)•h¯J:=∂¯J(1)(v)·h¯J(1)+⋯+∂¯J(r)(v)·h¯J(r).(b2)
There exists a group homomorphism∂^J=(∂^J(1),…,∂^J(r)):ℤJ→ℤr≅Kerq^J(called the  Tits defect of J) such that
(74)Φ^JcˇJ(v)=v+∂^J(v)•h^J=v+∂¯J(1)(v)·h^J(1)+⋯+∂^J(r)(v)·h^J(r),∀v∈ℤJ,∂^J∘Φ^J=∂^J, ∂^J=∂¯J∘hB=∂J∘hB′-1∘hB, and ∂^J(h)=0,  for all h∈Kerq^J, where one sets
(75)h^J=(h^J(1),…,h^J(r)),∂^J(v)•h^J:=∂^J(1)(v)·h^J(1)+⋯+∂^J(r)(v)·h^J(r).(c)
The Coxeter numbercJ of J is finite if and only if the incidence defect ∂J:ℤJ→ℤr is zero. In this case, cˇJ=cJ.(d)
Givenv∈ℤm≡ℤJ, the order sv:=|𝒪(v)| of the ΦJ-orbit 𝒪(v) is finite if and only if∂J(v)=0. Ifsv=|𝒪(v)| is finite, then sv divides cˇJ and there is a unique integer mv such that
(76)mv·h=v+ΦJ(v)+ΦJ2(v)+⋯+ΦJsv-1(v)=v+ΦJ-1(v)+ΦJ-2(v)+⋯+ΦJ-sv+1(v).(e)
The statement  (d) holds withΦJ and Φ^J  (resp., Φ¯J) interchanged.Proof.
For simplicity of presentation, we assume thatr=2. We recall from the proof of Proposition 9 that ℤ·hJ(1)⊕ℤ·hJ(2)=KerqJ=Ker[DqJ:ℤm→ℤm],  where m=|J|≥2 and
(77)DqJ:ℤm→ℤm,v↦DqJ(v)=(∂qJ∂x1(v),…,∂qJ∂xm(v)),
is the gradient group homomorphism. It follows that
(78)ℤmℤ·hJ(1)⊕ℤ·hJ(2)=ℤmKerqJ≅ImDqJ≅ℤm-2.
Denote by ϕ:ℤm→ℤm-2 the composite quotient epimorphism. Then, the form qJ induces the form q~J:ℤm-2→ℤ such that q~J(ϕ(x))=qJ(x), for all x∈ℤm. Moreover, the Coxeter transformation ΦJ:ℤm→ℤm induces a group automorphism Φ~J:ℤm-2→ℤm-2 such that
(79)Φ~J∘ϕ=ϕ∘ΦJ,q~J(Φ~J(y))=q~J(y),
for all y∈ℤm-2. It follows that q~J is positive definite, and there exists a minimal integer cˇJ≥1 such that Φ~JcˇJ is the identity map on ℤm-2. Hence, given v∈ℤm, the element ΦJcˇJ(v)-v lies in the kernel of qJ; that is, it has the form
(80)ΦJcˇJ(v)-v=∂J(1)(v)·hJ(1)+∂J(2)(v)·hJ(2),
where ∂J(1)(v),∂J(2)(v) are integers uniquely determined by v. Since ΦJ is a group homomorphism, then
(81)∂J(1)(v+v′)=∂J(1)(v)+∂J(1)(v′),∂J(2)(v+v′)=∂J(2)(v)+∂J(2)(v′);
that is, we have defined a pair of group homomorphisms ∂J(1),∂J(2):ℤJ→ℤ; hence, ∂J=(∂J(1),∂J(2)):ℤJ→ℤ2 is a group homomorphism. It is easy to see that ∂J has the properties required in (a), and (a) follows.
In view of Proposition5, the statements (b)–(e) are a consequence of (a). The reader is referred to [34, Theorem 4.17] for more details and a generalization.Corollary 19.
Assume thatJ is a finite nonnegative poset of corank r≥2.(a)
The Coxeter numbercJ of J is infinite if and only if the defect homomorphism ∂J:ℤJ→ℤr is nonzero, or, equivalently, if and only if the ΦJ-orbit 𝒪(ej) of some basis vector ej∈ℤJ is infinite.(b)
The Coxeter transformationΦJ is weakly periodic in the sense of Sato  [42]; that is, ΦJs-id is nilpotent, for some s≥1.Proof.
The statement (a) follows immediately from Theorem18. To prove (b), we check that (ΦJcˇJ-id)2=0.Remark 20.
(a) It was shown in [34, Example 5.18] that, for the one-peak garland I=𝒢3* of Theorem 17(d), we have(i)
∂I=∂^I=∂¯I=0 and cI=cˇI=4,(ii)
the setℛq^I of Tits roots of I lies on 22 sand-glass tubes; six of them are of rank two, and each of the remaining fourteen tubes is of rank four; see [34, pp. 459–461] for details.
(b) By Lemma8(a), the Coxeter number cJ is infinite, for every principal poset J.
(c) By Theorem17, there is no nonnegative connected poset J of corank 2, with |J|≤5. Moreover, a minimal such a poset is the garland
(82)
(d) We show in [43] that most of the nonnegative connected posets J of corank 2, with at most 15 elements, are of zero defect. We also show there that a smallest nonnegative connected poset J with nonzero defect has 8 elements and is one of the following two posets:
(83)
It is easy to check that(i)
cˇJ′=cˇJ′′=2,(ii)
coxJ′(t)=coxJ′′(t)=t8-4t6+6t4-4t2+1,(iii)
the coordinates of the Tits defect∂^J′=(∂^J′(1),∂^J′(2)):ℤ8→ℤ2 of J′, with respect to the ℤ-basis
(84)h^J′(1)=(2,0,-1,-1,1,1,2,0),h^J′(2)=(0,2,1,1,-1,-1,0,2)ofKerq^J′, are given by the formulae
(85)∂^J′(1)(x)=x2-12x3-12x4+12x5+12x6-x7,∂^J′(2)(x)=x1+12x3+12x4-12x5-12x6-x8,(iv)
the coordinates of the Tits defect∂^J′′=(∂^J′′(1),∂^J′′(2)):ℤ8→ℤ2 of J′′, with respect to the ℤ-basis
(86)h^J′′(1)=(0,-1,-1,0,1,1,0,0),h^J′′(2)=(1,0,0,1,0,0,1,1)
ofKerq^J′′, are given by
(87)∂^J′(1)(x)=x1+x2+x3+x4+x5+x6-x7-x8,∂^J′′(2)(x)=2x1+x2+x3+2x4+x5+x6-2x7-2x8.


## 5. An Example

In this section, we illustrate the results of Section3 by an example of a principal one-peak poset I of the Euclidean type DI=𝔻~4. We give a description of the set ℛq^I of roots of q^I and the mesh translation quiver Γ(ℛq^I,Φ^I) together with the decomposition (see (51))
(88)Γ(ℛq^I,Φ^I)=Γ(∂^I-ℛq^I,Φ^I)∪Γ(∂^I+ℛq^I,Φ^I)∪Γ(∂^I0ℛq¯I,Φ^I).
Let I be the one-peak garland
(89)
The incidence matrix CI, the Tits matrix C^I, and the Coxeter-Tits matrix C^oxI=-C^I·C^I-tr of I are the following:
(90)CI=[1011101111001010001100001],C^I=[1000-10100-11110-11101-100001],C^oxI=[01001100010001100101-1-111-1].
The Coxeter polynomial coxI(t), the Tits quadratic form q^I:ℤ5→ℤ, and the Coxeter-Tits transformation Φ^I:ℤ5→ℤ5 of I are
(91)coxI(t)=t5+t4-2t3-2t2+t+1=F𝔻~4(t),q^I(x)=x·C^I·xtr=x12+x22+x32+x42+x52+(x1+x2)(x3+x4)-(x1+x2+x3+x4)x5,Φ^I(x)=x·C^oxI=(x2-x5,x1-x5,x4+x5,x3+x5,x1+x2+x3+x4-x5),
for x=(x1,x2,x3,x4,x5)∈ℤI≡ℤ5. Note that the Φ^I-orbit 𝒪(e5) of e5 consists of two vectors e5 and Φ^I(e5)=(-1,-1,1,1,-1)=Φ^I-1(e5). Since
(92)q^I(x)=(x1+12x3+12x4-12x5)2+(x2+12x3+12x4-12x5)2+12(x3-x4)2+12x52,
then the form q^I:ℤ5→ℤ is positive semidefinite, is not positive definite,
(93)Kerq^I=ℤ·h^I,whereh^I=(1,1,-1,-1,0),
and Φ^I(h^I)=h^I. This means that q^I is principal, but not P-critical; see [44]. One easily shows that the reduced Coxeter number of I equals cˇI=2 and the Tits defect ∂^I:ℤ5→ℤ of I is given by ∂^I(x)=-(x1+x2+x3+x4), because Φ^I≠id and Φ^I2(v)=v+∂^I(v)·h^I, for any v∈ℤ5. The set ℛq^I of roots of q^I has the disjoint union decomposition (see (51))
(94)ℛq^I=∂^I-ℛq^I∪∂^I+ℛq^I∪∂^I0ℛq^I,
and ∂^I-ℛq^I, ∂^I+ℛq^I, ∂^I0ℛq^I are Φ^I-invariant subsets of ℛq^I. Obviously, the Φ^I-orbit 𝒪(v) of any v∈∂^I0ℛq^I is of length two, whereas the Φ^I-orbit 𝒪(w) of any vector w∈∂^I-ℛq^I∪∂^I+ℛq^I is infinite. By (92), a vector v=(v1,v2,v3,v4,v5)∈ℤ5 is a root of q^I:ℤ5→ℤ if and only if (2v1+v3+v4-v5)2+(2v2+v3+v4-v5)2+2(v3-v4)2+2v52=4. Hence, looking at all possible decompositions 4=a12+a22+2a32+2a42, with a1,a2,a3,,a4,∈ℤ, we show that v=(v1,v2,v3,v4,v5)∈ℤ5 is a root of q^I:ℤ5→ℤ if and only if v or v^:=-v is one of the vectors listed in Table 1 or in Table 2.Table 1
j
Φ
^
I
j
(
e
1
)
Φ
^
I
j
(
e
2
)
Φ
^
I
j
(
p
12
)
Φ
^
I
j
(
e
3
)
Φ
^
I
j
(
e
4
)
⋮
⋮
⋮
⋮
⋮
⋮
j
=
7
(
-
3
,
-
2,3
,
3,1
)
(
-
2
,
-
3,3
,
3,1
)
(
-
6
,
-
6,7
,
7,1
)
(
-
3
,
-
3,3
,
4,1
)
(
-
3
,
-
3,4
,
3,1
)
j
=
6
(
-
2
,
-
3,3
,
3,0
)
(
-
3
,
-
2,3
,
3,0
)
(
-
5
,
-
5,6
,
6,1
)
(
-
3
,
-
3,4
,
3,0
)
(
-
3
,
-
3,3
,
4,0
)
j
=
5
(
-
2
,
-
1,2
,
2,1
)
(
-
1
,
-
2,2
,
2,1
)
(
-
4
,
-
4,5
,
5,1
)
(
-
2
,
-
2,2
,
3,1
)
(
-
2
,
-
2,3
,
2,1
)
j
=
4
(
-
1
,
-
2,2
,
2,0
)
(
-
2
,
-
1,2
,
2,0
)
(
-
3
,
-
3,4
,
4,1
)
(
-
2
,
-
2,3
,
2,0
)
(
-
2
,
-
2,2
,
3,0
)
j
=
3
(
-
1,0
,
1,1
,
1
)
(
0
,
-
1,1
,
1,1
)
(
-
2
,
-
2,3
,
3,1
)
(
-
1
,
-
1,1
,
2,1
)
(
-
1
,
-
1,2
,
1,1
)
j
=
2
(
0
,
-
1,1
,
1,0
)
(
-
1,0
,
1,1
,
0
)
(
-
1
,
-
1,2
,
2,1
)
(
-
1
,
-
1,2
,
1,0
)
(
-
1
,
-
1,1
,
2,0
)
j
=
1
(
0,1
,
0,1
,
0
)
(
1,0
,
0,0
,
1
)
(
0,0
,
1,1
,
1
)
(
0,0
,
0,1
,
1
)
(
0,0
,
1,0
,
1
)
j
=
0
(
1,0
,
0,0
,
0
)
(
0,1
,
0,0
,
0
)
(
1,1
,
0,0
,
1
)
(
0,0
,
1,0
,
0
)
(
0,0
,
0,1
,
0
)
j
=
-
1
(
1,2
,
-
1
,
-
1,1
)
(
2,1
,
-
1
,
-
1,1
)
(
2,2
,
-
1
,
-
1,1
)
(
1,1
,
-
1,0
,
1
)
(
1,1
,
0
,
-
1,1
)
j
=
-
2
(
2,1
,
-
1
,
-
1,0
)
(
1,2
,
-
1
,
-
1,0
)
(
3,3
,
-
2
,
-
2,1
)
(
1,1
,
0
,
-
1,0
)
(
1,1
,
-
1,0
,
0
)
j
=
-
3
(
2,3
,
-
2
,
-
2,1
)
(
3,2
,
-
2
,
-
2,1
)
(
4,4
,
-
3
,
-
3,1
)
(
2,2
,
-
2
,
-
1,1
)
(
2,2
,
-
1
,
-
2,1
)
j
=
-
4
(
3,2
,
-
2
,
-
2,0
)
(
2,3
,
-
2
,
-
2,0
)
(
5,5
,
-
4
,
-
4,1
)
(
2,2
,
-
1
,
-
2,0
)
(
2,2
,
-
2
,
-
1,0
)
j
=
-
5
(
3,4
,
-
3
,
-
3,1
)
(
4,3
,
-
3
,
-
3,1
)
(
6,6
,
-
5
,
-
5,1
)
(
3,3
,
-
3
,
-
2,1
)
(
3,3
,
-
2
,
-
3,1
)
j
=
-
6
(
4,3
,
-
3
,
-
3,0
)
(
3,4
,
-
3
,
-
3,0
)
(
7,7
,
-
6
,
-
6,1
)
(
3,3
,
-
2
,
-
3,0
)
(
3,3
,
-
3
,
-
2,0
)
j
=
-
7
(
4,5
,
-
4
,
-
4,1
)
(
5,4
,
-
4
,
-
4,1
)
(
8,8
,
-
7
,
-
7,1
)
(
4,4
,
-
4
,
-
3,1
)
(
4,4
,
-
3
,
-
4,1
)
⋮
⋮
⋮
⋮
⋮
⋮Table 2
j
u
(
j
)
u
+
(
j
)
w
(
j
)
w
+
(
j
)
v
(
j
)
v
+
(
j
)
j
=
0
-
e
5
=
(
0,0
,
0,0
,
-
1
)
(
1,1
,
-
1
,
-
1,1
)
(
1,0
,
0
,
-
1,0
)
(
0,1
,
-
1,0
,
0
)
(
0,1
,
0
,
-
1,0
)
(
1,0
,
-
1,0
,
0
,
)
j
=
1
(
2,2
,
-
2
,
-
2,1
)
(
1,1
,
-
1
,
-
1
,
-
1
)
(
1,2
,
-
2
,
-
1,0
)
(
2,1
,
-
1
,
-
2,0
)
(
2,1
,
-
2
,
-
1,0
)
(
1,2
,
-
1
,
-
2,0
)
j
=
2
(
2,2
,
-
2
,
-
2
,
-
1
)
(
3,3
,
-
3
,
-
3,1
)
(
3,2
,
-
2
,
-
3,0
)
(
2,3
,
-
3
,
-
2,0
)
(
2,3
,
-
2
,
-
3,0
)
(
3,2
,
-
3
,
-
2,0
)
j
=
3
(
4,4
,
-
4
,
-
4,1
)
(
3,3
,
-
3
,
-
3
,
-
1
)
(
3,4
,
-
4
,
-
3,0
)
(
4,3
,
-
3
,
-
4,0
)
(
4,3
,
-
4
,
-
3,0
)
(
3,4
,
-
3
,
-
4,0
)
j
=
4
(
4,4
,
-
4
,
-
4
,
-
1
)
(
5,5
,
-
5
,
-
5,1
)
(
5,4
,
-
4
,
-
5,0
)
(
4,5
,
-
5
,
-
4,0
)
(
4,5
,
-
4
,
-
5,0
)
(
5,4
,
-
5
,
-
4,0
)
j
=
5
(
6,6
,
-
6
,
-
6,1
)
(
5,5
,
-
5
,
-
5
,
-
1
)
(
5,6
,
-
6
,
-
5,0
)
(
6,5
,
-
5
,
-
6,0
)
(
6,5
,
-
6
,
-
5,0
)
(
5,6
,
-
5
,
-
6,0
)
j
=
6
(
6,6
,
-
6
,
-
6
,
-
1
)
(
7,7
,
-
7
,
-
7,1
)
(
7,6
,
-
6
,
-
7,0
)
(
6,7
,
-
7
,
-
6,0
)
(
6,7
,
-
6
,
-
7,0
)
(
7,6
,
-
7
,
-
6,0
)
j
=
7
(
8,8
,
-
8
,
-
8,1
)
(
7,7
,
-
7
,
-
7
,
-
1
)
(
7,8
,
-
8
,
-
7,0
)
(
8,7
,
-
7
,
-
8,0
)
(
8,7
,
-
8
,
-
7,0
)
(
7,8
,
-
7
,
-
8,0
)
⋮
⋮
⋮
⋮
⋮
⋮
⋮(1)TheΦ^I-orbits in𝒫^I:=∂^I-ℛq^I. Since ∂^I(u)<0, if u∈{e1,e2,e3,e4} or u is the vector p12=(1,1,0,0,1), then the Φ^I-orbits of the vectors e1,e2,e3,e4,p12 lie in 𝒫^I:=∂^I-ℛq^I, because 𝒫^I is a Φ^I-invariant subset of ℛq^I. It is easy to see that the Φ^I-orbits consist of the vectors listed in Table 1.Throughout this section, we freely use theΦ^I-mesh terminology and notation introduced in [2, 34, 40].(2)Φ^I-mesh quiverΓ(𝒫^I,Φ^I)=Γ(∂^I-ℛq^I,Φ^I). It follows from our earlier remarks that the set 𝒫^I:=∂^I-ℛq^I of the negative defect roots of q^I splits into the five Φ^I-orbits 𝒪(e1), 𝒪(e2), 𝒪(e3), 𝒪(e4), 𝒪(p12). By applying the mesh toroidal algorithm defined in [2, 34], one constructs the following infinite Φ^I-mesh translation quiver of the negative defect roots of q^I; see Figure 2, where we set a^:=-a for any positive integer a≥1.Figure 2Figure 3
Sand-glass tube of rank(2,1).(3)Φ^I-mesh quiverΓ(𝒬^I,Φ^I)=Γ(∂^I+ℛq^I,Φ^I). Since the group isomorphism ℤI→ℤI, v↦-v, carries roots to roots, Φ^I-meshes to Φ^I-meshes, and Φ^I-orbits to Φ^I-orbits, then it defines the bijections ∂^I-ℛq^I→∂^I+ℛq^I and ∂^I0ℛq^I→∂^I0ℛq^I, because ∂^I(-v)=-∂^I(v). It follows that the set 𝒬^I:=∂^I+ℛq^I of the positive defect roots of q^I splits into the five Φ^I-orbits 𝒪(e^1), 𝒪(e^2), 𝒪(e^3), 𝒪(e^4), 𝒪(p^12), and one constructs the infinite Φ^I-mesh translation quiver
(95)Γ(𝒬^I,Φ^I)=Γ(∂^I+ℛq^I,Φ^I)
of the positive defect roots of q^I by interchanging any vector v in Γ(𝒫^I,Φ^I)=Γ(∂^I-ℛq^I,Φ^I) with its negative v^:=-v.(4)Φ^I-mesh quiverΓ(∂^I0ℛq^I,Φ^I). By the equality Φ^I2(v)=v-∂^I(v)·h^I, the Φ^I-orbit of any v∈∂^I0ℛq^I consists of two vectors v and Φ^I(v). Now, we show that the Φ^I-orbits in ∂^I0ℛq^I form a Φ^I-mesh translation quiver Γ(∂^I0ℛq^I,Φ^I).Note thatΦ^I(e5)=(-1,-1,1,1,-1), Φ^I2(e5)=e5, ∂^I(e5)=0, and ∂^I(Φ^I(e5))=0. It follows that the two-element Φ^I-orbits of e5 and -e5 lie in ∂^I0ℛq^I. Moreover, the vectors
(96)u+(1)=(1,1,-1,-1,-1),-u+(1),w(0)=(1,0,0,-1,0),w+(0)=Φ^I(w(0))=(0,1,-1,0,0),v(0)=(0,1,0,-1,0),v+(0)=Φ^I(v(0))=(1,0,-1,0,0)w+(1)=(2,1,-1,-2,0),v+(1)=(1,2,-1,-2,0)
belong to ∂^I0ℛq^I. It is easy to see that we have the following Φ^I2-mesh quivers of vectors in ∂^I0ℛq^I:
(97)
Note that the Φ^I-orbit of u+(1) consists of the following two vectors:
(98)u(1)=(2,-2,-2,-2,1),u+(1)=Φ^I(u(1)).
By (92), a vector v=(v1,v2,v3,v4,v5)∈ℤ5 is a root of q^I:ℤ5→ℤ of defect zero if and only if
(99)(2v1+v3+v4-v5)2+(2v2+v3+v4-v5)2+2(v3-v4)2+2v52=4,v1+v2+v3+v4=0.
It follows that v or -v belongs to any of the six series of roots presented in Table 2.Hence, we conclude that theΦ^I-orbits in the set ∂^I0ℛq^I form three Φ^I-mesh quivers 𝒯u, 𝒯w, 𝒯v, and each of them has the form of infinite two-surface tube of rank 2:
(100)
where a is one of the vectors
(101)u=u(0)=(0,0,0,0,-1),w=w(0)=(1,0,0,-1,0),v=v(0)=(0,1,0,-1,0).(5)Φ^I-mesh quiver  Γ(∂^I0ℛq^I∪Kerq^I,Φ^I). We recall that Kerq^I=ℤ·h^I, where h^I=(1,1,-1,-1,0). Note that
(102)Φ^I(h^I)=h^I,Φ^I(m·h^I)=m·h^I,foranym∈ℤ.
Obviously, the vectors lying in Kerq^I form the Φ^-mesh translation quiver 𝒯^h^I presented in (108).Now, we construct from theΦ^I-orbits in the set ∂I0ℛq^I∪Kerq^I an infinite Φ^I-mesh translation quiver. For this purpose, we note that the following six vectors
(103)-e5,e5,Φ^I(-e5),Φ^I(e5),-h^I,h^I
form two Φ^I-meshes of width 1. If we complete them by the three vectors
(104)0,u+(1):=(1,1,-1,-1,-1),-u+(1),
we get the Φ^I-mesh quiver
(105)
Analogously, we construct the following two Φ^I-mesh quivers
(106)
where
(107)u(1)=(2,-2,-2,-2,1),u+(1)=Φ^I(u(1)),w(0)=(1,0,0,-1,0),w+(0)=Φ^I(w(0))=(0,1,-1,0,0),v(0)=(0,1,0,-1,0),v+(0)=Φ^I(v(0))=(1,0,-1,0,0),w+(1)=(2,1,-1,-2,0),v+(1)=(1,2,-1,-2,0).
We recall that if v∈∂^I0ℛq^I, then v or -v is one of the vectors presented in Table 2. It follows that the Φ^I-orbits in ∂^I0ℛq^I∪ℤ·h^I form three infinite Φ^I-mesh sand-glass tubes 𝒯^u, 𝒯^w, 𝒯^v of rank (2,1), and each of them has the shape presented in (109)
(108)(109)
where a is one of the vectors
(110)u=u(0)=(0,0,0,0,-1),w=w(0)=(1,0,0,-1,0),v=v(0)=(0,1,0,-1,0).
Construct the disjoint union 𝒯^u∪𝒯^w∪𝒯^v of the tubes 𝒯^u, 𝒯^w, 𝒯^v, and note that each of them contains the tube 𝒯^h^I. By making the identification of the vectors m·h^I, with m∈ℤ, lying in the corresponding Φ^I-orbits, we get the quotient Φ^I-mesh translation quiver
(111)Γ(∂^I0ℛq^I∪Kerq^I,Φ^I)=𝒯^u∪𝒯^w∪𝒯^v≃
that has a shape of a threefold sand-glass tube of rank (2,2,2,1) in the sense of [40]. It is obtained from the disjoint union of three copies of the onefold sand-glass tube of rank (2,1) presented in Figure 3 (see also [34, Figure 5.8]) by making an obvious identification of their waist vectors.(
6
)
A  ℤ-congruence of the bigraph  Δ^I  with the Euclidean diagram  𝔻~4.  Since we have coxI(t)=F𝔻~4(t)=t5+t4-2t3-2t2+t+1 and speccI=specc𝔻~4, the Euclidean diagram 𝔻~4 is the diagram DI associated to I. A technique developed in [2, 17, 18, 34, 40] allows us to construct a ℤ-invertible matrix B∈Gl(5,ℤ) such that the following diagrams are commutative:
(112)
where b𝔻~4 and q𝔻~4 are the forms of the Euclidean diagram
(113)
defined by the  formulae b𝔻~4(x,y)=x1y1+x2y2+x3y3+x4y4+x5y5-(x1+x2+x3+x4)y5=x·Gˇ𝔻~4·ytr, q𝔻~4(x)=b𝔻~4(x,x), for x,y∈ℤ5, hB:ℤ5→ℤ5 is the group automorphism defined by the formula hB(x)=x·B, and
(114)B=[10-1-1000-10-1000-1-101-1-1000111],Gˇ𝔻~4=[1000-10100-10010-10001-100001].
It is easy to check that the equality Gˇ𝔻~4=B·C^I·Btr holds, and therefore the diagrams (112) are commutative. Furthermore, by the same technique, we construct another matrix
(115)B1=[-1000-1000-1000-1000-100-111001]
such that the equality Gˇ𝔻~4=B1·C^I·B1tr holds.


## 6. Concluding Remarks

6.1
It follows from Lemma3 and the results obtained recently in [3, 4] that for any connected positive (resp., principal) poset J, there exists a simply laced Dynkin diagram DJ∈{𝔸m,𝔻m,𝔼6,𝔼7,𝔼8}  (resp., a simply laced Euclidean diagram DJ), uniquely determined by J, such that the symmetric Gram matrices GJ, GDJ are ℤ-congruent.
Analogous Coxeter spectral classification of one-peak posetsI, with almost P-critical Tits form q^I:ℤI→ℤ, is obtained in [33] by a reduction to computer calculations.
6.2.
Although the Coxeter spectral classification problem for arbitrary finite posets remains unsolved, we have a solution for positive one-peak posets. Indeed, it follows from the results in [17] that for any one-peak positive poset J, there exists a simply laced Dynkin diagram DJ∈{𝔸s,𝔻n,𝔼6,𝔼7,𝔼8}(uniquely determined by J) such that speccJ=speccDJ, the nonsymmetric Gram matrices GˇJ, GˇDJ are ℤ-congruent, and the symmetric Gram matrices GJ, GDJ are ℤ-congruent.
6.3.
We can determine the diagramDJ as follows. Fix an upper-triangular numbering {a1,…,am} of elements of J. Then, the incidence matrix CJ∈𝕄m(ℤ) is upper-triangular, and the Euler matrix C¯J:=CJ-1 is also upper triangular. Then, the Euler edge-bipartite graph Δ¯J (33) is loop-free, and we have CΔ¯J=C¯J. Hence, the symmetric Gram matrices GΔ¯J, G¯J coincide, and, by Lemma 3, the poset J is positive (resp., principal) if and only if the bigraph ΔJ is positive (resp., principal). By applying to Δ¯J the inflation algorithm constructed in [4, 21] (see also [45]), we get (in a finite number of steps) an edge-bipartite graph DΔJ such that the symmetric Gram matrix GΔ¯J=G¯J is ℤ-congruent with the symmetric Gram matrix GDΔJ, and the edge-bipartite graph DΔJ has no dotted edges; that is, DΔJ is a (multi) graph. We set DJ:=DΔJ. It follows from the results in [3, 4] that DJ is a simply laced Dynkin diagram, if J is positive, and DJ is a simply laced Euclidean diagram, if J is principal. Moreover, the matrix G¯J is ℤ-congruent with GDJ. Since the incidence Gram matrix GJ of J is ℤ-congruent with the matrix G¯J (by Proposition 5), then the matrices GJ and GDJ are ℤ-congruent.
6.4.
Although we can apply in 6.3 the inflation algorithm to the incidence edge-bipartite graphΔJ, we use in the construction of DJ the Euler edge-bipartite graph Δ¯J, because the number of nonzero entries in the Euler matrix C¯J:=CJ-1 does not increase the number for the matrix CJ; see [28, Proposition 2.12]. It follows that the number of the dotted edges in Δ¯J does not increase the number of the dotted edges in ΔJ, and the use in 6.3 the bigraph Δ¯J reduces the time of calculation in the procedure Δ¯J↦DJ.



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*Source: 743734-2013-03-21.xml*