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 themainmotivations for the study is an application ofmatrix 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 + C tr


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][3][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 .
Here, we use the terminology and notation introduced in [1,4,[26][27][28].We denote by N ⊆ Z ⊆ Q the set of nonnegative integers, the ring of integers, and the rational number field.Given  ≥ 1, we view Z  as a free abelian group and denote by  1 , . . .,   the standard Z-basis of Z  .Given an index set , we denote by Z  the abelian group of all vectors  = (  ) ∈ , with integer coordinates   ∈ Z, by M  (Z) the Z-algebra of all square  by  integral matrices, and by  ∈ M  (Z) the identity matrix.In particular, M  (Z), with  ≥ 1, is the Z-algebra of all square  by  matrices.The group Gl (, Z) := { ∈ M  (Z) , det  ∈ {−1, 1}} ⊆ M  (Z) (1) is called the (integral) general linear group.We say that two square rational matrices ,   ∈ M  (Q) are Z-equivalent, or Z-congruent, (and denote  ∼ Z   ) if there is a matrix  ∈ Gl (, Z) such that   =  tr ⋅  ⋅ .By a poset  ≡ (, ⪯) we mean a finite partially ordered set  with respect to a partial order relation ⪯.Following [26], a poset  is called a one-peak poset if  has a unique maximal element * .A finite poset  is uniquely determined by its incidence matrix   ∈ M  (Z), that is, the square  ×  matrix, as follows: 1, for  ⪯ , 0, for   . ( International Journal of Mathematics and Mathematical Sciences Following an idea of Drozd [32] (developed in [27]), we have introduced in [1,28] the Tits matrix Ĉ ∈ M  (Z) of  to be the integral matrix with ĉ { { { { { 1,  =  or  ⪯ , ,  ∉ max , 0, ,  incomparable, or  ⪯  and ,  ∉ max , −1, if  ≺  and  ∈ max , where max  is the set of all maximal elements of .Usually, we equip the elements of  with a numbering; that is,  is viewed as  = { 1 , . . .,   },  = || ≥ 1. Throughout, we fix such a numbering and make the identifications M  (Z) ≡ M  (Z) and Z  ≡ Z  .The incidence matrix   ∈ M  (Z) ≡ M  (Z) and the Tits matrix Ĉ ∈ M  (Z) ≡ M  (Z) depend on the numbering of  1 , . . .,   .Namely, if   = {  1 , . . .,    } is obtained from  = { 1 , . . .,   } by a permutation  ∈ S  and σ ∈ Gl(, Z) is the permutation matrix of , then Note that any poset  admits an upper-triangular numbering  = { 1 , . . .,   }; that is,   ⪯   implies  ≤ .In this case,   ∈ M  (Z) is an upper-triangular matrix with 1 on the main diagonal, and, hence, det   = 1, and det    = 1, for any numbering   = {  1 , . . .,    }.Fix a numbering  1 , . . .,   of elements of .Following [1,28], by the Euler matrix of the poset  we mean the inverse of   .Following [3,4], we call the symmetric adjacency matrix and the characteristic polynomial of the poset .The set spec  of all  = || real roots of   () is defined to be the (real) spectrum of the poset .
We denote by   , q ,   : Z  ≡ Z  → Z the incidence quadratic form, the Tits quadratic form, and the Euler quadratic form of  defined by the formulae respectively, where J =  \ max , max  is the set of all maximal elements in , and Ĉ ∈ M  (Z) is the Tits matrix of ; see (27) and [1,28] for a definition.The matrices with rational coefficients, are called the symmetric incidence Gram matrix, the symmetric Tits-Gram matrix, and the symmetric Euler-Gram matrix of .The matrices with integer coefficients, are called the Tits adjacency matrix, and the Euler adjacency matrix of .: (11) then   () =   () =  4 − 5 2 − 4; that is, the characteristic polynomial   () of  coincides with the Euler-characteristic polynomial   () of .
(b) If  is the poset : (12) of the Dynkin type E 6 , then the characteristic polynomial   () of  does not coincide with the Euler-characteristic polynomial   () of , because Following [17,33], we introduce the following definition.
(b) We define a poset  to be principal if its incidence form   : Z  → Z is principal in the sense of [34,Definition 2.1]; that is,   is nonnegative, not positive, and the kernel is an infinite cyclic subgroup of Z  .Following the main idea of the Coxeter spectral analysis of acyclic edge-bipartite graphs (signed graphs) presented in [3,4], we study finite posets  (with a fixed numbering  = { 1 , . . .,   }) by means of the Coxeter spectrum of , that is, the set specc  of all  = || eigenvalues of the Coxeter matrix of , or equivalently, the set specc  of all  = || roots of the Coxeter polynomial see (31) and [1].It follows from (4) that the Coxeter spectrum specc  of  and the spectrum spec  of  do not depend on the numbering of the elements of the poset .
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  to their canonical forms.For simplicity of its presentation, we illustrate it in case when q () is the Tits quadratic form (7) of the poset  = { 1 , . . .,   , * , +}, with an uppertriangular partial order ⪯ such that  has precisely two maximal elements * := * +1 and + := + +2 .In this case, we have Fix a vector  = ( 1 , . . .,   ,  * ,  + ) ∈ N +2 ≡ N  , and consider the -vector space Mat   of all partitioned matrices of the form (compare with [27]) *  + (19) with coefficients in , where   * = 0 if   ⊀ * and  + = 0 if   ⊀ +.Consider the group G   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 th column block to th column block, if the relation   ⪯   holds in the poset  \ { * , +} (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    , with respect to the elementary transformations from the set G   , is controlled by the Tits quadratic form q in the following sense.For any  ∈ N  , there is only a finite number G   -canonical forms of matrices in Mat   if and only if the form q is weakly positive; that is, q () is positive, for all nonzero vectors  ∈ N  .Moreover, there is one-to-one correspondence between the irreducible G   -canonical forms in Mat   and the vectors  ∈ N  satisfying the equation q () = 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 (, ) =  ⋅ Ĉ ⋅  tr (26) and Z-bilinear Euler form   (, ) =  ⋅   ⋅  tr is given in [1, (1.3)].The reader is referred to [6][7][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  essentially uses the Coxeter spectra of the simply laced Euclidean diagrams presented in Figure 1.
One of the aims of the Coxeter spectral analysis of nonnegative finite posets is to study the question when the Coxeter type of a poset  determines the matrix   (and, hence, the poset ) uniquely, up to a Z-congruency.Here, we set č  = c  , if  is positive.In other words, we claim that, for any pair ,  of nonnegative one-peak posets, Ctype  = Ctype  if and only if the incidence matrices   and   are Z-congruent.We also study the problem related with the results proved by Horn and Sergeichuk [35], if for any Z-invertible matrix  ∈ M  (Z), there exists  ∈ M  (Z) such that  tr =  tr ⋅  ⋅  and  2 =  is the identity matrix; see [17,18].
The main results of the present paper on nonnegative posets  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 , established in Proposition 5; (2) a characterization of principal posets given in Section 3. We show that a connected poset  is principal if and only if there exists a simply laced Euclidean diagram  ∈ { D ,  ≥ 4, Ẽ6 , Ẽ7 , Ẽ8 } (25) such that the symmetric Gram matrix Moreover, we show in Section 3 that, given a connected principal poset , the Coxeter spectrum specc  is a subset of a unit circle S 1 = { ∈ C; || = 1}, 1 ∈ specc  , and any  ∈ specc  is a root of unity; (3) a Coxeter spectral classification result (Corollary 11) asserting that, given a pair ,  of one-peak principal posets with at most 13 elements, the following conditions are equivalent: (3a)  = , (3b) specc  = specc  , (3c) č  = č  and || = ||, (3d) the incidence matrix   ∈ M  (Z) is Z-congruent to the incidence matrix   ∈ M  (Z); that is, there is a Z-invertible matrix  such that   =  tr ⋅   ⋅ .
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  ≥ 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.

A Framework for the Coxeter Spectral Analysis of Finite Posets
The quadratic wanderings on finite posets  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   ∈ M  (Z) ≡ M  (Z) (2), with det   = 1 and a fixed numbering  = { 1 , . . .,   }, as well as the Euler matrix   :=  −1  , we study in [1,[26][27][28] the Tits wandering defined by the Tits matrix Ĉ ∈ M  (Z) ≡ M  (Z) of  (see [28, (3.6)]), that is, the Gram matrix of the Tits Z-bilinear form b : where max  is the set of all maximal elements in the poset  and J :=  \ max .We call q () := b (, ) =  ⋅ Ĉ ⋅  tr the Tits quadratic form of .
(ii) There is an edge (iii) There is an edge   ---  in Δ  , if   and   are not maximal in  and   ≺   or   ≺   holds in .There is an edge We call Δ  , Δ  , and Δ  the incidence bigraph of Δ, the Tits bigraph of Δ, and the Euler bigraph of Δ, respectively, (with respect to the numbering  = { 1 , . . .,   }).
The following simple lemma is of importance.
The following example shows that the correspondence   → Δ  defined in (33) does not preserve the Coxeter types of  and Δ  .In particular, it shows that the equality cox  () = cox Δ  () does not hold in general and the Coxeter polynomial cox Δ  () depends on the numbering of , whereas the Coxeter polynomial cox  () does not depend on the numbering of .
Example 7. Consider the poset  such that its Hasse quiver has the form By a permutation of the elements in , we get Note that the first numbering is upper-triangular, whereas the second one is not upper-triangular.

Principal Posets
We recall that a poset  is principal if its incidence unit form   is principal in the sense of [34, Definition 2.1]; that is,   : Z  → Z is nonnegative and not positive, and the kernel Ker   := { ∈ Z  ;   () = 0} is an infinite cyclic subgroup of Z  .We start with the following useful observation.(c) The Tits quadratic form q of  is nonnegative and Ker q = Z ⋅ ĥ, for some nonzero vector ĥ ∈ Z  .
Theorem 10.Let  be a finite principal poset, with a numbering { 1 , . . .,   } of elements of .Fix a nonzero vector (b1) There exists a group homomorphism   : Z  → Z (called the Euler defect of ) such that (b2) There exists a group homomorphism ∂ : Z  → Z (called the Tits defect of ) such that Proof.We recall from the proof of Proposition 9 that where  = || ≥ 2 and   : It follows that q is positive definite and there exists a minimal integer č  ≥ 1 such that Φ č   is the identity map on Z −1 .Hence, (a) follows, because the equalities   (h  ) = 0 and   (Φ  ()) =   (), for all  ∈ Z  , are almost obvious; see [34,Theorem 4.7].
To finish the proof of (b), we note that the equality   =  tr ⋅   ⋅  in (b4) implies that the matrices Cox  and Cox  are conjugate, and, hence, we get specc  = specc  ; 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 , with at most 13 elements and the associated Euclidean diagram , we construct (by a computer search) a Z-invertible matrix   such that Ǧ  =  tr  ⋅   ⋅   (compare with [17,18,33,43]).Hence, (b4) follows, and the proof is complete.
If  is a principal poset, then the sets International Journal of Mathematics and Mathematical Sciences of roots of the unit forms   , q , and   have the disjoint union decompositions where Note that the group isomorphism Z  → Z  ,   → v := −, restricts to the bijections 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  is principal, because It follows that   is nonnegative and Ker   = Z ⋅ h, where h = (1, 1, 1, 1, 1, 1, 1);   is critical in the sense of Ovsienko [24]; see also [38,44].Note that the Euler matrix   =  −1  of  and the inverse of the Coxeter-Euler matrix Cox  := − −1  ⋅  tr  have the forms Moreover, we have  Ẽ6 =  tr ⋅   ⋅ , and the matrix  :=  tr ⋅   ⋅  is a morsification of the Euclidean diagram Ẽ6 (see [34,40]), where (iv) the Φ  -orbit of any vector of defect zero in  0  R q 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 T 1,2 of rank 2 and on six sand-glass tubes of rank five.

Nonnegative Posets of Positive Corank
In the study of nonnegative posets, the following extensions of [34, Definition 2.2] are of importance.
(a) The form  is defined to be nonnegative of corank  ≥ 0, if  is nonnegative and the Q-rank rk Q   of the rational Gram matrix (b) The form  is defined to be nonnegative critical of corank  ≥ 1, if  is nonnegative of corank  ≥ 1 and each of the nonnegative quadratic forms  (1) , . . .,  () : is viewed as a subgroup of Z  .
(d)  is nonnegative critical of corank  = 1 if and only if  is -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.
Definition 15.Assume that  is a connected poset and   , q : Z  → Z are its incidence and Tits quadratic forms (6), respectively.
(a)  is defined to be nonnegative of corank  ≥ 0 if its incidence quadratic form   : Z  → Z (resp., one of the forms q and   ) is nonnegative and the free abelian subgroup Ker   of Z  is of Z-rank  (resp., Ker q ≅ Ker   ≅ Ker   is of Z-rank ); see (36).
(b)  is defined to be nonprincipal critical if the incidence quadratic form   : Z  → Z is nonnegative and not positive,  is not principal, and the quadratic form    : Z   → Z is principal or positive, for every proper subposet   of .
(c) A one-peak poset , with max  = { * }, is defined to be nonprincipal Tits-critical if the Tits quadratic form q : Z  → Z is nonnegative and not positive,  is not principal, and the Tits quadratic form q  : Z   → Z is principal or positive, for every proper subposet   of International Journal of Mathematics and Mathematical Sciences  containing the peak * .We call a nonprincipal Titscritical poset  exceptional, if the subposet  =  \ { * } is nonprincipal Tits-critical; see [33,34].
(d) A poset  is defined to be -hypercritical if  is not nonnegative and each of its proper subposet is nonnegative; see [34,Definition 2.2].
Remark 16.Assume that  is a poset and  * =  ∪ { * } is its one-peak enlargement.
(a) If  * is -hypercritical, then  is -critical in the sense of [14], but not conversely.
(b) By [43], many of the -critical posets  listed in [14, Table 2] are of corank at most two.
We frequently use the following important characterisation.
Theorem 17. Assume that  is a connected poset and   , q : Z  → Z are the incidence and the Tits quadratic forms of  (7), respectively.
(b4) || ≥ 6 and   : Z  → Z is nonnegative, and the group Ker   has a Z-basis h, h  such that there is no  ∈ , with ℎ  = ℎ   = 0.
(c) Let  be a one-peak poset , with max  = { * }.The following three conditions are equivalent.
Proof.(a) It is easy to check that any poset  with at most 5 elements is either positive or principal.Moreover, if  is nonnegative of corank two and || = 6, then  is the garland the Lagrange's algorithm yields It follows that  G 3 : Z 6 → Z is nonnegative and its kernel is a rank-two free abelian group of the form shown in (a).Hence, (a) follows.
(d) Note that and the Lagrange's algorithm yields It follows that qG * 3 : Z 7 → Z is nonnegative and its kernel is a rank-two free abelian group of the form shown in (d).Hence, the one-peak garland G * 3 is nonprincipal Tits-critical and exceptional.On the other hand, one shows by a computer search that G * 3 is the only one-peak poset that is nonprincipal Tits-critical and exceptional.This finishes the proof.
defined in the following extension of Theorem 10.
Proof.For simplicity of presentation, we assume that  = 2.

(110)
Construct the disjoint union T ∪ T ∪ T of the tubes T , T , T , and note that each of them contains the tube Tĥ  .By making the identification of the vectors  ⋅ ĥ , with  ∈ Z, lying in the corresponding Φ -orbits, we get the quotient Φmesh translation quiver

Γ ( ∂0
R q ∪ Ker q , Φ ) = T ∪ T ∪ T ≃ (111) 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.(33) is loop-free, and we have  Δ  =   .Hence, the symmetric Gram matrices  Δ  ,   coincide, and, by Lemma 3, the poset  is positive (resp., principal) if and only if the bigraph Δ  is positive (resp., principal).By applying to Δ  the inflation algorithm constructed in [4,21] (see also [45]), we get (in a finite number of steps) an edge-bipartite graph Δ  such that the symmetric Gram matrix  Δ  =   is Z-congruent with the symmetric Gram matrix  Δ  , and the edgebipartite graph Δ  has no dotted edges; that is, Δ  is a (multi) graph.We set  := Δ  .It follows from the results in [3,4] that  is a simply laced Dynkin diagram, if  is positive, and  is a simply laced Euclidean diagram, if  is principal.Moreover, the matrix   is Z-congruent with   .Since the incidence Gram matrix   of  is Z-congruent with the matrix   (by Proposition 5), then the matrices   and   are Z-congruent.

Concluding Remarks
6.4.Although we can apply in 6.3 the inflation algorithm to the incidence edge-bipartite graph Δ  , we use in the construction of  the Euler edge-bipartite graph Δ  , because the number of nonzero entries in the Euler matrix   :=  −1  does not increase the number for the matrix   ; see [28,Proposition 2.12].It follows that the number of the dotted edges in Δ  does not increase the number of the dotted edges in Δ  , and the use in 6.3 the bigraph Δ  reduces the time of calculation in the procedure Δ   → .

Lemma 8 .Proposition 9 .
Assume that  is a connected principal poset.(a) The Coxeter number c  of  is infinite.(b) The Coxeter spectrum specc  is a subset of a unit circle S 1 = { ∈ C; || = 1}, 1 ∈ specc  , and any  ∈ specc  is a root of unity.(c)If Ker   = Z ⋅ h, then Ker q = Z ⋅ ĥ and Ker   = Z ⋅ h, where(i) h = h ⋅   , h = ĥ ⋅ , ĥ = h ⋅   ⋅  −1 , (ii)   =  tr  ,  = [  J 0 0  ],and J =  \ max  are as in Proposition 5. Proof.(a) By Proposition 5 (d2), c  is independent of the numbering of .Then, without loss of generality, we may suppose that the numbering of  is upper-triangular.Then, by Lemma 3(d) and Proposition 5(d1), the Coxeter number c  coincides with the Coxeter number of the principal edgebipartite graph Δ  associated with  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).Let  be a connected poset,  = || ≥ 2, and let   , Ĝ ,   , ∈ M  (Q) be the symmetric incidence Gram matrix of , the symmetric Tits-Gram matrix of , and the symmetric Euler-Gram matrix of , respectively.The following five conditions are equivalent.(a) The poset  is principal.(b) The Gram matrix   is positive indefinite of rank −1.

( a )
If  is nonnegative of corank two, then  contains at least 6 elements, and || = 6 if and only if  is the garland

( b )
The following four conditions are equivalent.(b1) The poset  is nonprincipal critical.(b2) || ≥ 6 and the form   : Z  → Z is nonnegative critical of corank two.

Proof.
The statement (a) follows immediately from Theorem 18.To prove (b), we check that (Φ č   − ) 2 = 0. Remark 20.(a) It was shown in [34, Example 5.18] that, for the one-peak garland  = G * 3 of Theorem 17(d), we have (i)   = ∂ =   = 0 and c  = č  = 4, (ii) the set R q of Tits roots of  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 Lemma 8(a), the Coxeter number c  is infinite, for every principal poset .

Figure 2 we
Figure 2
(35)me that  is nonnegative.The Coxeter type of  is defined to be the pair Ctype  := (specc  , c  ) if  is positive, and the triple Ctype  := (specc  , c  , č  ) if  is not positive, where č  is the reduced Coxeter number of  in the sense of Theorems 10 and 18.The following proposition shows that equality(35)holds.Let  be a finite poset, with a fixed numbering  = { 1 , . . .,   }, let   , q ,   : Z  → Z be the incidence, Tits, and Euler quadratic form of , and let Φ  , Φ , Φ  : Z  → Z  be the corresponding Coxeter transformations.(a) The following equalities hold Ĉ =  ⋅   ⋅   and    =   ⋅   ⋅   , and the following diagrams are commutative ) is called the Coxeter polynomial of the poset .(c) The Coxeter spectrum of  is the set specc  ⊆ C of all  = || eigenvalues of the matrix Cox  , or, equivalently, the set specc  of all  = || roots of the Coxeter polynomial cox  ().(d) The order c  := ord (Φ  ) of the Coxeter transformation Φ  : Z  → Z  is called the Coxeter number of the poset .In other words, c  is the minimal integer  ≥ 1 such that Φ   = .We set c  = ∞, if Φ and Φ  op = Φ −1  . ) of  does not depend on the numbering  = { 1 , . . .,   }. (d3) The Coxeter spectrum specc  is a subset of a unit circle S 1 = { ∈ C; || = 1}, and any  ∈ specc  is a root of unity.(d4) The poset  is positive if and only if 1 ∉ specc  .Proof.The first equality Ĉ =  ⋅   ⋅  tr is obvious, and the second one  tr  =   ⋅   ⋅  tr follows by a direct calculation.d2) It is sufficient to note that the incidence matrix   is upper triangular.Hence,   = Ǧ Δ  and Cox Δ  = Cox  .
d) The Euler quadratic form   of  is nonnegative and Ker   = Z ⋅ h, for some nonzero vector h ∈ Z  .(e):Z→Z  ,   →   () = (    1 () , ...,     ()) ,(39)is the gradient group homomorphism of   , then Ker   = Ker [  : Z  → Z  ] and the subgroup Ker   of Z  is of rank  − rk   and consists of all integral solutions of the system 2 ⋅ (c) The Coxeter number c  of  is infinite, and the incidence defect   : Z  → Z is nonzero.(d) Given  ∈ Z  , the order s  := |O()| of the Φ  -orbit O() is finite if and only if   () = 0.If s  = |O()| is finite, then s  divides č  and there is a unique integer   such that Theorem 4.7, Corollary 4.15]for more details.If  is a principal connected poset with at most 13 elements, then its Coxeter spectrum specc  is a subset of a unit circle S 1 = { ∈ C; || = 1}, 1 ∈ specc  , and any  ∈ specc  is an th root of unity, where  ≤ č and || = ||, (b4) the incidence matrix   ∈ M  (Z) is Z-congruent to the incidence matrix   ∈ M  (Z); that is, there is a Z-invertible matrix  such that   =   ⋅   ⋅ .Proof.(a) By Lemma 8, specc  ⊆ S 1 and 1 ∈ specc  .Assume that  is the associated Euclidean diagram of , as in Proposition 9 where  () ( 1 , . . .,  −1 ,  +1 , . . .,   ) =  ( 1 , . . .,  −1 , 0,  +1 , . . .,   ) . is nonnegative of corank  = 0 if and only if  is positive, and  is nonnegative of corank one if and only if  is principal.(c)  is nonnegative critical of corank  ≥ 1 if and only if  is nonnegative and, for any  ∈ {1, . . ., }, the abelian subgroup Z , ∩ Ker  of Z  is free of rank at most  − 1, where Lemma 14. Assume that  ≥ 2,  ≥ 0, and  : Z  → Z is an integral quadratic form.(a) is nonnegative of corank  ≥ 0 if and only if  is nonnegative and the subgroup Ker  of the abelian group Z  is free of rank .(b) (c) The Coxeter number c  of  is finite if and only if the incidence defect   : Z  → Z  is zero.In this case, č  = c  .(d) Given  ∈ Z  ≡ Z  , the order s  := |O()| of the Φ  -orbit O() is finite if and only if   () = 0.If s  = |O()| is finite, then s  divides č  and there is a unique integer   such that The Coxeter number c  of  is infinite if and only if the defect homomorphism   : Z  → Z  is nonzero, or, equivalently, if and only if the Φ  -orbit O(  ) of some basis vector   ∈ Z  is infinite.(b) The Coxeter transformation Φ  is weakly periodic in the sense of Sato ,  2 ,  3 , ,  4 , ∈ Z, we show that  = ( 1 ,  2 ,  3 ,  4 ,  5 ) ∈ Z 5 is a root of q : Z 5 → Z if and only if  or v := − is one of the vectors listed in Table1or in Table2.{ 1 ,  2 ,  3 ,  4 } or  is the vector p 12 = (1, 1, 0, 0, 1), then the Φ -orbits of the vectors  1 ,  2 ,  3 ,  4 , p 12 lie in P := ∂−  R q , because P is a Φ -invariant subset of R q .It is easy to see that the Φ -orbits consist of the vectors listed in Table1.International Journal of Mathematics and Mathematical Sciences (1)The Φ -orbits in P := ∂−  R q .Since ∂ () < 0, if  ∈ [17]rnational Journal of Mathematics and Mathematical SciencesFigure 3: Sand-glass tube of rank (2, 1).principal) poset , there exists a simply laced Dynkin diagram  ∈ {A  , D  , E 6 , E 7 , E 8 } (resp., a simply laced Euclidean diagram ), uniquely determined by , such that the symmetric Gram matrices   ,   are Z-congruent.Analogous Coxeter spectral classification of one-peak posets , with almost -critical Tits form q : Z  → Z, is obtained in[33]by a reduction to computer calculations.6.2.Although the Coxeter spectral classification problemfor 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 onepeak positive poset , there exists a simply laced Dynkin diagram  ∈ {A  , D  , E 6 , E 7 , E 8 } (uniquely determined by ) such that specc  = specc  , the nonsymmetric Gram matrices Ǧ  , Ǧ  are Zcongruent, and the symmetric Gram matrices   ,   are Z-congruent.6.3.We can determine the diagram  as follows.Fix an upper-triangular numbering { 1 , . . .,   } of elements of .Then, the incidence matrix   ∈ M  (Z) is uppertriangular, and the Euler matrix   :=  −1  is also upper triangular.Then, the Euler edge-bipartite graph Δ [3,4]t follows from Lemma 3 and the results obtainedrecently in[3,4]that for any connected positive (resp.,