Algorithmic computation of principal posets using Maple and Python

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Volume 17(2014). Number 1. pp. 33 – 69 c

Journal “Algebra and Discrete Mathematics”

Algorithmic computation of principal posets

using Maple and Python

Marcin Gąsiorek, Daniel Simson and Katarzyna Zając

A b s t r ac t . We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classiﬁcation prob-lems. One of the main aims of the paper is to study ﬁnite posets I that are principal, i.e., the rational symmetric Gram matrix GI :=

1

2[CI +C

tr

I ]∈MI(Q) ofI is positive semi-deﬁnite of corank

one, whereCI ∈MI(Z) is the incidence matrix ofI. With any such

a connected posetI, we associate a simply laced Euclidean diagram DI ∈ {Aen,Den,Ee6,Ee7,Ee8}, the Coxeter matrix CoxI :=−CI·C−

tr I ,

its complex Coxeter spectrumspeccI, and a reduced Coxeter

num-ber ˇcI. One of our aims is to show that the spectrumspeccI of any

such a posetIdetermines the incidence matrixCI (hence the poset

I) uniquely, up to aZ-congruence. By computer calculations, we ﬁnd a complete list of principal one-peak posetsI (i.e.,I has a unique maximal element) of cardinality≤15, together withspeccI, ˇcI, the

incidence defect ∂I :Z I

→Z, and the Coxeter-Euclidean typeDI. In case when DI ∈ {Aen,Den,Ee6,Ee7,Ee8} andn:= |I|is relatively

small, we show that given such a principal poset I, the incidence matrix CI is Z-congruent with the non-symmetric Gram matrix

ˇ

GDI ofDI,speccI =speccDI and ˇcI = ˇcDI. Moreover, given a

pair of principal posets Iand J, with|I|=|J| ≤15, the matrices CI andCJ areZ-congruent if and only ifspeccI =speccJ.

Supported by Polish Research Grant NCN 2011/03/B/ST1/00824. 2010 MSC:06A11, 15A63, 68R05, 68W30.

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1. Introduction

Throughout, we freely use the terminology and notation introduced in [45], [47], [50], [51]. We denote byNthe set of non-negative integers, byZ the ring of integers, and byQ⊂R⊂Cthe ﬁeld of the rational, the real, and the complex numbers, respectively. We viewZn, withn≥1, as a free abelian group. Bye1, . . . , en we denote the standardZ-basis of the

groupZn_{. Given}_n_≥_{1, we denote by}_M

n(Z) theZ-algebra of all square

nby nmatricesA= [aij], withaij ∈Z, and by E ∈Mn(Z) the identity

matrix. Given a ﬁnite set J, we denote by M_J(Z) the Z-algebra of all square J byJ matrices. The group

Gl(n,Z) :={A∈Mn(Z),detA∈ {−1,1}} ⊆Mn(Z)

is called the (integral)general linear group. We say that two square rational matricesA, A′_∈_M

n(Q) areZ-congruent (and denote byA∼Z A′)

if there exists aZ-invertible matrixB ∈Gl(n,Z) such thatA′₌_Btr_·_A_·_B.

By a poset J ≡(J,) we mean a ﬁnite partially ordered setJ with respect to a partial order relation . Following [43],J is called a one-peak poset if it has a unique maximal element∗. Given a posetJ, with m=|J|, we denote by

CJ = [cij]∈MJ(Z)≡Mm(Z) (1.1)

the incidence matrix of J, with cij = 1, for all i j, and cij = 0

otherwise. The rational matrix

GJ := 1₂[CJ +CJtr]∈MJ(Q)≡Mm(Q) (1.2)

is called the symmetric Gram matrix ofJ. Following [50] and [51], we call the symmetric matrixAdJ :=CJ +CJtr −2·E theadjacency matrix

of J, and

PJ(t) = det(t·E−AdJ)∈Z[t], (1.3)

the characteristic polynomialof the poset J. We say that the poset J is Z-bilinear equivalent to a poset J′ _{(and we write} _J _≈

Z J′) if

CJ ∼ZCJ′.

We defineJ to be non-negative (resp. positive) if the rational symmet-ric Gram matrix GJ(1.2) is positive semi-definite (resp. positive definite).

If J is connected and the symmetric Gram matrix GJ is positive

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quadratic form qJ(x) = x·CJ ·xtr is non-negative and the subgroup

KerqJ :={v∈ZJ;qJ(v) = 0} ofZJ is inﬁnite cyclic.

Our study is inspired by important applications of the quadratic forms and edge-bipartite graphs in constructing linear algebra invariants that measure a geometric complexity of Nazarova-Roiter matrix problems over a field K and in the study of module categories and their derived categories, see the monographs [1], [11], [43], [53], and the articles [2]-[9], [12]-[24], [27], [29]-[34], [39]-[42], [45]-[52], and [54], [55]. In particular, our study is inspired by a well-known result of Drozd [10], by the Coxeter spectral analysis of loop-free edge-bipartite graphs developed in [50], and the Coxeter spectral classification technique of finite posets introduced in [44], [45], and [51], see also [26], [28], [36]-[38], and [56].

In the present paper we are mainly interested in the class of non-negative posetsJ; in particular, in the class of principal posets. We study them by applying our recent results on the Coxeter spectral classiﬁcation of loop-free edge-bipartite graphs deﬁned in [50] (see also [51]) as follows.

Anedge-bipartite graph(bigraph, for short), withn≥2 vertices, is a pair ∆ = (∆0,∆1 = ∆−1 ⊔∆1+), where ∆0 ={a1, . . . , an}is a set of vertices

and ∆1 is a ﬁnite set of edges such that |∆−₁(ai, aj)| · |∆+1(ai, aj)| = 0,

for all ai 6=aj ∈ ∆0. Edges in ∆1−(ai, aj) and ∆+₁(ai, aj) are visualized

as continuousai——–aj, and dotted onesai- - - -aj, respectively. We say

that ∆ is loop-free if ∆1(ai, ai) is empty, for all ai ∈ ∆0. We denote

byUBigr_n the set of all connected loop-free edge-bipartite graphs, with n≥2 vertices.

We view any ﬁnite graph ∆ = (∆0,∆1) as an edge-bipartite graph by setting ∆−₁(ai, aj) = ∆1(ai, aj) and ∆1+(ai, aj) =∅, forai, aj ∈∆0.

A non-symmetricGram matrixof ∆∈ UBigr_nis the matrix

ˇ G_∆=

   

1 d∆

12 . . . d∆1n 0 1 . . . d∆

2n .

.

. ... . .. ... 0 0 . . . 1

  

∈Mn(Z),

where d∆_ij =−|∆−₁(ai, aj)|, if there is an edgeai——–aj andi≤j,d∆ij =

|∆+₁(ai, aj)|, if there is an edge ai- - - - aj and i≤j. We set d∆ij = 0, if

the set ∆1(ai, aj) is empty orj < i. The rational matrix

G∆:= 1

2[ ˇG∆+ ˇG

tr

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is called the symmetric Gram matrix of ∆. The Gram quadratic formof ∆∈ UBigr_n is deﬁned by the formula

q∆(x) =q∆(x1, . . . , xn) =x21+· · ·+x2n+

X

i<j

d∆_ijxixj=x·Gˇ∆·xtr=x·G∆·xtr.

We call ∆∈ UBigr_n, with n≥2 numbered vertices,positive(resp. non-negative of corank s≥1), if its symmetric Gram matrixG_∆∈M_n(Q) is positive definite (resp. positive semi-definite of rankn−s). Moreover, we call ∆∈ UBigr_n principal if the matrixG∆ is positive semi-definite of rankn−1, see [50]. Note that a non-negative loop-free bigraph ∆ is of coranks≥1 if and only if the kernel

Kerq_∆:={v ∈Zn; q_∆(v) =v·Gˇ_∆·vtr _{= 0}_{} ⊆}_Zn

of q∆ : Zn → Z is a free subgroup of Zn of Z-rank s. Obviously, ∆ is principal if and only if ∆ is non-negative loop-free and Kerq∆=Z·h, with h6= 0.

The matrix Ad∆:= ˇG∆+ ˇGtr∆−2·E is called the symmetric

adja-cency matrix of a loop-free edge-bipartite graph ∆∈ UBigrn, and the

spectrumof ∆ is the set spec_∆⊂Rofn real roots of the polynomial

P∆(t) = det(t·E−Ad∆)∈Z[t],

called the characteristic polynomialof the edge-bipartite graph ∆. Following [50], with any principal posetJ, we have associated in [51] a loop-free edge-bipartite principal graph ∆J, and a simply laced Euclidean

diagram DJ, that is, one of the graphs presented in the following table.

Ta b l e 1 . 1 . Simply laced Euclidean diagrams

e

An: 1 2

n+ 1

n−1 n

n≥1,

e

Dn: 1

2

3

n+ 1

n−1 n

n≥4,

e

E₆: 1

4 5

2 3 6 7

e

E₇: 1 2

5

3 4 6 7 8

e

E₈: 1

4

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We recall thatDJ is the simply laced Euclidean diagram D∆e J obtained

from ∆J by applying the inﬂation algorithm ∆J 7→ D∆e J presented in

[28, Algorithm 5.4] and [50, Algorithm 3.1] (see also [52]). Consequently, we have the passage

J 7→∆J 7→DJ :=D∆e J.

We study the Coxeter spectral properties of any principal poset J by means of the Coxeter spectral properties of the associated simply laced Euclidean diagramDJ ∈ {Ae_n,De_n,Ee₆,Ee₇,Ee₈}.

Following [45], with any posetJ, we associate theCoxeter matrix

CoxJ :=−CJ·CJ−tr,

with det CoxJ = (−1)m, where m=|J|andCJ−tr = (CJtr)−1. The

Cox-eter spectrum specc_J ⊆CofJ is deﬁned to be the set of allm=|J| complex roots of theCoxeter polynomial

coxJ(t) = det(t·E−CoxJ)∈Z[t],

theCoxeter number cJ ≥2 is the minimal integer such that Cox

c_J

J =E,

and theCoxeter transformationof J is the group automorphism

ΦJ :ZJ →ZJ, ΦJ(v) :=v7→v·CoxJ,

see [45] and [51] for details. IfJ is non-negative, the Coxeter spectrum

specc_J lies on the unit circleS1 ={z∈C; |z|= 1}, consists of roots of unity, and 1∈/ specc_J if and only if J is positive, see [50, Lemma 2.1] and [51]. In this case we have associated with J (see [47] and [51]) a reduced Coxeter number ˇcJ and theincidence defect homomorphism

e

∂J :ZJ →KerqJ ⊆ZJ such that

Φˇc_J

J (v) =v+∂eJ(v), for all v∈ZJ,

where KerqJ := {v ∈ ZJ; qJ(v) = 0} is the kernel of the incidence

quadratic formqJ :ZJ →Z deﬁned by the formula

qJ(x) =

X

j∈J

x2_j +X

i≺j

xixj =x·CJ ·xtr.

Since J is assumed to be non-negative, the quadratic form qJ is

non-negative and KerqJ is a subgroup of ZJ, see [47]. If, in addition, the

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ZJ of the form KerqJ =Z·hJ, wherehJ is a non-zero vector in KerqJ

uniquely determined by J, up to multiplication by −1. In this case, e

∂J(v) =∂J(v)·hJ, where ∂J :ZJ →Zis a group homomorphism, called

theincidence defect of J, see [51].

The following characterisation of principal posets obtained in [51, Proposition 9] is of importance.

Theorem 1.4.Assume thatJ is a connected poset with m=|J| ≥2 and letGJ ∈MJ(Q) be the symmetric incidence Gram matrix of J (1.2). The

following four conditions are equivalent. (a) The poset J is principal.

(b) The incidence symmetric Gram matrixGJ is positive semidefinite

of rank m−1.

(c) The incidence quadratic formqJ :ZJ →Z ofJ is non-negative

and KerqJ =Z·h, for some non-zero vector h∈ZJ.

(d) There exists a simply laced Euclidean diagram

DJ ∈ {Ae_s, s≥3,De_n, n≥4,Ee6,Ee7,Ee₈}

(uniquely determined by J) such that the incidence symmetric Gram matrixGJ is Z-congruent to the symmetric Gram matrix GDJ ∈MDJ(Q)

of the Euclidean diagram DJ, that is, there exists a Z-invertible matrix B ∈Gl(m,Z) such that GDJ =Btr·GJ ·B.

Proof. Apply [51, Proposition 3.2] and a characterisation of principal loop-free edge-bipartite graphs given in [50].

One of the aims of the Coxeter spectral analysis of ﬁnite posets is to study the following problem.

Problem 1.5.When the Coxeter spectrumspecc_J of a posetJ deter-mines the incidence matrix CJ (hence the poset J) uniquely, up to a Z-congruence.

In connection with Problem 1.5, and a problem studied by Horn and Sergeichuck in [25], we also consider the problem if for any Z-invertible matrixA∈Mn(Z), there exists B ∈Mn(Z) such thatAtr =Btr·A·B

and B2 ₌_E _{(the identity matrix).}

We would like to note that the following problem remains unsolved.

Problem 1.6. Show that DJ 6= Ae_m−1, if J is a one-peak connected principal poset, withm=|J| ≥5.

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Theorem 1.7.(a) Assume that I is a poset of the shape De(1)m,s,p,Dem,s(2),

e

Dm,s,p(3) ,Dem,s,p(4) ,De(5)m,s,p,r,Dem,s,p,r(6) or De(7)m,s listed in Table 1.11, and |I| =

m≥5. Then I is principal and the associated Euclidean graph DI of I is the diagram De_m.

(b) If I is any of the one-peak posets listed in Tables4.1 and 4.2 then I is principal and the associated Euclidean graphDI of I is the diagram e

E_m−1, where m=|I|.

Theorem 1.8. Assume that I is a one-peak principal poset with m:= |I| ≤15 and DI is its associated Euclidean diagram.

(a) m≥5 andDI is not the diagramAe_m−1. In particular, form= 5, we have DI =De₄ and I is one of the four posets:

e

D(1)₄_,₀_,₅:

2 1 4 3 5 e

D(2)₄_,₀:

1

2

3

4 5

e

D₄(4)_,₁_,₄: 1

4 2 3

5

e

D(6)₄_,₀_,₄_,₂:

3 4 2 1

5

(b) If m ≥6 and DI = De_m−1 then I is one of the 2.115 posets of the shapes presented in Table 1.11. In particular, for m = 6, we have

DI =De₅ andI is one of the 13 posets:

e

D₅(1),0,5:

2 1 4 3 5 6 e

D₅(1),0,6:

2 1 5 3 4 6 e

D₅(1),1,6:

1 3 2 5 4 6 e

D(2)₅,0:

1

2 3

4 5 6

e

D(2)₅,1:

1 2 3 4 5 6 e

D₅(3),2,2:

1 2

3

4 5

6

e

D₅(4)_,₁_,₄:

1 4 3 2 5 6 e

D₅(4)_,₁_,₅: 1 2

3

4 5 6

e

D₅(4)_,₂_,₅:

1 2

5 3

4 6

e

D₅(5),0,3,6:

1 2 3 4 5 6 e

D₅(6),0,5,3:

1 2 3 4 5 6 e

D(6)₅,0,5,2:

1 2 3 4 5 6 e

D(7)₅_,₃:

4 1

5 2 3

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(c) Ifm = 7 and DI =Ee₆ then I is one of the 31 posets presented in Table 4.1.

(d) If m= 8 and DI =Ee₇ then I is one of the 132 posets presented in Table 4.2.

(e) If m= 9 andDI =Ee₈ then I is one of the posetsJEe8

1 ,. . . , J422Ee8 listed in [21]. In particular, we have

JEe8

1 : ₁

2 3 4 5 6 7 8 9

JEe8 31 :

1 2

3 5

4 6

7 8 9

JEe8

48 : 1 2

6

7

3 4 5 8 9 JEe8

147: 1

3

2 4 6

5 7

8 9

JEe8 412:

1 4

2

6

7

3 5 8

9 JEe8

422: 2

1

3

7 4 6

5 8 9

(f) The total number of principal posetsJ (not necessarily one-peak ones), withm = |J| ≤15, equals 158.448 and the number # J of such posets J of the Coxeter-Euclidean type DJ ∈ {Dem−1,Eem−1} is listed in the following tables.

n=|J| n= 5 n= 6 n= 7 n= 8 n= 9 n= 10

DJ De₄ De₅ eD6

e

E6

e

D7

e

E7

e

D8

e

E8

e

D₉

# J 8 30 92₈₄ 227₄₇₀ _{2 102}577 1.357

n=|J| n= 11 n= 12 n= 13 n= 14 n= 15 DJ De₁₀ De₁₁ De₁₂ De₁₃ De₁₄

# J 3.217 7.371 16.897 38.069 85.561

Theorem 1.9. Assume that I is a one-peak principal poset with 5 ≤ m:=|I| ≤15, DI is its associated Euclidean diagram, GˇDI is the

non-symmetric Gram matrix of the graph DI, and ΦI : ZI → ZI is the

incidence Coxeter transformation of I. Denote by

RI:=RqI ={v∈Z

I_; _q

I(v) = 1}

the set of roots of the incidence quadratic form qI :ZI →Z.

(a) There exists aZ-invertible matrix B ∈Mm(Z) such that GˇDI =

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(b) The Coxeter number cI ofI is infinite, the incidence defect

ho-momorphism ∂I :ZI →Zis non-zero and the set ∂I0RI∪KerqI admits

a ΦI-mesh translation quiver Γ(∂I0RI∪KerqI,ΦI) of a sand-glass tube

shape (in the sense of [46]-[47]), where

∂_I0RI={v∈ZI; qI(v) = 1 and ∂I(v) = 0}.

(c) There exists a Z-invertible matrix C∈Mm(Z) such that C2 =E

and Ctr

I =Ctr·CI·C.

The proofs of Theorems 1.7-1.9 are given in Sections 2 and 3. We ﬁnish this section by a result that relates the Coxeter spectrum

specc_I with the usual spectrum spec_I of a posetI, compare with [50, Proposition 2.4(c)].

Theorem 1.10. Assume thatI ={1, . . . , m} is an arbitrary poset and PI(t) := det(t·E−AdI) is the characteristic polynomial of the Euler

adjacency matrix AdI :=C tr

I +CI−2E of I, where CI :=CI−1 is the

Euler matrix of I, see [45].

(a) If the Hasse quiver of I (see [43])is a tree then

coxI(t2) =tm·PI(t+1_t).

(b) Assume that I is a one-peak poset and m := |I| ≤ 15. If I is positive or I is principal then coxI(t2) =tm·PI(t+1_t) if and only if the

Hasse quiver ofI is a tree.

Proof. Since detCI= 1, the Euler matrixCI :=C_I−1 lies inMm(Z) and

Ctr

I =CItr ·CI·CI.

(a) Assume that the Hasse quiver of I is a tree. Without loss of gen-erality, we may assume that the points ofI ={a1, . . . , am}are numbered

in such a way thatai aj impliesi≤j in the natural order. Let ∆I be

the Euler edge-bipartite graph associated withI, see [51, (33)].

By the deﬁnition of ∆I, the non-symmetric Gram matrix of the

edge-bipartite graph ∆I coincides with the Euler matrixCI. Hence, Cox_∆_I =

CoxI = −C_I−tr ·CI and cox_∆

I(t) = det(t·E −CoxI) = coxI(t), see [50] and [51, Corollary 6]. Since the Hasse quiver ofI is a tree then, by [44, Proposition 2.12], the Euler matrixCI :=CI−1 of I has the form

CI :=CI−1=

     

1 c−₁₂ . . . c−₁_m 0 1 . . . c−₂_m

..

. . .. ... 0 0 . . . 1

     ∈

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with

• c−₁₁=. . .=c−

mm = 1,

• c−_ab =−1, if there is an arrowa→bin the Hasse quiverQI of I,

• c−_ab = 0, if a6≺bor there is a path a≺j1 ≺. . .≺js−1 ≺js =b of

lengths≥2 in the Hasse quiverQI.

It follows from [51, (33)] that the Euler edge-bipartite graph ∆I is a

tree and, by [50, Proposition 2.4(c)], we get

coxI(t2) = cox_∆_I(t2) =tm·P_∆_I(t+1_t) =tm·PI(t+ 1_t).

(b) Assume thatI is a one-peak poset,m:=|I| ≤15, andI is positive orI is principal. If the Hasse quiver QI of I is a tree, the statement (a)

applies. To prove the converse implication, assume to the contrary that the Hasse quiver QI of I is not a tree. If I is positive, I is one of the

non-tree shape posets described in [18, Theorem 5.2(e)], see also [19]. IfI is principal,I is one of the non-tree shape poset described in Theorem 1.8. By a case by case inspection of the posets from our lists, a straightforward computer calculations show that coxI(t2)6=tm·PI(t+1_t).

For example, if I,I′_,_I′′_{, and}_J_,_J′_,_J′′ _{are the positive and principal}

posets

I =

1 2

3 4

5 6

∗ I′ =

1 2 3 4

5 6 7

∗

I′′₌ 1 2

3 4

5 6 7

∗

J =

1

2

3

4 5

6

7

∗ J′ = 1 2

3

4 5

6

∗ J′′= 1

2

3 4

5 6

7 8

∗

then we have

L coxL(t2) tm·PL(t+ 1_t)

I t14+t12−t8−t6 t14−3t12−8t11−13t10−18t9−21t8−22t7−21t6−18t5

+t2_{+ 1} ₋₁₃_t4₋₈_t3₋₃_t2_{+ 1}

I′

t16₊_t14₊_t2_{+ 1} _t16₋_t14₋₂_t13₋₃_t12₋₆_t11₋₇_t10₋₈_t9₋₈_t8₋₈_t7₋₇_t6 −6t5₋₃_t4₋₂_t3₋_t2_{+ 1}

I′′

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J t14+t12−2t8−2t6 t14−5t12−12t11−20t10−28t9−36t8−40t7−36t6−28t5

+t2_{+ 1} ₋₂₀_t4₋₁₂_t3₋₅_t2_{+ 1}

J′

t16₊_t14₋_t10₋₂_t8 _t16₋₃_t14₋₈_t13₋₁₄_t12₋₂₀_t11₋₂₅_t10₋₂₈_t9₋₃₀_t8₋₂₈_t7 −t6₊_t2_{+ 1} ₋₂₅_t6₋₂₀_t5₋₁₄_t4₋₈_t3₋₃_t2_{+ 1}

J′′

t18₊_t16₋_t14₋_t12 _t18₋₇_t16₋₂₄_t15₋₅₃_t14₋₈₈_t13₋₁₂₁_t12₋₁₄₄_t11₋₁₅₆_t10 −t6₋_t4₊_t2_{+ 1} ₋₁₆₀_t9₋₁₅₆_t8₋₁₄₄_t7₋₁₂₁_t6₋₈₈_t5₋₅₃_t4₋₂₄_t3₋₇_t2_{+ 1}

This ﬁnishes the proof of Theorem 1.10.

Ta b l e 1 . 1 1 . Seven inﬁnite series of principal one-peak posets of Coxeter-Euclidean typeDe_m

e

D(1)m,s,p:

1 s

s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

0≤s, s+ 5≤p≤m+ 1,

4≤m

e

D(2)m,s:

1 s

s+ 1

s+ 2

s+ 4 m−1

∗ m+ 1

m

s+ 3

0≤s,4≤m, s+ 4≤m

e

D(3)m,s,p: p+ 1 s

s+ 1

s+ 2

s+ 3

m−1

∗ m+ 1

m

1 p

1< p≤s≤m−3,

5≤m, delete•_sifs=p

e

D(4)m,s,p:

1 s

s+ 1

s+ 2 p+ 1

∗ m+ 1

m

p s+ 3

1≤s, s+ 3≤p≤m,

4≤m

e

D(5)m,s,p,r:

1 s

s+ 1

s+ 2

s+ 3

∗ m+ 1

p p+ 1

p+ 2

r m

r−1

p+ 3

0≤s, s+ 3≤p,

6≤p+ 3≤r≤m+ 1,

delete•_p+3ifp+ 3 =r

e

D(6)m,s,p,r:

1 s

s+ 1

s+ 2 p+ 1

∗ m+ 1

r r+ 2

r+ 1

r+ 3 p s+ 3

m 0≤s, s+ 2≤r, 4≤r+ 2≤p≤m,

delete•_r_ifr=s+ 2

e

D(7)m,s:

1 s

s+ 1

∗ m+ 1

m

2

m−1

5≤m,1< s < m−2

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Remark 1.13. Let I be a one-peak principal poset I such that |I| ≤ 15. It follows from Theorem 1.9 that the Coxeter polynomial coxI(t) ∈ Z[t] coincides with the Coxeter polynomial coxDI(t) of the simply laced

Euclidean diagramDI ∈ {De_n, n≥4,Ee6,Ee7,Ee₈}associated withI, where

coxDI(t) =

        

tn+1_+tn₋_tn−1₋_tn−2₋_t3₋_t2_+t+1,_if_DI₌_D_e

n,

t7+t6−2t4−2t3+t+ 1, ifDI=Ee6, t8₊_t7₋_t5₋_2t4₋_t3₊_t_{+ 1,} _if_DI₌_E_e

7, t9+t8−t6−t5−t4−t3+t+ 1, ifDI=Ee₈,

(1.13)

see [33] and [45]. In particular, forn= 4 andn= 5, we have cox_e_D

4(t) =t

5₊_t4₋_2t3₋_2t2₊_t_{+ 1,}

cox_e_D

5(t) =t

6₊_t5₋_t4₋_2t3₋_t2₊_t_{+ 1.}

2. Proof of Theorems 1.7 and 1.8

In this section we present the proof of Theorems 1.7 and 1.8 that are main results of the paper. First, we note that every posetJ = ({1, . . . , m},) is Z-bilinear equivalent to a posetJ′ _{with vertices numbered in such a}

way that i j implies i ≤ j in a natural order and the matrix CJ′ is upper triangular. Therefore, without loss of generality, we can assume that the matrixCJ ∈MJ(Z) of any posetJ has an upper triangular form.

Following [51], we identify a poset J = (J,) with its acyclic edge-bipartite graph ∆J (usually viewed as a signed graph in the sense of

[56]) without continuous edges, and with the dotted edges •_i- - - -•_j, for

all i ≺ j. We recall from [51] that ∆J is uniquely determined by its

non-symmetric Gram matrix ˇG∆J ≡CJ.

One of our main tools is the second step ∆ 7→ ∆′ _:= _t−

ab∆ of the

inﬂation algorithm [28, Algorithm 5.4] and [50, Algorithm 3.1] (see also [52]) that associates to any loop-free principal edge-bipartite graph ∆, with a ﬁxed dotted edge•_a- - - -•_bin ∆₁(a, b),a6=b, the loop-free principal

edge-bipartite graph ∆′:=t−_ab∆ as follows:

• we set ∆0 = ∆′0 and replace the dotted edge •a- - - -•b in ∆ by a

continuous one •_a——–•_b in ∆′,

• given c6=ain ∆0 = ∆0′ such that ∆1(a, c)6=∅, we deﬁne ∆′1(b, c) to be the set with exactly d∆_bc′ dotted edges ifd∆_bc′ >0, and exactly −d∆_bc′ continuous edges if d∆_bc′ ≤0, where d_bc∆′ :=d∆_bc−d∆_ac·d∆_ab′, • each of the remaining edges of ∆1becomes an edge in ∆′₁, i.e., we set

∆′₁+(a′_{, b}′_{) = ∆}+

1(a′, b′) and ∆

′−

1 (a′, b′) = ∆

−

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The main idea of the inﬂation algorithm is to reduce the number of dotted edges in ∆. It is shown in [50] that ˇG∆′ =∇((T−

ab)tr·Gˇ∆·T

−

ab),

where

T_ab−= [tij],with tij =

    

1, ifi=j or (i, j)6= (a, b), −1, if (i, j) = (a, b),

0, otherwise,

and the operationC7→ ∇(C) = [c▽

ij] (introduced in [47, (4.5)]) associates

to any square matrixC = [cij]∈Mm(R) the upper triangular matrix

∇(C) = [c▽

ij] =

     

c▽ 11 c

▽

12 . . . c ▽ 1m

0 c▽

22 . . . c ▽ 1m

..

. ... . .. ... 0 0 . . . c▽

mm

     ∈

M_m(R)

by settingc▽

ij =cij +cji, fori < j,c▽ij = 0, fori > j, and c

▽

jj =cjj, for

j= 1, . . . , m.

Note that the inﬂation matrix T_ab− = [tij] ∈ Mm(Z) is the identity

matrix with an elementtab changed to−1.

Proof of Theorem 1.7. (a) Assume that

I ∈ {De(1)m,s,p,Dem,s(2),Dem,s,p(3) ,Dem,s,p(4) ,De(5)m,s,p,r,Dem,s,p,r(6) ,De(7)m,s}

with |I|=m+ 1≥5, as listed in Table 1.11. Our aim is to construct a matrixBI ∈Mm+1(Z) such that

( ˇGtr_e

Dm+ ˇGeDm) =B

tr

I ·(CItr+CI)·BI.

Since detCI = 1, the matrixCI isZ-invertible andCI−1=C

−1

I ·CItr·

C_I−tr. Note that, for B′′

I :=C

−tr

I , we have (C

−tr I +C

−1

I ) =B

′′tr

I ·(CItr+

CI) ·B′′I. Therefore, it is suﬃcient to construct a Z-invertible matrix

B′

I ∈Gl(m+ 1,Z) such that

B_I′tr·(C_I−tr+C_I−1)·B_I′ = ( ˇGtr_e

Dm+ ˇGeDm), because then, forBI :=BI′′·BI′ ∈Mm+1(Z), we have

( ˇGtr_e

Dm+ ˇGeDm) =B ′tr I ·(C

−tr I +C

−1

I )·B

′

I

=B_I′tr·(B_I′′tr·(C_Itr+CI)·BI′′)·B

′

I

= (BI′′·BI′)tr·(CItr +CI)·(BI′′·BI′)

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We construct the matrixB_I′ ∈M_m+1(Z) in six cases considered below, in the form of the product of inﬂation matrices T_ab−= [t′

ij]∈Mm+1(Z), by applying the inﬂation algorithm procedure.

First, we note that our assumptions imply that the matrix C_I−1 is upper triangular and coincides with the non-symmetric Gram matrix ˇG_∆

I that deﬁnes the Euler acyclic edge-bipartite graph ∆I ofI, see [51].

Case 1◦ _{Assume that} _I ₌_D_e(1)

m,s,p. The Euler bigraph ∆I of I, with

ˇ G_∆

I =C −1

I , has the shape

∆I =

1 s

s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

To simplify the notation, we set

t(1)_j :=t−_{j s}₊₃◦t−_{j s}₊₄, t(2)_j :=t−_{j s}₊₁◦t−_{j s}₊₂, T_j(1) :=T_{j s}−₊₃·T_{j s}−₊₄ andT_j(2) :=T_{j s}−₊₁·T_{j s}−₊₂.

The passage ∆I7→D∆I and the construction of the matrixBI′∈Mm+1(Z) can be illustrated as follows.

∆I

t(1)_s =⇒

1 s−1 s s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

t(1)_s−1

=⇒ 1 s−2 s

s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

t(1)_s−2 =⇒ · · ·t

(1) 1

=⇒ 1 s

s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

t(2)

p

=⇒ · · ·t (2)

m+1 =⇒ · · ·t

(2)

s+5

=⇒ 1 s s+ 1

s+ 2

s+ 3

s+ 4

p m

∗ m+ 1

p−1

s+ 5

Summing up, D∆I =Dem and we deﬁneB′I∈Gl(m+ 1,Z) to be the

matrix

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Case 2◦ _{Assume that} _I ₌ _D_e(2)

m,s. The Euler bigraph ∆I of I, with

ˇ G_∆

I =C −1

I , has the shape

∆I =

1 s

s+ 1

s+ 2

s+ 4 m−1

∗ m+ 1

m

s+ 3

We sett(1)_j =t−_{j s}₊₄ and T_j(1)=T_{j s}−₊₄. The passage ∆I 7→D∆I and the

construction of the matrixB′

I∈Mm+1(Z) can be illustrated as follows:

∆I

t(1)_s

=⇒ 1 s−1 s

s+ 1

s+ 2

s+ 4 m−1

∗ m+ 1

m

s+ 3

t(1)_s−2 =⇒ · · ·t

(1) 1

=⇒ 1 s

s+ 1

s+ 2

s+ 4 m−1

∗ m+ 1

m

s+ 3

We have D∆I =Dem and we deﬁneBI′ to be the matrix

B_I′ :=T₁(1)·T₂(1)· · ·T_s(1)−1·Ts(1)

Case3◦_{Assume that}_I_{is one of the posets}_I1₌_D_e(3)

m,s,pandI2 =Dem,s,p(4) .

Then D∆I =Dem andBI′ is deﬁned to be the matrix

BI′1 :=T

(3)

p−1· · ·T (3) 2 ·T

(3) 1 ·T

(1)

p+1·T (1)

p+2· · ·T (1)

s−1·Ts(1), ifI =I1 and

B_I′₂ :=T_s(2)₊₃· · ·T_p(2)−1·Tp(2)·T

(2)

1 · · ·Ts(2)·T

(3)

m−1· · ·T (3) 2 ·T

(3)

1 , ifI =I2,

whereT_j(1) =T_{j s}−₊₃,T_j(2) =T_{j p}−₊₁ and T_j(3)=T_{j m}− ₊₁.

Case 4◦ _If_I ₌_D_e(5)

m,s,p,r, thenD∆I =Dem and BI′ is deﬁned to be the

matrix

BI′ :=T

(2)

p+3· · ·T (2)

r−1·T (2)

m+1·Tm(2)· · ·T

(2)

r+1·Tr(2)·T

(1) 1 · · ·T

(1)

s−1·Ts(1),

whereT_j(1) =T_{j s}−₊₃ and T_j(2) =T_{j r}−.

Case 5◦ _If_I ₌_D_e(6)

m,s,p,r, thenD∆I =Dem and BI′ is deﬁned to be the

matrix

B_I′ :=T_s(2)₊₃· · ·T_r(2)−1·Tr(2)·T

(1) 1 · · ·T

(1)

s−1·Ts(1),

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Case 6◦ _If _I ₌ _D_e(7)

m,s, then D∆I = Dem and BI′ is deﬁned to be the

matrix

B′_I:=T_s(1)−2· · ·T (1) 2 ·T

(1) 1 ·T

(2)

s+3· · ·Tm(2)·T

(2)

m+1·T

−

m+1 1,

where T_j(1)=T_{j s}− and T_j(2)=T_{j s}−₊₁.

To ﬁnish the proof of (a), we recall from [50] that the Euclidean diagrams are principal and the existence of a Z-equivalence of a ﬁnite posetI with a Euclidean diagram Dem implies thatI is principal.

(b) The proof is a computational one and we proceed similarly as in the proof of (a). As earlier, we identify the poset I with its acyclic edge-bipartite graph ∆I and apply the inﬂation algorithm to ∆I. In this

way we obtain a matrix BI deﬁning theZ-equivalence of the posetI with

the Euclidean diagramEem, wherem=|I| −1. HenceI is principal and

the proof is complete.

Proof of Theorem 1.8. Assume thatm≤15 and I is a principal one-peak poset. We compute a complete list of such posets I, with |I| = m≤15, and their Coxeter-Euclidean typesDI by applying the inﬂation algorithm [50] and Algorithm 3.1 described in Section 3. In particular, the computations show that:

(a) there is no such a poset I thatDI =Ae_m−1,

(b) if DI =Dem−1, thenI is one of the posets listed in Theorem 1.7, (c) if 7≤m≤9 andDI =Ee_m−1, thenI is one of the posets listed in (c), (d), and (e) of Theorem 1.8.

3. Algorithms

In this section, we outline a description of computational algorithms we use in the proofs of Theorems 1.8 and 1.9. First, we discuss an algorithm used in computation of all principal posets with at most 15 elements, compare with [37] and [38].

Algorithm 3.1. Input: An integer 1≤n≤15.

Output:Finite setsprinc[1], . . . ,princ[n] of all connected non-negative posets of corank 1 encoded in the form of their incidence matrices. That is, the set princ[i] contains the principal posets with ivertices.

Step 1◦ _{Initialize the set}_princ_{[1] with the matrix} h ₁ i_∈_M

1(Z). Step 2◦ _{For every}_m _{from 2 to}_n:

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Step 2.2◦ For every posetJ ∈ princ[m−1], generate a list of all possible extensions of J to a poset with m vertices. In other words, generate a listWJ ∋w of all vectorsw= [w2, . . . , wm]∈ {0,1}m−1 such

that the matrix

CJw = "

1 w 0 CJ

#

= [cij]∈Mm(Z)

is an incidence matrix of a posetJw(matrix with the following transitivity

property:cij = 1 andcjs= 1 implies cis= 1, for 1≤i, j, s≤m).

Step 2.3◦ _{For every poset} _J _∈ _princ_[m₋_{1] and for every vector}

w∈ WJ, construct the matrix CJw ∈ Mm(Z) and add the poset Jw to the listcandidatem if the symmetric matrix CJ_w+C

tr

Jw is non-negative of corank at most 1, (by checking, for example, if all diagonal minors of the matrix are non-negative - the extended Sylvester criterion, see [16]). Step 2.4◦ _{Construct the set}_princ_{[m] by selecting non isomorphic}

posets from the listcandidatem (using the Hasse digraph representation

in order to test poset isomorphism).

Step 3◦ _{Remove from the sets} _princ_{[1], . . . ,}_princ_{[n] the matrices of}

posetsJw that are not connected (using a graph search algorithm, such

as breadth-ﬁrst search – BFS) or have the symmetric matrixCJw+C

tr Jw positive deﬁnite (for example, by checking if all principal minors ofCJw+ Ctr

Jw are positive - Sylvester criterion).

Step4◦ _{Return the sets} _princ_{[1], . . . ,}_princ_{[n] as a result.}

Remark 3.2. (a) Note that posets J, with |J| ≤ 3, need not to be checked, because any such a poset has the shape ◦, ◦ ◦, ◦ → ◦, ◦ ◦ ◦, ◦ ◦ → ◦, ◦ → ◦ → ◦, ◦ → ◦ ← ◦or ◦ ← ◦ → ◦and is positive. (b) Note that Step 2.3◦ _{and Step 2.4}◦ _{can be done simultaneously}

by adding to the setprinc[m] only these non-negative posets that have Hasse digraphs not isomorphic to the posets that are already in the set

princ[m].

(c) In our implementation of the algorithm we use theigraph pack-age (http://igraph.sourceforge.net/) to test digraph isomorphism in step 2.4◦_.

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(e) It is easy to eﬃciently implement the algorithm in a parallel environment (for example, Step 3◦ _{can be executed in parallel without}

need of any synchronisation).

In the proof of Theorem 1.9 we determine theZ-congruenceCI∼ZGˇDI,

for any principal poset I,|I| ≤15, of the Coxeter-Euclidean type

DI∈ {De_m, m≥4,Ee₆,Ee₇,Ee₈}.

We do it by constructing a matrixB ∈M_n(Z) withn=|I|, such thatCI =

Btr_·_G_ˇ

DI·B. We essentially use the following theorem and Procedure 3.6.

Theorem 3.3. (a) Let Dbe one of the Euclidean diagrams De_m, m≥4, e

E₆,Ee₇,Ee₈, with vertices numbered as in Introduction. LetRD be the set of

roots of the Euler quadratic formqD ofD. Then there exists a uniqueΦD

-mesh translation quiver Γ(RD,ΦD) of ΦD-orbits of the set RD (called a

ΦD-mesh geometry of roots ofD) such thatΓ(RD,ΦD)admits a principal

Coxeter Φ∆-orbit configuration Γ ˇ

GD

D of simple roots of the form presented

below(see also [47] and [48, Table 4.7]).

(b) If I is a connected principal poset, with n ≥ 5 elements, of the Coxeter-Euclidean type DI and B ∈Gl(n,Z) is such a matrix that

ˇ

GDI =B·GˇI·Btr, then the following diagram is commutative

Zn −−−−−→ΦD Zn

h

 

y≃ _h

  y≃

Zn −−−−−→ΦI Zn

(3.4)

where h=hB is the group isomorphism defined byhB(x) =x·B.

If RI :=RqI is the set of roots of the unit quadratic formqI :Z

I _→_Z

and Γ(RDI,ΦDI) is ΦDI-mesh geometry of roots of DI (with

princi-pal Coxeter ΦDI-orbit configuration Γ

ˇ

GDI

DI ), then the group isomorphism

h=hB carries it to the ΦI-mesh geometryΓ(RI,ΦI) (with a principal

Coxeter ΦI-orbit configuration ΓCII), induces the mesh translation quiver

isomorphism

h: Γ(RDI,ΦDI)

≃

−→Γ(RI,ΦI)

and the quiver isomorphism h : ΓGˇDI

DI

≃

−→ΓCI

I . Moreover, the matrix B

has the form

B=   

h(e1)

h(e2)

. . .

h(en)  

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Proof. Apply [46, Section 5], [46, Theorems 4.7 and Proposition 4.8] and their proofs, see also [47], [48, Lemma 4.3], and [50].

ΓG

e

E6 Γ

G

e

E7

be3 be2 be1 e4 e6 e7

e5

e8 e7 e6

e5

be4 be3 be2 be1

ΓG

e

E8

ΓG

e

D6

b

e3 be2 be1

e4

e9 e8 e7 e6 e5

be1 e e3 be2 ee4 ee5 e7 e6

b

ej=−ej e

ej=be1+· · ·+bej

Procedure 3.6.Assume thatI is a poset of the Coxeter-Euclidean type DI∈ {De_n−1,Ee6,Ee7,Ee8}, with n=|I|.

To construct a matrixB that deﬁnes aZ-equivalenceCI ∼ZGˇDI, we

proceed as follows.

1◦ _{First, by applying the mesh toroidal algorithm described in [46,}

Proposition 4.5] and [47] (see also [48, Lemma 4.6 and Algorithm 4.8.2]), we construct the ΦI-mesh translation quiver Γ(RI,ΦI), with a principal

Coxeter ΦI-orbit conﬁguration ΓCII of simple roots ofqI, together with a

mesh quiver isomorphismh: ΓGˇDI

DI →Γ CI

I .

2◦ _{Next, we deﬁne the matrix}_B _{by setting}

B =   

h(e1)

h(e2)

.. .

h(en)   ,

as in (3.5) and [48, Lemma 4.3].

3◦ _{Finally, we check the matrix equality ˇ}_G

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Remark 3.7.Procedure 3.6 has implementations in Maple and Python, with an assistance of programming and graphics in Java.

Algorithm 3.8. Input:A non-negative posetI, withnelements, encoded in the form of incidence matrix CI ∈Mn(Z).

Output: Reduced Coxeter number ˇcI ∈Z.

Step 1◦ _{Initialize the symbolic vector}_v _{:= [v}

1, . . . , vn].

Step 2◦ _{Calculate the Coxeter matrix Cox}

I :=−CI·CI−tr.

Step 3◦ _For_r_{= 1,}_2,_{3, . . .}

Step 3.1◦ _Calculate _w_:=_v_·_Coxr I−v.

Step3.2◦ _If_q

I(w) :=w·CI·wtr equals zero then stop the calculations

and return ˇcI as a result.

Remark 3.9.By [51, Theorem 18], Algorithm 3.8 returns ˇcI in a ﬁnite

number of steps.

We use the following algorithm computing the incidence defect of any principal poset.

Algorithm 3.10. Input: A principal poset I, with nelements, encoded in the form of incidence matrix CI ∈Mn(Z).

Output: The incidence defect ∂I :Zn→Z.

Step 1◦ Initialize the symbolic vector v:= [v1, . . . , vn] and calculate

the Coxeter matrix CoxI:=−CI·C_I−tr.

Step 2◦ Using Algorithm 3.8, compute the reduced Coxeter number ˇ

cI ∈Zand calculate the vectorw= [w1, . . . , wn] :=v·CoxˇcI−v.

Step3◦ _{Compute the vector}_h_∈_Zn_{such that Ker}_q

I =Z·hby solving

inZ the system of Z-linear equations (CI+CItr)·vtr = 0 (for example,

using the procedureisolve of the LinearAlgebra package in MAPLE). Step 4◦ _{Solve the symbolic system of linear equations}_λ_·_h

I=w(in Z), for the unknown λ, and return computedλas a result.

Outline of proof of Theorem 1.9. Assume thatI is a one-peak prin-cipal poset with m:=|I| ≤15. By Theorem 1.8, I is one of the four ﬁve elements posets listed in 1.8(a), one of the 2.115 posets with 6≤m≤15 of the shape listed in Table 1.11, one of the 31 posets listed in Table 4.1, one of the 132 posets listed in Table 4.2, or one of the 422 posets listed in [21]. By applying Algorithms 3.8 and 3.10, for each of the posets I from this ﬁnite list, we calculate its reduced Coxeter number ˇcI and the

incidence defect ∂I:ZI →Z. In particular, we show that∂I is non-zero.

It follows from [51, Theorem 1.18 (c)] that the Coxeter number cI ofI is

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(a) By applying Procedure 3.6 as in [17, Section 7], for each of the posetsI from the ﬁnite list described above, we construct aZ-invertible matrixBI such that ˇGDI =BI·CI·BItr. By settingB :=BItr we get the

required equality ˇGDI =Btr·CI·B.

The proof is long and a computational one. Here we do not present a complete proof, but we illustrate its idea on examples of several posets. The proof for the remaining posets of the list is analogous.

First we illustrate an application of Procedure 3.6 for the posetsDe(7)₅,3 andJEe6

29 presented in Theorem 1.8 (b) and Table 4.1, respectively. 1◦ _{If we enumerate the elements of the poset}_J _:=_D_e(7)

5,3 from Theorem 1.8 (b) as follows

J =

1 2 3

4 5

6

then the incidence matrix CJ ∈ M6(Z), the Coxeter matrix and the incidence quadratic formqJ :Z6 →Zare given as follows

CJ =

    

1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1

   

 , CoxJ =     

0 0 0 1 0 −1 0 0 0 0 1 −1

−1 1 0 0 1 −1 0 1 0 0 0 −1 0 0 1 0 0 −1

−1 0 1 −1 1 −1     ,

qJ(x) = x21+x22+x23+x24+x52+x26+x1x2+x3(x1+x2+x4) +(x1+x4)x5+ (x1+x2+x3+x4+x5)x6

= x1+1₂x2+1₂x3+1₂x5+ 1₂x6 2

+3₄x2+1₃x3−1₃x5+1₃x6 2

+2₃x3+3₄x4−1

4x5+ 14x6 2

+5₈x4+x5+3₅x62+2₅x2₆.

It follows thatqJ :Z6 → Z is non-negative and KerqJ = Z·hJ, where

hJ = [1,0,−1,1,−1,0]; hence J is principal. By Theorem 1.7, the poset

J is of Euclidean typeDJ =De₅. A routine calculations show that

• coxJ(t) =t6+t5−t4−2t3−t2+t+ 1 = cox_e_D

5(t),

• cˇJ = 6, cJ =∞,

• ∂eJ(v) = 2(v1+v2+v3+v4+v5)·hJ,∂J(v) = 2(v1+v2+v3+v4+v5),

and the setRJ of roots of qJ has the decomposition

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where the ﬁnite set

RredJ ={v∈Z6;v1 = 0 and qJ(v) = 1}

is a reducer ofRJ in the sense of [47]. One shows that

|Rred

J |= 40,|∂J0RredJ |= 10,|∂

−

JR red

J |=|∂J+R red J |= 15,

where∂_J0Rred J ,∂

−

JRredJ , and∂J+RredJ is the subset ofRredJ consisting of the

zero-defect vectors, negative-defect vectors, and positive-defect vectors, respectively. It is easy to see that the negative-defect part∂−_JRJ ofRJ is

the set

∂_J−RJ =∂_J−RredJ +Z·hJ

and admits the following ΦJ-translation quiver structure Γ(∂_J−RJ,ΦJ):

00b1001 0000b10 1b1b11b11 10b10b10

0b10b101 000b100 00b10b11 00b11b11 1b1b11b21 1b100b10 2b1b21b21 10b21b11

b100000 0b100b11 1b100b11 1b1b10b11 00b1000 10b21b21 10b11b21 2b1b22b31

0000b11 0b10000 10b10b11 00b11b10 O1

O2,O3

O4,O5

O6

Hence, by applying Procedure 3.6 and using the vectors in Γ(∂_J−RJ,ΦJ)

marked by the framed boxes, we obtain theZ-invertible matrix

BJ =

    

0 0 0 0 1 0

0 0 1 −1 1 −1 0 0 0 1 −1 0 0 1 −1 0 0 0 0 −1 0 0 0 0 1 −1 0 0 −1 1

    

such that ˇGDJ = ˇG_e_D

5 =BJ ·

ˇ

GJ·BJtr.

2◦ _{If we enumerate the points of the poset}_I _:=_JEe6

29 from Table 4.1 as follows

I =

1 2

3 4

5 6 7

(23)

then the incidence matrix CI ∈ M7(Z), the Coxeter matrix and the

incidence quadratic formqI:Z7 →Zhave the forms

CI =

      

1 1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1

      

, CoxI=

      

0 0 1 0 1 0 −1

−1 0 0 1 0 1 −1

−1 1 0 0 0 1 −1

−1 1 1 0 0 1 −1

−1 1 0 1 0 0 −1

−1 1 0 1 1 0 −1

−2 1 0 1 0 1 −1        ,

qI(x) =x21+x22+x23+x24+x52+x26+x27+x1x2+ (x1+x3)x4 + (x1+x5)x6+ (x1+x2+x3+x4+x5+x6)x7

=x1+1₂x2+ 1₂x4+1₂x6+1₂x7 2

+3₄x2−1₃x4−1₃x6+ 1₃x7 2

+x3+1₂x4+1₂x7 2

+₁₂5 x4− 4₅x6+1₅x7 2

+x5+1₂x6+1₂x7 2

+₂₀3 (x6+x7)2.

It follows that qI :Z7 → Z is non-negative and KerqI = Z·hI, where

hI = [1,−1,0,−1,0,−1,1]; henceI is principal. By theorem 1.7 the poset

I is of Euclidean type DI =Ee₆. A routine calculations show that

• coxI(t) =t7+t6−2t4−2t3+t+ 1 = cox_e_E

6(t),

• cˇI = 6,cI =∞,

• ∂eI(v) = (v1−v7)·hI,∂I(v) =v1−v7,

and the setRI of roots ofqI has the decomposition

RI =RredI +Z·hI,

where the ﬁnite set

Rred

I ={v∈Z7;v1= 0 and qJ(v) = 1}

is a reducer ofRI in the sense of [47]. One shows that

|RredI |= 72,|∂I0RredI |= 14,|∂

−

I R red

I |=|∂I+R red I |= 29,

where∂_I0Rred I ,∂

−

I RredI , and∂I+RredI is the subset ofRredI consisting of the

zero-defect vectors, negative-defect vectors, and positive-defect vectors, respectively. It is easy to see that the negative-defect part∂−_IRI ofRI is

the set

∂_I−RI =∂_I−RredI +Z·hI

(24)

b10b11b110 b1100000 0b100001 00b10b101

b10b11b111 b21b11b110 b1000001 0b1b10b102

b1000010 0000b101 b10b11000 000b1001

b21b11010 b1000b111 b10b11b101 b10b10001

b21b11b111 b20b11b111 b10b10b102 b1b1b10b102

b2101b110 b10b11001 b10b10b111 b1000b101

b1001000 00b10001 b1000b110 00000b11 O1

O2

O3

O4

O5

O6

O7

Hence, by applying Procedure 3.6 and using the vectors in Γ(∂_I−RI,ΦI)

distinguished by the framed boxes, we obtain theZ-invertible matrix

BI =

      

1 0 0 0 1 −1 0 0 0 1 0 0 0 −1 1 0 0 −1 0 0 0

−1 0 −1 1 0 0 0 0 0 0 0 −1 0 1 0 −1 0 0 0 0 1

−1 1 0 0 0 0 0       

such that ˇGDI = ˇG_e_E

6 =BI·

ˇ GI·BItr.

Now we collect a ﬁnal eﬀect of the computational Procedure 3.13 (presented below) applied to principal posets of type Ee₈. For example,

consider the principal posets

I1:=J1Ee8,I2 :=J31Ee8,I3 :=J48Ee8,I4 :=J147Ee8,I5 :=J412Ee8,I6 :=J422Ee8 from Theorem 1.8 (e). Using Procedure 3.13, computer calculations yield the followingZ-invertible matrices B1, . . . , B6 ∈M₉(Z):

B1 =      

5 3 4 −2 −1 −2 −2 −1 −2 5 4 3 −1 −2 −2 −1 −2 −2 6 3 4 −2 −2 −1 −2 −2 −2

−8 −5 −5 2 3 2 3 2 3

−3 −1 −2 1 1 1 0 1 1

−2 −2 −2 1 1 0 1 1 1

−3 −2 −1 1 0 1 1 1 1

−2 −1 −2 0 1 1 1 1 1

−3 −2 −2 1 1 1 1 1 0

    

, B2 =      

1 1 0 −2 −2 −1 1 1 0 1 1 −1 −1 −2 −1 1 0 1 1 1 −1 −2 −1 −1 0 1 1

−2 −1 1 2 2 2 −1 −1 −1 0 −1 1 1 0 0 0 0 0

−1 0 0 1 1 0 0 0 −1 0 0 0 0 1 1 0 −1 0 0 −1 0 1 1 1 −1 0 −1

−1 −1 1 1 1 0 0 −1 0

(25)

B3=      

0 0 1 0 0 0 1 0 −1 0 0 0 0 1 1 1 −1 −1 0 0 0 1 0 0 0 0 −1 0 0 0 −1 0 −1 −1 0 2 0 0 0 0 −1 0 0 0 1 0 0 0 −1 0 0 −1 1 0 0 1 −1 0 0 −1 0 0 1 1 −1 0 0 0 0 0 0 0

−1 0 0 0 0 0 0 0 0

    

, B4=      

0 2 2 −1 −1 −2 −1 0 0 1 1 1 0 0 −2 −2 0 1 0 1 2 0 −1 −2 −2 1 0

−1 −2 −2 0 1 3 3 −1 0 0 0 −1 0 0 1 1 0 0 0 −1 −1 0 1 1 1 0 −1 0 −1 −1 0 0 2 1 −1 0 0 −1 −1 1 0 0 1 0 0

−1 0 −1 0 1 1 0 0 0

     ,

B5=      

2 2 2 1 −1 −1 −2 −1 −1 1 3 1 1 −1 −1 −2 −1 −1 1 3 2 2 −1 −1 −3 −2 0

−2 −4 −2 −2 1 1 4 2 1 0 −1 −1 −1 0 1 1 1 0

−1 −2 −1 0 1 0 1 1 0 0 −1 −1 −1 1 0 1 0 1

−1 −1 0 −1 0 1 1 1 0

−1 −2 −1 −1 1 1 2 0 0

    

, B6=      

1 1 1 0 1 −1 0 −1 −1 1 2 1 0 0 0 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1

−1 −2 −1 −1 −1 1 1 1 2

−1 −1 0 0 −1 0 1 1 0 0 0 −1 0 0 0 0 0 1 0 −1 −1 −1 0 1 0 1 0

−1 0 0 0 0 0 0 0 1

−1 −1 −1 0 0 1 1 0 0

     ,

for the posetI1,I2,I3,I4,I5, andI6, respectively. One checks that ˇGDI =

ˇ G_e_E

8 =Bj·

ˇ GIj ·B

tr

j , forj = 1, . . . ,6.

(b) By (a) and [46, Theorems 4.7 and 4.8], for each of the posets I from the ﬁnite list described above, there is a Z-invertible matrix B such that ˇGDI =B·GˇI·Btr and, by [46, Proposition 4.8], we have the

commutative diagram (3.4), withD=DI andh=hB. Hence, in view of

our remarks made in the ﬁrst part of proof, (b) is a consequence of the results in [46, Section 5].

(c) Given a principal posetI such thatm :=|I| ≤15, ﬁx a matrix BI ∈Gl(m,Z) such that ˇGDI =Btr_I ·CI·BI, as in (a). By applying the

technique used in [17, Section 7], one can construct a matrixC∈Gl(m,Z) such that ˇGtr

DI =Ctr·GˇDI·C andC2 =E. Then

ˇ

Gtr_DI =B_Itr·C_Itr·BI and ˇGtrDI =Ctr·GˇDI·C=Ctr·BItr·CI·BI·C.

Hence we get

Ctr I =B

−tr

I ·Ctr·BItr·CI·BI·C·BI−1=C tr

·CI·C,

whereC =BI·C·BI−1. It follows thatC ∈Gl(m,Z) andC

2

=E. This ﬁnishes the proof of (c) and of Theorem 1.9.

Corollary 3.11.Assume that I is a principal one-peak poset such that m =|I| ≤ 15. Then there exists a ΦI-mesh translation quiver of roots

Γ(RI,ΦI) satisfying the conditions stated in Theorem 3.3 (b).

Proof. LetDI be the Coxeter-Euclidean type of I. By Theorem 1.9 (a), there exists a matrixB ∈Gl(m,Z) such that ˇGDI =B·GˇI·Btr and the

diagram (3.4) is commutative, whereh=hB is the group isomorphism

deﬁned by hB(x) = x·B. Therefore the corollary is a consequence of