A horizontal mesh algorithm for posets with positive Tits form

Збережено в:

Бібліографічні деталі
Дата:	2016
Автори:	Kaniecki, M., Kosakowska, J., Malicki, P., Marczak, G.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2016
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/155733
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	A horizontal mesh algorithm for posets with positive Tits form / M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 240-261. — Бібліогр.: 34 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-155733
record_format	dspace
spelling	irk-123456789-1557332019-06-18T01:26:45Z A horizontal mesh algorithm for posets with positive Tits form Kaniecki, M. Kosakowska, J. Malicki, P. Marczak, G. 2016 Article A horizontal mesh algorithm for posets with positive Tits form / M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 240-261. — Бібліогр.: 34 назв. — англ. 1726-3255 2010 MSC:68R10, 05C50, 06A07, 15A63. http://dspace.nbuv.gov.ua/handle/123456789/155733 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
format	Article
author	Kaniecki, M. Kosakowska, J. Malicki, P. Marczak, G.
spellingShingle	Kaniecki, M. Kosakowska, J. Malicki, P. Marczak, G. A horizontal mesh algorithm for posets with positive Tits form Algebra and Discrete Mathematics
author_facet	Kaniecki, M. Kosakowska, J. Malicki, P. Marczak, G.
author_sort	Kaniecki, M.
title	A horizontal mesh algorithm for posets with positive Tits form
title_short	A horizontal mesh algorithm for posets with positive Tits form
title_full	A horizontal mesh algorithm for posets with positive Tits form
title_fullStr	A horizontal mesh algorithm for posets with positive Tits form
title_full_unstemmed	A horizontal mesh algorithm for posets with positive Tits form
title_sort	horizontal mesh algorithm for posets with positive tits form
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/155733
citation_txt	A horizontal mesh algorithm for posets with positive Tits form / M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 2. — С. 240-261. — Бібліогр.: 34 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT kanieckim ahorizontalmeshalgorithmforposetswithpositivetitsform AT kosakowskaj ahorizontalmeshalgorithmforposetswithpositivetitsform AT malickip ahorizontalmeshalgorithmforposetswithpositivetitsform AT marczakg ahorizontalmeshalgorithmforposetswithpositivetitsform AT kanieckim horizontalmeshalgorithmforposetswithpositivetitsform AT kosakowskaj horizontalmeshalgorithmforposetswithpositivetitsform AT malickip horizontalmeshalgorithmforposetswithpositivetitsform AT marczakg horizontalmeshalgorithmforposetswithpositivetitsform
first_indexed	2025-07-14T07:58:25Z
last_indexed	2025-07-14T07:58:25Z
_version_	1837608369677926400
fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 22 (2016). Number 2, pp. 240–261 © Journal “Algebra and Discrete Mathematics” A horizontal mesh algorithm for posets with positive Tits form Mariusz Kaniecki, Justyna Kosakowska, Piotr Malicki and Grzegorz Marczak Communicated by D. Simson Abstract. Following our paper [Fund. Inform. 136 (2015), 345–379], we define a horizontal mesh algorithm that constructs a Φ̂I -mesh translation quiver Γ(R̂I , Φ̂I) consisting of Φ̂I -orbits of the finite set R̂I = {v ∈ Z I ; q̂I(v) = 1} of Tits roots of a poset I with positive definite Tits quadratic form q̂I : ZI → Z. Under the assumption that q̂I : ZI → Z is positive definite, the algorithm constructs Γ(R̂I , Φ̂I) such that it is isomorphic with the Φ̂D-mesh translation quiver Γ(RD, ΦD) of Φ̂D-orbits of the finite set RD of roots of a simply laced Dynkin quiver D associated with I. 1. Introduction The paper is mainly devoted to the existence of a Φ̂I -mesh root system Γ(R̂I , Φ̂I) in the sense of [30], that is, a Φ̂I -mesh translation quiver Γ(R̂I , Φ̂I) consisting of Φ̂I -orbits of the set R̂I = {v ∈ Z I ; q̂I(v) = 1} of Tits roots of a finite poset I = (I, �) with positive quadratic Tits form q̂I : ZI → Z, where Φ̂I : Z I → Z I is the Coxeter-Tits transformation associated with I in [9, 28, 29, 34]. The reader is also referred to [14], [16], and [30]-[34] for analogous existence mesh root system theorems in the setting of positive edge-bipartite graphs and non-negative posets. 2010 MSC: 68R10, 05C50, 06A07, 15A63. Key words and phrases: poset, combinatorial algorithm, Dynkin diagram, mesh geometry of roots, quadratic form. M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 241 Our interest in the Φ̂I -mesh analysis of Φ̂I -orbits of the set R̂I of Tits roots is motivated by applications of matrix representations of posets in representation theory, where a matrix representation of a partially ordered set T = {p1, . . . , pn}, with a partial order �, means a block matrix M = [M1\|M2\| . . . \|Mn] (over a field K) of size d∗ × (d1, . . . , dn) determined up to all elemen- tary row transformations, elementary column transformations within each of the substrips M1, M2, . . . , Mn, and additions of linear combina- tions of columns of Mi to columns of Mj , if pi ≺ pj , see Nazarova and Roiter [22]. In [9], Drozd proves that T has only a finite number of direct-sum-indecomposable representetions if and only if its quadratic Tits form q(x1, . . . , xn, x∗) = x2 1 + · · ·+x2 n +x2 ∗ + ∑ pi≺pj xixj −x∗(x1 + · · ·+xn) (1.1) is weakly positive (i.e., q(a1, . . . , an, a∗) > 0, for all non-zero vectors (a1, . . . , an, a∗) with integer non-negative coefficients). In this case, there exists an indecomposable representation M of size d∗ × (d1, . . . , dn) if and only if (d1, . . . , dn, d∗) is a root of q, i.e., q(d1, . . . , dn, d∗) = 1, see [10] and [26, Chapter 10] for more details. In [5,6], Bondarenko and Stepochkina give a complete list of posets T with positive Tits form q(x1, . . . , xn, x∗); it consists of four infinite series and 108 exceptional posets, up to duality (see also [11,12] for an alternative proof). Throughout this paper, we assume that I = (I, �) is a poset (i.e., a finite partially ordered set). We denote by max I the set of all maximal elements of I and let I− = I \ max I. For i, j ∈ I, we write i ≺ j if i � j and i 6= j. Moreover, for i, j ∈ I, we write i → j, if i ≺ j and there is no s in I such that i ≺ s ≺ j. We denote by Z the ring of integers and by MI(Z) the ring of I by I square matrices with integer coefficients. Usually we define a poset I by presenting its Hasse quiver H(I) = (H0(I), H1(I)), with the set of vertices H0(I) = I and the set H1(I) of arrows i → j defined earlier, for i, j ∈ I. 242 A horizontal mesh algorithm for posets Following [26,28,29,34], with any poset I, we associate the incidence matrix CI = [cij ] ∈ MI(Z) and the Tits matrix ĈI ∈ MI(Z), where cij = { 1 if i � j, 0 otherwise, (1.2) and ĈI = [ Ctr I− U 0 E ] , (1.3) where U = [uiw]i∈I−;w∈max I and uiw = { −1 if i � w, 0 otherwise, (1.4) Following [11,32,34], we call a poset I positive, if the symmetric Gram matrix GI := 1 2(ĈI + Ĉtr I ) is positive definite. The following two sets of vectors associated with a poset I are playing an important role in the representation theory of algebras: the set of Tits roots R̂I := {v ∈ Z n; v · ĈI · vtr = 1} (1.5) and the set of Euler roots RI := {v ∈ Z n; v · CI · vtr = 1} (1.6) of a poset I, where CI = C−1 I (1.7) see [10, 21,24,26]. We recall from [30] that the sets of Tits roots R̂I and Euler roots RI of I are finite, if I is positive. Moreover, if I is assumed to be connected then the sets R̂I and RI are irreducible and reduced root systems in the sense of Bourbaki, see [24, p. 40] and [16], for more details. By [29, Corollary 1.8], given a positive poset I, the root systems R̂I and RI are isomorphic, and we denote by DI the common Coxeter-Dynkin type of these root systems. One should note that DI is one of the simply laced Dynkin diagrams (see [24, p. 40] and [16]) Am : •1−−−−•2−−−−•3−−−− . . . −−−−•−−−−•m (m vertices, m > 1); Dm : •2 \| •1−−−−•3 −−−−•4−−−− . . . −−−−•−−−−•m (m vertices, m > 4); E6 : •4 \| •1−−−−•2−−−− •3 −−−−•5−−−−•6; M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 243 E7 : •4 \| •1−−−−•2−−−− •3 −−−−•5−−−−•6−−−−•7; E8 : •4 \| •1−−−−•2−−−− •3 −−−−•5−−−−•6−−−−•7−−−−•8; It follows from [16] that the Dynkin diagram DI can be determined by applying the inflation algorithm constructed in [20] and [32]. We recall from [29] that the square matrix ĈoxI := −ĈI · Ĉ−tr I ∈ Mn(Z), (1.8) is called the Coxeter-Tits matrix of I. Here Ĉtr I is the transpose of ĈI , and we set Ĉ−tr I := (Ĉtr I )−1. The charactristic polynomial coxI(t) := det(t · E − ĈoxI) ∈ Z[t], (1.9) of ĈoxI is called the Coxeter polynomial of I, the group isomorphism Φ̂I : Zn → Z n, x 7→ Φ̂I(x) := x · ĈoxI , (1.10) is called the Coxeter-Tits transformation of I, and the Coxeter number cI of I is the minimal integer r > 1 such that Φ̂r I is the identity map on Z n. If Φ̂r I 6= id, for all r > 1, we set cI = ∞. Recall also that the matrix CoxI := −CI · C −tr I ∈ Mn(Z), (1.11) is called the Coxeter-Euler matrix of I, and the group isomorphism ΦI : Zn → Z n, x 7→ ΦI(x) := x · CoxI , (1.12) is called the Coxeter-Euler transformation of I. Following an idea introduced in [30, 31], we study in the paper a Φ̂I -mesh root system structure Γ(R̂I , Φ̂I) on the set of roots R̂I ⊆ Z n of any connected positive poset I, with n > 2 vertices, where Φ̂I : Zn → Z n is the Coxeter-Tits transformation defined by the Tits matrix ĈI ∈ Mn(Z) of I. One of the main aims of the paper is to present a combinatorial algorithm that constructs a Φ̂I -mesh root system structure Γ(R̂I , Φ̂I) (see Definition 2.13) on the finite set of all Φ̂I -orbits of the irreducible root system R̂I . Moreover, in Corollary 4.6, we prove that the Coxeter polynomial coxI(t) and the Coxeter number cI of such poset I depend only on the simply laced Dynkin type DI of R̂I and coxI(t) coincides 244 A horizontal mesh algorithm for posets with the Coxeter polynomial coxDI(t) of the Dynkin diagram DI, see [29, Example 3.12]. The idea of construction of our horizontal mesh algorithm is inspired by the method of a construction of postprojective component in some categories of modules (see [7,8,15,26]). However, this well-known method computes only a mesh quiver consisting of the positive vectors. In the present paper we show that our modification of this algorithm computes a Φ̂I -mesh root system structure Γ(R̂I , Φ̂I) for the set R̂I of all roots (not only positive roots). We recall that one of the motivations for the study of a Φ̂I -mesh root system structure Γ(R̂I , Φ̂I) comes from the poset representation theory (see [9, 10,21,24,26,28,29,34]). The sets of roots and Tits roots are playing an important role in many areas of mathematics. In the representation theory of finite dimensional algebras over a field the roots control categories of indecomposable modules for a large classes of algebras (see [1–3,24,25]), while in the theory of Lie groups and Lie algebras they are connected with root spaces (see [4, 13]). Moreover, they control linear bases, generators and relations of Ringel-Hall algebras (see [18,19]). Recall that in [17] the Tits roots were applied to get a classification of two-peak sincere posets of finite prinjective type. Therefore, it is of importance to have efficient combinatorial algorithms that compute roots, Tits roots and Φ̂I -mesh root system structures. 2. Preliminaries Throughout this paper all posets are assumed to be connected. 2.1. Unit quadratic forms associated with a poset Let I be a poset. By a Tits quadratic form and an Euler quadratic form of I we mean the unit quadratic forms q̂I , qI : ZI → Z defined by the formulae q̂I(x) = x · ĈI · xtr, qI(x) = x · CI · xtr. It is easy to see that q̂I(x) = ∑ i∈I x2 i + ∑ i≺j∈I− xjxi − ∑ w∈max I ∑ i≺w xixw. (2.1) M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 245 Note also that the Tits quadratic form q(x1, . . . , xn, x∗) (1.1) of a partially ordered set T = {p1, . . . , pn} (defined by Drozd [9]) coincides with the Tits form q̂I(x1, . . . , xn, x∗) (2.1) of the one-peak poset I = T ∗ ∪ {∗} obtained from T by adding a unique maximal element ∗. Recall from [29, Corollary 1.8] that one of the quadratic forms q̂I , qI is positive if and only if both of them are positive. Moreover, in this case we have qI(x) = ∑ i∈I x2 i − ∑ i→j xixj + ∑ i◭j c• ijxixj , (2.2) where the relation i ◭ j holds if there exists a minimal commutativity relation w′ − w′′ in I, where w′, w′′ are paths with the source i and the terminus j and c• ij is the maximal number of linearly independent minimal commutativity relations w′ − w′′ in I with the source i and the terminus j, see Corollary 1.8, Remark 3.5 and Proposition 4.2 in [29]. Remark 2.3. Let I be a positive poset. The formula (2.2) implies that the matrix CI = (cij) satisfies the non-cycle condition defined in [14]. Let us recall this definition. With a poset I we associate the biquiver QI = (Q0, Q1) with the set of vertices Q0 = I. Moreover, there are −cij solid arrows i // j , if cij < 0 and cij broken arrows i // j , if cij > 0. Let Q = (Q0, Q1) be a biquiver. (a) We say that a (unoriented) cycle (x1, x2, . . . , xn, x1) in Q is simple if for all i, j ∈ {1, . . . , n}, i 6= j we have xi 6= xj . (b) We say that a simple cycle (x1, x2, . . . , xn, x1) is chordless if for any arrow (xi, xj) we have i = j ± 1 (wherein 1 ≡ n + 1). (c) Further, consider a simple cycle in Q of the form // �� // // OO (2.4) The biquiver Q satisfies the non-cycle condition, if every simple chordless cycle in Q containing a broken arrow has the form (2.4). (d) Given a poset I the matrix CI = (cij) satisfies the non-cycle condi- tion, if the biquiver QI satisfies this condition. For all i ∈ I, denote by p̂i the Tits-projective vector associated with i, i.e. p̂i is defined by the formula p̂i(j) =    1 for i = j; 1 for i � j ∈ max I; 0 otherwise. (2.5) 246 A horizontal mesh algorithm for posets Let P̂ = P̂(I) = {p̂i ; i ∈ I} be the set of all Tits-projective vectors. For all i ∈ I, denote by r̂i the Tits-radical vector associated with i, i.e. r̂i is defined by the formula r̂i(j) =    1 for all i → j; 1 for i ≺ j ∈ max I; 0 otherwise. (2.6) Let R̂ad = R̂ad(I) = {r̂i ; i ∈ I} be the set of all Tits-radical vectors. Let i ∈ I and let r̂i be the corresponding Tits-radical vector. Consider the convex subposet I-supp(r̂i) = conv.hull{j ∈ I ; r̂i(j) 6= 0} of I. Let I1, . . . , Iki be the set of all connected components of the Hasse quiver of I-supp(ri). We define the vectors r̂1 i , . . . , r̂ki i by the following formula: r̂t i(j) = { r̂i(j) if i ∈ It; 0 otherwise (2.7) for all t ∈ {1, . . . , ki}. We denote by R̂adcomp the set of vectors r̂1 i , . . . , r̂ki i , where i ∈ I. It is known that p̂i ∈ R̂I and r̂ j i ∈ R̂I , for all i, j, see [23,26,27]. Denote by pi the Euler-projective vector associated with i, i.e. pi is defined by the formula pi(j) = { 1 for all i � j; 0 otherwise. (2.8) Let P = P(I) = {pi ; i ∈ I} be the set of all Euler-projective vectors. For all i ∈ I, denote by ri the Euler-radical vector associated with i, i.e. ri is defined by the formula: ri = pi − ei. (2.9) M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 247 Let Rad = Rad(I) = {ri ; i ∈ I} be the set of all Euler-radical vectors. Let i ∈ I and let ri be the corresponding Euler-radical vector. Consider the convex subposet I-supp(ri) = {j ∈ I ; ri(j) 6= 0} of I. Let I1, . . . , Iki be the set of all connected components of the Hasse quiver of I-supp(ri). We define the vectors r1 i , . . . , rki i by the following formula: rt i(j) = { ri(j) if i ∈ It; 0 otherwise (2.10) for all t ∈ {1, . . . , ki}. We denote by Radcomp the set of vectors r1 i , . . . , rki i , where i ∈ I. It is known that pi ∈ RI and r j i ∈ RI , for all i, j, see [14,23,26,27]. 2.2. Mesh translation quivers in Z n We recall from [30,31] the following definitions (see also [14]). They are inspired by the definition of the Auslander-Reiten quiver of an algebra (see [1, 2]). Let Φ : Zn → Z n be a group automorphism (e.g. the Coxeter-Tits transformation Φ̂I or the Coxeter transformation ΦI of a poset I). A Φ- orbit Φ − Orb(v) = {Φk(v)}k∈Z of a vector v ∈ Z n will be visualised as an infinite graph: . . . Φ(v) v Φ−1(v) Φ−2(v) . . . Definition 2.11. Let Φ : Zn → Z n be a non-trivial group automorphism (e.g. the Coxeter-Tits transformation Φ̂I or the Coxeter transformation ΦI of a poset I). We say that the vectors u, v1, . . . , vs, w ∈ Z n form a Φ-mesh starting from u and terminating at w, if the following two conditions are satisfied: (i) u = Φ(w) and u + w = s∑ i=1 vi, (ii) the vectors v1, . . . , vs are pairwise different, lie in pairwise different orbits of Φ and none of them lies in the Φ-orbit of u. 248 A horizontal mesh algorithm for posets A Φ-mesh we visualise as the following triangular quiver: v1 �� v2 $$ Φ(w) = u �� BB 99 w vs @@ (2.12) Definition 2.13. Let n > 2, let Φ : Zn → Z n be a non-trivial group automorphism and let R be a Φ-invariant subset of Z n (e.g. R = R̂I if Φ = Φ̂I or R = RI if Φ = ΦI). We say that R admits a geometry of Φ-mesh quiver if there exists a quiver ~R = ( ~R0, ~R1) with ~R0 = R, such that ~R together with the bijection Φ : R → R induced by Φ is a triangular translation quiver Γ(R, Φ) (see [1, IV.4.7]) with the following property: for every vector w ∈ R, the full convex subquiver containing the vertices w and Φ(w) is a Φ-mesh of the form (2.12), and if v′ 1 �� v′ 2 ## Φ(w) = u �� CC :: w v′ s′ @@ is a Φ-mesh, then s′ = s and v1 = v′ 1, . . . , vs = v′ s, up to permutation of the set {1, . . . , s}. Definition 2.14. Let Γ(R, Φ) be a Φ-mesh quiver in Z n as in Definition 2.13. A slice in Γ(R, Φ) is a full convex connected subquiver Σ = (Σ0, Σ1) of Γ(R, Φ) such that for any v ∈ R the set Φ − Orb(v) ∩ Σ0 contains exactly one element. Example 2.15. Consider the posets I and I ′ defined by the following Hasse quivers: 1 2 AA 3 OO 4, ^^ 1 2 OO 4 @@ 3 ^^ M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 249 respectively. Note that the set R̂I ⊆ Z 4 of Tits roots of I consists of 24 vectors. One easily see that the set R̂I admits the following geometry of Φ̂I -mesh quiver Γ(R̂I , Φ̂I) (we identify the vectors in frames): 1010 �� 1101 �� 0010 �� 1̂0̂10 �� 1̂̂10̂1 �� 00̂10 �� 1000 �� @@ �� 2111 @@ �� 1111 @@ �� 1̂000 @@ �� 2̂̂1̂1̂1 @@ �� 1̂̂1̂1̂1 @@ �� 1000 1100 @@ 1011 @@ 0100 @@ 1̂̂100 @@ 1̂0̂1̂1 @@ 0̂100 @@ 1001 GG 1110 GG 0001 GG 1̂00̂1 GG 1̂̂1̂10 GG 000̂1 GG Moreover the set R̂I′ of Tits roots of I ′ consists of 24 vectors and admits the following geometry of Φ̂I′-mesh quiver Γ(R̂I′ , Φ̂I′) (we identify the vectors in frames): 1101 �� 0010 �� 0̂101 �� 1̂00̂1 �� 00̂10 �� 010̂1 �� 1100 �� @@ �� 1011 @@ �� 0̂111 @@ �� 1̂̂100 @@ �� 1̂0̂1̂1 @@ �� 01̂1̂1 @@ �� 1100 0101 @@ 0001 @@ 0̂110 @@ 1̂0̂10 @@ 000̂1 @@ 01̂10 @@ 1000 GG 0100 GG 1̂100 GG 1̂000 GG 0̂100 GG 1̂1̂1̂1 GG 1000 GG Here we set â = −a, for a ∈ N. In the algorithm presented in Section 3 first we look for a slice canditate Σ in Γ(R̂I , Φ̂I). Then the remaining part of Γ(R̂I , Φ̂I) is easy to compute. In Γ(R̂I , Φ̂I) presented in Example 2.15 the quiver 1010 1000 && 88 �� 1100 1001 is a slice. Applying definition of a Φ̂I -mesh we can construct now the Φ̂I -mesh translation quiver Γ(R̂I , Φ̂I) by knitting Φ̂I -meshes as follows: 1010 $$ a 1000 && 88 �� b >> �� e @@ �� 1100 :: c >> 1001 DD d GG Indeed, we have 250 A horizontal mesh algorithm for posets b = (1 0 1 0) + (1 1 0 0) + (1 0 0 1) − (1 0 0 0), a = b − (1 0 1 0), c = b − (1 1 0 0), d = b − (1 0 0 1), e = a + c + d − b, and so on. Note that Φ̂I(a) = (1010), Φ̂I(b) = (1000), Φ̂I(c) = (1100), Φ̂I(d) = (1001), and Φ̂I(e) = b. 3. A horizontal mesh algorithm The idea of construction of a horizontal mesh algorithm that we present in this section is inspired by a construction of the postprojective component of the Auslander-Reiten quiver of an algebra or a poset (see [7, 8, 15]). We would like to stress that the algorithm (I, P̂, R̂ad, R̂adcomp, k) 7→ Γ̂ := Γ(R̂I , Φ̂I) presented below, called a horizontal mesh algorithm, associates to an arbi- trary poset I, with initial data P̂, R̂ad, R̂adcomp, k, a Φ̂I -mesh translation quiver Γ(R̃I , Φ̂I) such that Γ̂ defines a Φ̂I -mesh root system structure Γ(R̂I , Φ̂I) on the set R̂I of Tits roots of I, in case when I is positive (see Theorem 4.4 for a proof). The algorithm is a modification of a correspond- ing horizontal mesh algorithm presented in [14], for positive edge-bipartite graphs. Algorithm 3.1. Input: A system (I, P̂, R̂ad, R̂adcomp, k), where • I = (I, �) is a poset such that I = {1, . . . , n}, • P̂ = {p̂1, . . . , p̂n} is the set of Tits-projective vectors, • R̂ad = {r̂1, . . . , r̂n} is the set of Tits-radical vectors, • R̂adcomp = {r̂1 1, . . . , r̂k1 1 , . . . , r̂1 n, . . . , r̂kn n }, where r̂ j i are defined by formula 2.7, • k ∈ N. Output: The quiver Γ̂ = Γ(R̂I , Φ̂I). Step 1. Inductively, we construct the following data: • ordered lists L̂[i], for any i = 1, . . . , n; • quivers Ĝi = (Ĝi 0, Ĝi 1), for i = 0, 1, 2, . . .; • quivers Γ̂i = (Γ̂i 0, Γ̂i 1), for i = 0, 1, 2, . . .; • sets P̂0 ⊆ P̂1 ⊆ . . . ⊆ P̂k ⊆ P̂ = {p̂1, . . . , p̂n}; in the following way. Step 1.1. For any i = 1, . . . , n, we put L̂[i] := [p̂i]. M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 251 Step 1.2. Let P̂0 = Ĝ0 0 = {p̂i ∈ P̂ ; i ∈ max I} and Γ̂0 0 = Γ̂0 1 = Ĝ0 1 = ∅. Step 1.3. We put Ĉ1 = {p̂i ; r̂i 6= 0 and r̂ j i ∈ Ĝ0 0 for all j = 1, . . . , ki}, P̂1 := Ĝ1 0 := Ĝ0 0 ∪ Ĉ1 and Γ̂1 0 = Γ̂1 1 = ∅ Ĝ1 1 = {r̂ j i → p̂i ; for all p̂i ∈ Ĉ1 and all j = 1, . . . , ki}. Step 1.4. Assume that, for i = 0, . . . , m − 1, m > 2, data Ĝi, Γ̂i, P̂i are constructed. We set P̂ ′ m = {p̂i ∈ P̂ \ P̂m−1 ; r̂i 6= 0 and r̂ j i ∈ Ĝm−1 0 for all j = 1, . . . , ki} and P̂m = P̂ ′ m ∪ P̂m−1. We define Ĉm = P̂ ′ m ∪ {z = −x + ∑ x→y y ; y ∈ Ĉm−1}, Ĝm 0 = Ĝm−1 0 ∪ Ĉm and Ĝm 1 = {r̂ j i → p̂i ; for all p̂i ∈ Ĉm and all j = 1, . . . , ki}∪ ∪{y → z ; for all y such that z = −x + ∑ x→y y}. Moreover, if P̂m 6= P̂, z = −x + ∑ x→y y and x ∈ L̂[i], then we add z at the end of the list L̂[i] and delete the first element of the list L̂[i]. If P̂m 6= P̂, then we set Γ̂m 0 = Γ̂m 1 = ∅; otherwise we set Γ̂m 0 = Γ̂m−1 0 ∪ Ĉm and Γ̂m 1 = Γ̂m−1 1 ∪ {y → z ; for all y → z ∈ Ĝm 1 such that y, z ∈ Γ̂m−1 0 ∪ Γ̂m 0 }. Moreover, if P̂m = P̂, z = −x + ∑ x→y y and x ∈ L̂[i], then we add z at the end of the list L̂[i]. Step 2. If m = k, we finish and set Γ̂ = Γ̂k. 252 A horizontal mesh algorithm for posets Remark 3.2. In this algorithm the set P̂ of Tits-projective and the set R̂ad of Tits-radical can be replaced by the set P of Euler-projective vectors and the set Rad of Euler-radical vectors, respectively, i.e. as an input we put (I, P, Rad, Radcomp, k). In this way, we obtain an algorithm that for a positive poset I constructs a ΦI -mesh root system structure Γ(RI , ΦI), see Theorem 4.4. In the description of Algorithm 3.1 with input (I, P, Rad, Radcomp, k) the data computed in Step 1 we denote by adding a dash over a corre- sponding symbol (e.g. we replace L̂[i] by L[i], Γ̂k by Γ k etc.). We illustrate Algorithm 3.1 by the following example. Example 3.3. Consider the following poset 1 2 @@ 3 ^^ 4 ^^ @@ Note that P̂ = {p̂1 = (1, 0, 0, 0), p̂2 = (1, 1, 0, 0), p̂3 = (1, 0, 1, 0), p̂4 = (1, 0, 0, 1)}, R̂ad = {r̂2 = (1, 0, 0, 0), r̂3 = (1, 0, 0, 0), r̂4 = (1, 1, 1, 0)} and R̂adcomp = {r̂1 2 = r̂2, r̂1 3 = r̂3, r̂1 4 = r̂4}. We set k = 5. Applying Algorithm 3.1 to (I, P̂, R̂ad, R̂adcomp, k) we get 1100 %% 0010 %% 00̂11 1000 99 �� 1110 %% 99 �� 0001 99 %% �� 1001 99 1̂000 1010 BB 0100 BB 0̂101 Indeed: m=0: P̂0 = Ĝ0 0 = {p̂1 = (1, 0, 0, 0)}; Γ̂0 0 = Γ̂0 1 = Ĝ0 1 = ∅; L̂[1] = [p̂1], L̂[2] = [p̂2], L̂[3] = [p̂3], L̂[4] = [p̂4]. m=1: Ĉ1 = {p̂2 = (1, 1, 0, 0), p̂3 = (1, 0, 1, 0)}, P̂1 = Ĝ1 0 = {p̂1, p̂2, p̂3}, Ĝ1 1 = {(p̂1, p̂2), (p̂1, p̂3)}, Γ̂1 0 = Γ̂1 1 = ∅. L̂[1] = [p̂1], L̂[2] = [p̂2], L̂[3] = [p̂3], L̂[4] = [p̂4]. m=2: P̂ ′ 2 = ∅, P̂2 = P̂1, Ĉ2 = {p̂2 + p̂3 − p̂1 = (1, 1, 1, 0)}, Ĝ2 0 = Ĝ1 0 ∪ Ĉ2, Ĝ2 1 = {(p̂2, (1, 1, 1, 0)), (p̂3, (1, 1, 1, 0))}, Γ̂2 0 = Γ̂2 1 = ∅. L̂[1] = [(1, 1, 1, 0)], L̂[2] = [p̂2], L̂[3] = [p̂3], L̂[4] = [p̂4]. M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 253 m=3: P̂ ′ 3 = {p̂4 = (1, 0, 0, 1)}, P̂3 = {p̂1, p̂2, p̂3, p̂4}, Ĉ3 = {p̂4, (0, 1, 0, 0), (0, 0, 1, 0)}, Ĝ3 0 = Ĝ2 0 ∪ Ĉ3, Ĝ3 1 = {((1, 1, 1, 0), p̂4), ((1, 1, 1, 0), (0, 1, 0, 0)), ((1, 1, 1, 0), (0, 0, 1, 0))}, Γ̂3 0 = Ĉ3, Γ̂3 1 = ∅. L̂[1] = [(1, 1, 1, 0)], L̂[2] = [(0, 0, 1, 0)], L̂[3] = [(0, 1, 0, 0)], L̂[4] = [p̂4]. m=4: P̂ ′ 4 = ∅, P̂4 = P̂, Ĉ4 = {(0, 0, 0, 1)}, Ĝ4 0 = Ĝ3 0 ∪ Ĉ3, Ĝ4 1 = {(p̂4, (0, 0, 0, 1)), ((0, 1, 0, 0), (0, 0, 0, 1)), ((0, 0, 1, 0), (0, 0, 0, 1))}, Γ̂4 0 = Γ̂3 0 ∪ Ĉ3, Γ̂4 1 = Ĝ4 1. L̂[1] = [(1, 1, 1, 0), (0, 0, 0, 1)], L̂[2] = [(0, 0, 1, 0)], L̂[3] = [(0, 1, 0, 0)], L̂[4] = [p̂4]. m=5: P̂ ′ 5 = ∅, P̂5 = P̂ , Ĉ5 = {(0, 0, 1̂, 1), (1̂, 0, 0, 0), (0, 1̂, 0, 1)}, Ĝ5 0 = Ĝ4 0 ∪ Ĉ4, Ĝ5 1 = {((0, 0, 0, 1), (0, 0, 1̂, 1)), ((0, 0, 0, 1), (1̂, 0, 0, 0)), ((0, 0, 0, 1), (0, 1̂, 0, 1))}, Γ̂5 0 = Γ̂4 0∪Ĉ4, Γ̂5 1 = Γ̂4 1∪Ĝ5 1. L̂[1] = [(1, 1, 1, 0), (0, 0, 0, 1)], L̂[2] = [(0, 0, 1, 0), (0, 0, 1̂, 1)], L̂[3] = [(0, 1, 0, 0), (0, 1̂, 0, 1)], L̂[4] = [p̂4, (1̂, 0, 0, 0)]. Now we apply Algorithm 3.1 to (I, P, Rad, Radcomp, k). Note that P = {p1 = (1, 0, 0, 0), p2 = (1, 1, 0, 0), p3 = (1, 0, 1, 0), p4 = (1, 1, 1, 1)}, Rad = {r1 = (0, 0, 0, 0), r2 = (1, 0, 0, 0), r3 = (1, 0, 0, 0), r4 = (1, 1, 1, 0)}, and Radcomp = {r1 1 = r1, r1 2 = r2, r1 3 = r3, r1 4 = r4}. We set k = 5 and get 1100 && 0010 && 0101 1000 88 �� 1110 %% 88 �� 0111 88 %% �� 1111 99 1̂000 1010 BB 0100 BB 0011 Indeed: m=0: P0 = G 0 0 = {p1 = (1, 0, 0, 0)}; Γ 0 0 = Γ 0 1 = G 0 1 = ∅; L[1] = [p1], L[2] = [p2], L[3] = [p3], L[4] = [p4]. m=1: C1 = {p2 = (1, 1, 0, 0), p3 = (1, 0, 1, 0)}, P1 = G 1 0 = {p1, p2, p3}, G 1 1 = {(p1, p2), (p1, p3)}, Γ 1 0 = Γ 1 1 = ∅. L[1] = [p1], L[2] = [p2], L[3] = [p3], L[4] = [p4]. m=2: P ′ 2 = ∅, P2 = P1, C2 = {p2 + p3 − p1 = (1, 1, 1, 0)}, G 2 0 = G 1 0 ∪ C2, G 2 1 = {(p2, (1, 1, 1, 0)), (p3, (1, 1, 1, 0))}, Γ 2 0 = Γ 2 1 = ∅. L[1] = [(1, 1, 1, 0)], L[2] = [p2], L[3] = [p3], L[4] = [p4]. m=3: P ′ 3 = {p4 = (1, 1, 1, 1)}, P3 = {p1, p2, p3, p4}, C3 = {p4, (0, 1, 0, 0), (0, 0, 1, 0)}, G 3 0 = G 2 0 ∪ C3, G 3 1 = {((1, 1, 1, 0), p4), ((1, 1, 1, 0), (0, 1, 0, 0)), ((1, 1, 1, 0), (0, 0, 1, 0))}, Γ 3 0 = C3, Γ 3 1 = ∅. L[1] = [(1, 1, 1, 0)], L[2] = [(0, 0, 1, 0)], L[3] = [(0, 1, 0, 0)], L[4] = [p4]. 254 A horizontal mesh algorithm for posets m=4: P ′ 4 = ∅, P4 = P, C4 = {(0, 1, 1, 1)}, G 4 0 = G 3 0 ∪ C3, G 4 1 ={(p4, (0, 1, 1, 1)), ((0, 1, 0, 0), (0, 1, 1, 1)), ((0, 0, 1, 0), (0, 1, 1, 1))}, Γ 4 0 = Γ 3 0 ∪ C3, Γ 4 1 = G 4 1. L[1] = [(1, 1, 1, 0), (0, 1, 1, 1)], L[2] = [(0, 0, 1, 0)], L[3] = [(0, 1, 0, 0)], L[4] = [p4]. m=5: P ′ 5 = ∅, P5 = P, C5 = {(0, 1, 0, 1), (1̂, 0, 0, 0), (0, 0, 1, 1)}, G 5 0 = G 4 0 ∪ C4, G 5 1 = {((0, 1, 1, 1), (0, 1, 0, 1)), ((0, 1, 1, 1), (1̂, 0, 0, 0)), ((0, 1, 1, 1), (0, 0, 1, 1))}, Γ 5 0 = Γ 4 0 ∪ C4, Γ 5 1 = Γ 4 1 ∪ G 5 1. L[1] = [(1, 1, 1, 0), (0, 1, 1, 1)], L[2] = [(0, 0, 1, 0), (0, 1, 0, 1)], L[3] = [(0, 1, 0, 0), (0, 0, 1, 1)], L[4] = [p4, (1̂, 0, 0, 0)]. 4. Correctness of Algorithm 3.1 Following [28,29], we define a group isomorphism σ0 I : ZI → Z I (4.1) by the formula σ0 I (x) = x · C0 I , where C0 I is the reduced incidence matrix C0 I = [ CI− 0 0 E ] , (4.2) where E is the identity matrix. By [29, Proposition 3.13], σ0 I gives Z- equivalence of q̂I and qI , i.e. qI(σ0 I (x)) = q̂(x). Lemma 4.3. For any poset I and for all i ∈ I, we have (σ0 I )−1(pi) = p̂i and (σ0 I )−1(ri) = r̂i, where p̂i, r̂i, pi and ri are projective and radical vectors defined in Section 2.1, and σ0 I : ZI → Z I is the isomorphism (4.1). Proof. The proof is straightforward. Theorem 4.4. Assume that I is a connected positive poset. Let Γ̂ be the quiver constructed by Algorithm 3.1 with input (I, P̂, R̂ad, R̂adcomp, k) with k large enough (e.g. k = \|R̂I \|). The following conditions are satisfied. (a) P̂ = ⋃ k P̂k, in particular there exists m such that P̂m = P̂. (b) The sequence Γ̂0 ⊆ Γ̂1 ⊆ . . . stabilizes. (c) R̂I = ⋃ m Γ̂m 0 . (d) L̂[1], . . . , L̂[n] are the Φ̂I-orbits in R̂I of the Coxeter-Tits transfor- mation Φ̂I . (e) The Φ̂I-mesh translation quiver Γ̂ = Γ(R̂I , Φ̂I) defines a Φ̂I-mesh root system structure Γ(R̂I , Φ̂I) on the set R̂I of Tits roots of I. M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 255 Proof. Assume that I is a connected positive poset. We apply Algo- rithm 3.1 to the system (I, P, Rad, Radcomp, k) defined in Remark 3.2. The formula (2.2) implies that the Euler matrix CI = C−1 I satisfies the non-cycle condition defined in [14], see Remark 2.3. Therefore, [14, Theo- rem 4.13] and [14, Section 5] yield: (ā) P = ⋃ k Pk, in particular there exists m such that Pm = P. (b̄) The sequence Γ 0 ⊆ Γ 1 ⊆ . . . stabilizes. (c̄) RI = ⋃ m Γm 0 . (d̄) L[1], . . . , L[n] are the ΦI -orbits in RI of the Coxeter transformation ΦI . (ē) The ΦI -mesh translation quiver Γ = Γ(RI , ΦI) defines a ΦI -mesh root system structure Γ(RI , ΦI) on the set RI of Euler roots of I. By Lemma 4.3, we have (σ0 I )−1(pi) = p̂i and (σ0 I )−1(ri) = r̂i, see (4.1). It is easy to verify that the automorphism (σ0 i )−1 sends P, Rad, Radcomp to P̂, R̂ad, R̂adcomp, respectively. From [29, Proposition 3.13], it follows that Φ̂I = (σ0 I )−1 ◦ ΦI ◦ σ0 I . Now, applying the linearity of σ0 I , it is easy to deduce that the conditions (ā)-(ē) imply the conditions (a)-(e), and the theorem follows. Remark 4.5. It follows from the proof of Theorem 4.4 that the Φ̂I - mesh quiver Γ(RI , Φ̂I) is the image of Γ(RI , ΦI) via the automorphism (σ0 I )−1 : ZI → Z I (4.1). We refer also to [11,12] for a discussion of ΦI -mesh quivers of one-peak posets. Corollary 4.6. Let I be a positive connected poset and let DI be the Coxeter-Dynkin type of the root system R̂I . The Coxeter polynomial coxI(t) is equal to the Coxeter polynomial coxDI(t) of the Dynkin diagram DI and the Coxeter number cI is equal to the Coxeter number cDI of the Dynkin diagram DI; they are listed in [29, Example 3.12]. Proof. By [14, Theorem 1.10] there exists a Z-invertible matrix B ∈ MI(Z) such that CoxDI = B · CoxI · B−1, where CoxDI is the Coxeter matrix associated with the simply laced Dynkin diagram DI. Moreover by [29, Proposition 3.13], we have ĈoxI = C0 I · CoxI · (C0 I )−1. Now it is easy to deduce that coxI(t) = coxDI(t) and cI = cDI . Example 4.7. Consider the poset I given by the Hasse quiver 1 2oo 3oo 4oo (4.8) 256 A horizontal mesh algorithm for posets By applying Algorithm 3.1 to (I, P, Rad, Radcomp, k = 6) we get the Φ̂I -mesh quiver Γ(RI , ΦI): 1111 �� 1̂000 �� 0̂100 �� 00̂10 �� 000̂1 �� 1111 1110 �� AA 0111 AA �� 1̂̂100 AA �� 0̂1̂10 AA �� 00̂1̂1 AA �� 1110 AA 1100 �� AA 0110 AA �� 0011 AA �� 1̂̂1̂10 AA �� 0̂1̂1̂1 AA �� 1100 AA 1000 AA 0100 AA 0010 AA 0001 AA 1̂̂1̂1̂1 AA 1000 AA where vectors in frames lying in the same orbit are identified. Moreover, by applying Algorithm 3.1 to (I, P̂, R̂ad, R̂adcomp, k = 6) we get the Φ̂I -mesh quiver Γ(R̂I , Φ̂I): 1001 �� 1̂000 �� 0̂100 �� 01̂10 �� 001̂1 �� 1001 1010 �� AA 0001 AA �� 1̂̂100 AA �� 00̂10 AA �� 010̂1 AA �� 1010 AA 1100 �� AA 0010 AA �� 0̂101 AA �� 1̂0̂10 AA �� 000̂1 AA �� 1100 AA 1000 AA 0100 AA 0̂110 AA 00̂11 AA 1̂00̂1 AA 1000 AA Note that the Φ̂I -mesh quiver Γ(RI , ΦI) is isomorphic with the Φ̂I - mesh quiver Γ(R̂I , Φ̂I) via the authomorphism σ0 I : Z4 → Z 4 (4.1). Example 4.9. Consider the poset I given by the Hasse quiver 1 2 @@ 3 ^^ 4 OO 77 5 OO (4.10) By applying Algorithm 3.1 we get the Φ̂I -mesh quiver Γ(RI , ΦI): 00100 �� 11111 �� 1̂0000 �� 00010 �� 1̂̂1̂1̂10 �� 00̂10̂1 �� 0̂10̂10 �� 11000 �� 00100 11110 �� 00101 �� 01010 �� 1̂̂1000 �� 00̂100 �� 1̂̂1̂1̂1̂1 �� 10000 �� 000̂10 �� 11110 11100 �� CC II 11211 �� CC II 01111 �� CC II 1̂0010 �� CC II 1̂̂1̂100 �� CC II 1̂̂1̂2̂1̂1 �� CC II 0̂1̂1̂1̂1 �� CC II 100̂10 �� CC II 11100 CC �� II 11101 �� CC 01110 �� CC 00111 �� CC 1̂0̂100 �� CC 1̂̂1̂10̂1 �� CC 0̂1̂1̂10 �� CC 00̂1̂1̂1 �� CC 10100 �� CC 11101 10101 CC 01000 CC 00110 CC 00001 CC 1̂0̂10̂1 CC 0̂1000 CC 00̂1̂10 CC 0000̂1 CC 10101 CC M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 257 and the Φ̂I -mesh quiver Γ(R̂I , Φ̂I): 00100 �� 10̂111 �� 1̂0000 �� 0̂1̂110 �� 1̂00̂10 �� 0000̂1 �� 001̂10 �� 11000 �� 00100 10010 �� 00001 �� 00̂110 �� 1̂̂1000 �� 00̂100 �� 1̂01̂1̂1 �� 10000 �� 011̂10 �� 10010 11100 �� CC II 10011 �� CC II 00̂111 �� CC II 1̂̂1̂110 �� CC II 1̂̂1̂100 �� CC II 1̂00̂1̂1 �� CC II 001̂1̂1 �� CC II 111̂10 �� CC II 11100 CC �� II 11001 �� CC 00010 �� CC 0̂1̂111 �� CC 1̂0̂100 �� CC 1̂̂100̂1 �� CC 000̂10 �� CC 011̂1̂1 �� CC 10100 �� CC 11001 10001 CC 01000 CC 0̂1010 CC 00̂101 CC 1̂000̂1 CC 0̂1000 CC 010̂10 CC 0010̂1 CC 10001 CC Note that the Φ̂I -mesh quiver Γ(RI , ΦI) is isomorphic with the Φ̂I - mesh quiver Γ(R̂I , Φ̂I) via the authomorphism σ0 I : Z5 → Z 5 (4.1). Example 4.11. Consider the poset I given by the Hasse quiver 1 3oo yy 5oo 6oo tt2 4oo By applying Algorithm 3.1 to (I, P, Rad, Radcomp, k = 24) we get the Φ̂I -mesh quiver Γ(RI , ΦI): 111111 !! 001000 !! 000010 !! 000101 !! 0̂10̂100 !! 1̂0̂1000 !! 0̂1̂10̂10 �� 112111 == �� 001010 == �� 000111 == �� 0̂10001 == �� 1̂̂1̂1̂100 == �� 1̂̂1̂20̂10 == �� 101010 !! 011111 !! 0̂10000 !! 000011 !! 1̂̂1̂10̂10 !! 0̂1̂1̂100 !! 112110 FF == !! 112121 FF == !! 001111 FF == !! 0̂10011 FF == !! 1̂̂1̂1001 FF == !! 1̂̂2̂2̂1̂10 FF == !! 1̂̂1̂2̂1̂10 JJ DD �� 011110 == !! 101111 == !! 001011 == !! 1̂̂1̂1000 == !! 0̂1̂1001 == !! 1̂̂1̂1̂1̂10 == !! 011010 == 000100 == 101011 == 1̂00000 == 0̂1̂1000 == 000001 == 1̂̂1̂1̂1̂1̂1 DD �� 000̂100 !! 1̂0̂10̂1̂1 !! 100000 !! 011000 !! 00000̂1 !! 111111 !! 0̂1̂1̂1̂10 == �� 1̂0̂1̂1̂1̂1 == �� 00̂10̂1̂1 == �� 111000 == �� 01100̂1 == �� 111110 == �� 112111 1̂0̂10̂10 !! 0̂1̂1̂1̂1̂1 !! 010000 !! 0000̂1̂1 !! 111010 !! 011100 !! 101010 JJ DD �� 1̂̂1̂2̂1̂2̂1 FF == !! 00̂1̂1̂1̂1 FF == !! 0100̂1̂1 FF == !! 11100̂1 FF == !! 122110 FF == !! 112110 FF == !! 1̂̂1̂2̂1̂1̂1 == !! 00̂10̂10 == !! 000̂1̂1̂1 == !! 01000̂1 == !! 111100 == !! 112010 == !! 011110DD 00̂1000 == 0000̂10 == 000̂10̂1 == 010100 == 101000 == 011010 == where vectors in frames lying in the same orbit are identified. 258 A horizontal mesh algorithm for posets Moreover, by applying Algorithm 3.1 to (I, P̂, R̂ad, R̂adcomp, k = 24) we get the Φ̂I -mesh quiver Γ(R̂I , Φ̂I): 110001 !! 001000 !! 00̂1010 !! 0000̂11 !! 0̂10̂100 !! 1̂0̂1000 !! 0̂100̂10 �� 111001 == �� 000010 == �� 00̂1001 == �� 0̂10̂1̂11 == �� 1̂̂1̂1̂100 == �� 1̂̂1̂10̂10 == �� 100010 !! 010001 !! 0̂10000 !! 00̂1̂101 !! 1̂̂100̂10 !! 0̂1̂1̂100 !! 111110 FF == !! 110011 FF == !! 000001 FF == !! 0̂1̂1̂101 FF == !! 1̂̂1̂1̂1̂11 FF == !! 1̂̂2̂1̂1̂10 FF == !! 1̂̂1̂1̂1̂10 DD JJ �� 010110 == !! 100001 == !! 000̂101 == !! 1̂̂1̂1000 == !! 0̂1̂1̂1̂11 == !! 1̂̂10̂1̂10 == !! 010010 == 000100 == 100̂101 == 1̂00000 == 0̂1̂1000 == 000̂1̂11 == 1̂̂1000̂1 DD �� 000̂100 !! 1̂0010̂1 !! 100000 !! 011000 !! 00011̂1 !! 110001 !! 0̂10̂1̂10 == �� 1̂0000̂1 == �� 00010̂1 == �� 111000 == �� 01111̂1 == �� 110110 == �� 111001 1̂000̂10 !! 0̂1000̂1 !! 010000 !! 00110̂1 !! 110010 !! 011100 !! 100010 JJ DD �� 1̂̂100̂1̂1 FF == !! 00000̂1 FF == !! 01110̂1 FF == !! 11111̂1 FF == !! 121110 FF == !! 111110 FF == !! 1̂̂1̂100̂1 == !! 0000̂10 == !! 00100̂1 == !! 01011̂1 == !! 111100 == !! 111010 == !! 010110DD 00̂1000 == 0010̂10 == 00001̂1 == 010100 == 101000 == 010010 == Note that the Φ̂I -mesh quiver Γ(RI , ΦI) is isomorphic with the Φ̂I - mesh quiver Γ(R̂I , Φ̂I) via the authomorphism σ0 I : Z6 → Z 6 (4.1). Example 4.12. Consider the poset I given by the Hasse quiver 1 3oo xx 5oo xx 6oo 2 4oo By applying Algorithm 3.1 to (I, P, Rad, Radcomp, k = 24) we get theΦ̂I -mesh quiver Γ(RI , ΦI): 111111 !! 001000 !! 000110 !! 101011 !! 1̂00000 !! 0̂1̂1000 !! 000̂100 �� 112111 == �� 001110 == �� 101121 == �� 001011 == �� 1̂̂1̂1000 == �� 0̂1̂1̂100 == �� 011110 !! 101111 !! 001010 !! 000111 !! 0̂10̂100 !! 1̂0̂1000 !! 112110 FF == !! 112221 FF == !! 102121 FF == !! 001121 FF == !! 0̂10011 FF == !! 1̂̂1̂1̂100 FF == !! 1̂̂1̂2̂1̂10 JJ DD �� 101110 == !! 112121 == !! 001111 == !! 0̂10010 == !! 000011 == !! 1̂̂1̂1̂1̂10 == !! 000100 == 101010 == 011111 == 0̂10000 == 000010 == 000001 == 1̂̂1̂1̂1̂1̂1 DD M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 259 �� 1̂0̂10̂10 !! 0̂1̂1̂1̂1̂1 !! 010000 !! 0000̂10 !! 00000̂1 !! 111111 !! 1̂0̂1̂1̂10 == �� 1̂̂1̂2̂1̂2̂1 == �� 00̂1̂1̂1̂1 == �� 0100̂10 == �� 0000̂1̂1 == �� 111110 == �� 112111 0̂1̂1̂1̂10 !! 1̂0̂1̂1̂1̂1 !! 00̂10̂10 !! 000̂1̂1̂1 !! 010100 !! 101000 !! 011110 JJ DD �� 1̂̂1̂2̂2̂2̂1 FF == !! 1̂0̂2̂1̂2̂1 FF == !! 00̂1̂1̂2̂1 FF == !! 0100̂1̂1 FF == !! 111100 FF == !! 112110 FF == !! 1̂̂1̂2̂1̂1̂1 == !! 00̂1̂1̂10 == !! 1̂0̂1̂1̂2̂1 == !! 00̂10̂1̂1 == !! 111000 == !! 011100 == !! 101110DD 00̂1000 == 000̂1̂10 == 1̂0̂10̂1̂1 == 100000 == 011000 == 000100 == where vectors in frames lying in the same orbit are identified. Moreover, by applying Algorithm 3.1 to (I, P̂, R̂ad, R̂adcomp, k = 24) we get the Φ̂I -mesh quiver Γ(R̂I , Φ̂I): 110001 !! 001000 !! 00̂1010 "" 100̂101 !! 1̂00000 !! 0̂1̂1000 !! 000̂100 �� 111001 == �� 000010 == �� 10̂1̂111 << �� 000̂101 == �� 1̂̂1̂1000 == �� 0̂1̂1̂100 == �� 010010 !! 100001 !! 000̂1100 "" 00̂1001 !! 0̂10̂100 !! 1̂0̂1000 !! 111010 FF == !! 110011 FF == !! 100̂111 EE << "" 00̂1̂111 FF == !! 0̂1̂1̂101 FF == !! 1̂̂1̂1̂100 FF == !! 1̂̂1̂10̂10 DD JJ �� 100010 == !! 110̂111 == !! 000001 << "" 0̂1̂1̂110 == !! 00̂1̂101 == !! 1̂̂100̂10 == !! 000100 == 100̂110 == 010001 << 0̂10000 == 00̂1̂110 == 0000̂11 == 1̂̂1000̂1 DD �� 1̂001̂10 !! 0̂1000̂1 !! 010000 !! 0011̂10 !! 00001̂1 !! 110001 !! 1̂000̂10 == �� 1̂̂101̂1̂1 == �� 00000̂1 == �� 0111̂10 == �� 00110̂1 == �� 1̂10010 == �� 111001 0̂100̂10 !! 1̂0000̂1 !! 0001̂10 !! 00100̂1 !! 010100 !! 101000 !! 010010 JJ DD �� 1̂̂100̂1̂1 FF == !! 1̂001̂1̂1 FF == !! 0011̂1̂1 FF == !! 01110̂1 FF == !! 111100 FF == !! 111010 FF == !! 1̂̂1̂100̂1 == !! 0000̂10 == !! 1̂011̂1̂1 == !! 00010̂1 == !! 111000 == !! 011100 == !! 100010DD 00̂1000 == 0010̂10 == 1̂0010̂1 == 100000 == 011000 == 000100 == Note that the Φ̂I -mesh quiver Γ(RI , ΦI) is isomorphic with the Φ̂I - mesh quiver Γ(R̂I , Φ̂I) via the authomorphism σ0 I : Z6 → Z 6 (4.1). References [1] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1. Techniques of Representation Theory, London Math. Soc. Student Texts 65, Cambridge Univ. Press, Cambridge-New York, 2006. [2] M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge Univ. Press, 1995. 260 A horizontal mesh algorithm for posets [3] M. Barot, A characterization of positive unit forms, II, Bol. Soc. Mat. Mexicana (3) 7 (2001), 13–22. [4] M. Barot, D. Kussin and H. Lenzing, The Lie algebra associated to a unit form, J. Algebra 296 (2007), 1–17. [5] V. M. Bondarenko and M. V. Stepochkina, On posets of width two with positive Tits form, Algebra and Discrete Math. 2 (2005), 20–35. [6] V. M. Bondarenko and M. V. Stepochkina, (Min, max)-equivalence of partially ordered sets and quadratic Tits form (in Russian, English Summary), Zb. Pr. Inst. Mat. NAN Ukr. 2, No. 3, 2005, 18–58 (Zbl. 1174.16310). [7] K. Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), 1–12. [8] S. Brenner, Unfoldings of algebras, Proc. London Math. Soc. (3) 62 (1991), 242–274. [9] J. A. Drozd, Coxeter transformations and representations of partially ordered sets, Funct. Anal. Appl. 8(1974), 219–225. [10] P. Gabriel and A. V. Roiter, Representations of Finite Dimensional Algebras, in: Algebra VIII, Encyclopaedia of Math. Sci., vol. 73, Springer-Verlag, 1992. [11] M. Ga̧siorek and D. Simson, One-peak posets with positive Tits quadratic form, their mesh quivers of roots, and programming in Maple and Python, Linear Algebra Appl. 436 (2012), 2240–2272. [12] M. Ga̧siorek and D. Simson, A computation of positive one-peak posets that are Tits sincere, Colloq. Math. 127 (2012), 83–103. [13] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag, New York-Heilderberg-Berlin, 1972. [14] M. Kaniecki, J. Kosakowska, P. Malicki and G. Marczak, A horizontal mesh algorithm for a class of edge-bipartite graphs and their matrix morsifications, Fund. Inform. 136 (2015), 345–379. [15] S. Kasjan and J. A. de la Peña, Constructing the preprojective components of an algebra, J. Algebra 179 (1996), 793–807. [16] S. Kasjan and D. Simson, Mesh algorithms for Coxeter spectral classification of Cox-regular edge-bipartite graphs with loops, I. Mesh root systems, Fundam. Inform. 139 (2015), 153-184. [17] J. Kosakowska, A classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations, Colloq. Math. 87 (2001), 27–77. [18] J. Kosakowska, A specialization of prinjective Ringel-Hall algebra and the associated Lie algebra, Acta Mathematica Sinica, English Series, 24 (2008), 1687–1702. [19] J. Kosakowska, Lie algebras associated with quadratic forms and their applications to Ringel-Hall algebras, Algebra and Discrete Math. 4 (2008), 49–79. [20] J. Kosakowska, Inflation algorithms for positive and principal edge-bipartite graphs and unit quadratic forms, Fund. Inform. 119 (2012), 149–162. [21] J. Kosakowska and D. Simson, On Tits form and prinjective representations of posets of finite prinjective type, Comm. Algebra, 26 (1998), 1613–1623. [22] L. A. Nazarova and A.V. Roiter, Representations of partially ordered sets, in Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 28(1972), 5–31 (in Russian); English version: J. Soviet Math. 3 (no. 5) (1975), 585–606. M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak 261 [23] J. A. de la Peña and D. Simson, Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences, Trans. Amer. Math. Soc. 329 (1992), 733–753. [24] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. [25] M. Sato, Periodic Coxeter matrices and their associated quadratic forms, Linear Algebra Appl. 406 (2005), 99–108. [26] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon & Breach, London (1992). [27] D. Simson, Posets of finite prinjective type and a class of orders, J. Pure Appl. Algebra 90 (1993), 71-103. [28] D. Simson, Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories, Colloq. Math. 115 (2009), 259–295. [29] D. Simson, Integral bilinear forms, Coxeter transformations and Coxeter polyno- mials of finite posets, Linear Algebra Appl. 433 (2010), 699–717. [30] D. Simson, Mesh geometries of root orbits of integral quadratic forms, J. Pure Appl. Algebra 215 (2011), 13–34. [31] D. Simson, Mesh algorithms for solving principal Diophantine equations, sand-glass tubes and tori of roots, Fund. Inform. 109 (2011), 425–462. [32] D. Simson, A Coxeter-Gram classification of positive simply laced edge-bipartite graphs, SIAM J. Discrete Math. 27 (2013), 827–854. [33] D. Simson, A framework for Coxeter spectral analysis of edge-bipartite graphs, their rational morsifications and mesh geometries of root orbits, Fund. Inform. 124 (2013), 309-338. [34] D. Simson and K. Zając, A framework for Coxeter spectral classification of finite posets and their mesh geometries of roots, Intern. J. Math. Mathematical Sciences, Volume 2013, Article ID 743734, 22 pages, doi: 10.1155/2013/743734. Contact information M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland E-Mail(s): kanies@mat.umk.pl, justus@mat.umk.pl, pmalicki@mat.umk.pl, lielow@mat.umk.pl Web-page(s): www.mat.umk.pl/∼justus, www.mat.umk.pl/∼pmalicki, www.mat.umk.pl/∼lielow Received by the editors: 22.12.2015 and in final form 05.01.2016.

A horizontal mesh algorithm for posets with positive Tits form

Репозитарії

Схожі ресурси