© 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
A b s t r ac t . Following our paper [Fund. Inform. 136 (2015), 345–379], we define a horizontal mesh algorithm that constructs aΦbI-mesh translation quiver Γ(RbI,ΦbI) consisting of ΦbI-orbits of the finite setRbI ={v ∈ZI ; qbI(v) = 1} of Tits roots of a poset I with positive definite Tits quadratic form bqI : ZI → Z. Under
the assumption thatqbI :ZI →Zis positive definite, the algorithm
constructs Γ(RbI,ΦbI) such that it is isomorphic with theΦbD-mesh translation quiver Γ(RD,ΦD) ofΦbD-orbits of the finite setRDof roots of a simply laced Dynkin quiverDassociated with I.
1. Introduction
The paper is mainly devoted to the existence of a ΦbI-mesh root
system Γ(RbI,ΦbI) in the sense of [30], that is, aΦbI-mesh translation quiver
Γ(RbI,ΦbI) consisting ofΦbI-orbits of the setRbI ={v∈ZI ; qbI(v) = 1} of
Tits roots of a finite poset I = (I,) with positive quadratic Tits form
b
qI : ZI → Z, where ΦbI : ZI → ZI 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.
Our interest in theΦbI-mesh analysis ofΦbI-orbits of the setRbI of Tits
roots is motivated by applications of matrix representations of posets in representation theory, where a matrix representation of a partially ordered setT ={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, elemenelemen-tary column transformations within each of the substripsM1, 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∗) =x12+· · ·+x2n+x2∗+ X 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 representationM of sized∗×(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 ofI and let I−=I\max I. Fori, j ∈I, we write i≺j if ij and i=6 j. Moreover, for i, j∈I, we write i→j, if
i≺j and there is no sin I such thati≺s≺j. We denote byZ the ring of integers and byMI(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
Following [26, 28, 29, 34], with any poset I, we associate the incidence matrix CI = [cij]∈MI(Z) and the Tits matrix CbI ∈MI(Z), where
cij = (
1 ifij,
0 otherwise, (1.2)
and
b CI=
" Ctr
I− U
0 E
#
, (1.3)
where U = [uiw]i∈I−;w∈maxI and
uiw =
(
−1 ifiw,
0 otherwise, (1.4)
Following [11,32,34], we call a posetI positive, if the symmetric Gram matrixGI := 12(CbI+CbItr) is positive definite.
The following two sets of vectors associated with a posetI are playing an important role in the representation theory of algebras: the set ofTits roots
b
RI :={v∈Zn; v·CbI·vtr = 1} (1.5)
and the set ofEuler roots
RI :={v∈Zn; v·CI·vtr = 1} (1.6)
of a poset I, where
CI =CI−1 (1.7)
see [10, 21, 24, 26]. We recall from [30] that the sets of Tits roots RbI and
Euler rootsRI ofI are finite, ifI is positive. Moreover, ifI is assumed to
be connected then the sets RbI andRI 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 systemsRbI
andRIare isomorphic, and we denote byDI 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 (mvertices, m>1);
Dm : •|2
•1−−−− •3−−−−•4−−−− . . .−−−−•−−−−•m (mvertices, m>4);
E6 : •|4
E7: •|4
•1−−−−•2−−−− •3−−−−•5−−−−•6−−−−•7;
E8: •|4
•1−−−−•2−−−− •3−−−−•5−−−−•6−−−−•7−−−−•8;
It follows from [16] that the Dynkin diagramDI can be determined by applying the inflation algorithm constructed in [20] and [32].
We recall from [29] that the square matrix
b
CoxI:=−CbI·CbI−tr ∈Mn(Z), (1.8)
is called theCoxeter-Tits matrix of I. Here Cbtr
I is the transpose of CbI,
and we setCbI−tr := (Cbtr
I )−1. The charactristic polynomial
coxI(t) := det(t·E−Coxb I)∈Z[t], (1.9)
ofCoxb I is called theCoxeter polynomial of I, the group isomorphism b
ΦI:Zn→Zn, x7→ΦbI(x) :=x·Coxb I, (1.10)
is called theCoxeter-Tits transformation of I, and theCoxeter number
cI ofI is the minimal integerr >1 such that ΦbrI is the identity map on
Zn. IfΦbr
I 6=id, for all r>1, we set cI =∞.
Recall also that the matrix
CoxI :=−CI·CI−tr ∈Mn(Z), (1.11)
is called theCoxeter-Euler matrix of I, and the group isomorphism ΦI:Zn→Zn, x7→ΦI(x) :=x·CoxI, (1.12)
is called theCoxeter-Euler transformation of I.
Following an idea introduced in [30, 31], we study in the paper a
b
ΦI-mesh root system structure Γ(RbI,ΦbI) on the set of rootsRbI ⊆Zn of
any connected positive posetI, with n>2 vertices, where ΦbI :Zn→Zn
is the Coxeter-Tits transformation defined by the Tits matrixCbI ∈Mn(Z)
ofI.
One of the main aims of the paper is to present a combinatorial algorithm that constructs a ΦbI-mesh root system structure Γ(RbI,ΦbI)
(see Definition 2.13) on the finite set of allΦbI-orbits of the irreducible
root systemRbI. Moreover, in Corollary 4.6, we prove that the Coxeter
polynomial coxI(t) and the Coxeter numbercI of such poset I depend
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 ΦbI-mesh root system structure Γ(RbI,ΦbI) for the set RbI of all roots
(not only positive roots).
We recall that one of the motivations for the study of a ΦbI-mesh root
system structure Γ(RbI,ΦbI) 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 andΦbI-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
b
qI, qI :ZI →Z
defined by the formulae
b
qI(x) =x·CbI·xtr, qI(x) =x·CI·xtr.
It is easy to see that
b qI(x) =
X i∈I
x2i + X
i≺j∈I− xjxi−
X w∈maxI
X i≺w
Note also that the Tits quadratic form q(x1, . . . , xn, x∗) (1.1) of a
partially ordered setT ={p1, . . . , pn}(defined by Drozd [9]) coincides with
the Tits formqbI(x1, . . . , xn, x∗) (2.1) of the one-peak posetI =T∗∪ {∗}
obtained fromT by adding a unique maximal element∗.
Recall from [29, Corollary 1.8] that one of the quadratic formsqbI, qI
is positive if and only if both of them are positive. Moreover, in this case we have
qI(x) =X
i∈I
x2i −X i→j
xixj + X i◭j
c•ijxixj, (2.2)
where the relation i◭j holds if there exists a minimal commutativity relationw′−w′′ in I, wherew′, w′′ are paths with the sourcei and the terminusjandc•ij is the maximal number of linearly independent minimal commutativity relationsw′−w′′ in I with the sourceiand the terminus
j, see Corollary 1.8, Remark 3.5 and Proposition 4.2 in [29].
Remark 2.3. LetI 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 posetI we associate the biquiver
QI = (Q0, Q1) with the set of vertices Q0 = I. Moreover, there are
−cij solid arrows i //j , ifcij <0 andcij 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) inQ issimple
if for all i, j∈ {1, . . . , n}, i6=j we have xi 6=xj.
(b) We say that a simple cycle (x1, x2, . . . , xn, x1) ischordless if for any
arrow (xi, xj) we have i=j±1 (wherein 1≡n+ 1).
(c) Further, consider a simple cycle inQ of the form
/
/
// //
O
O (2.4)
The biquiver Q satisfies the non-cycle condition, if every simple chordless cycle in Qcontaining a broken arrow has the form (2.4). (d) Given a poset I the matrixCI = (cij) satisfies thenon-cycle
condi-tion, if the biquiver QI satisfies this condition.
For alli∈I, denote bypbi theTits-projective vector associated with i,
i.e.pbi is defined by the formula
b pi(j) =
1 for i=j;
1 for ij ∈max I;
0 otherwise.
Let
b
P =Pb(I) ={pbi; i∈I}
be the set of all Tits-projective vectors.
For all i∈I, denote byrib the Tits-radical vector associated with i, i.e.rbi is defined by the formula
b ri(j) =
1 for all i→j;
1 for i≺j ∈max I;
0 otherwise.
(2.6)
Let
d
Rad =Rad(d I) ={bri ; i∈I}
be the set of all Tits-radical vectors.
Let i∈I and letrbi be the corresponding Tits-radical vector. Consider
the convex subposet
I-supp(bri) =conv.hull{j ∈I ; rib(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 rb1i, . . . ,br ki
i by the following
formula:
b rti(j) =
( b
ri(j) if i∈It;
0 otherwise (2.7)
for allt∈ {1, . . . , ki}. We denote byRaddcomp the set of vectorsrb1i, . . . ,rb ki i ,
where i∈I.
It is known that pbi ∈RbI and rbji ∈RbI, 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 ij;
0 otherwise. (2.8)
Let
P =P(I) ={pi; i∈I}
be the set of all Euler-projective vectors.
For all i∈I, denote byri theEuler-radical vector associated with i,
i.e.ri is defined by the formula:
Let
Rad = Rad(I) ={ri; i∈I}
be the set of all Euler-radical vectors.
Leti∈Iand letribe the corresponding Euler-radical vector. Consider
the convex subposet
I-supp(ri) ={j ∈I ; ri(j)6= 0}
ofI. LetI1, . . . , Iki be the set of all connected components of the Hasse
quiver of I-supp(ri). We define the vectors r1i, . . . , r ki
i by the following
formula:
rti(j) =
(
ri(j) if i∈It;
0 otherwise (2.10)
for allt∈ {1, . . . , ki}. We denote by Radcomp the set of vectorsr1i, . . . , r ki i ,
wherei∈I.
It is known thatpi∈ RI and rji ∈ RI, for all i, j, see [14, 23, 26, 27].
2.2. Mesh translation quivers inZn
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 → Zn be a group automorphism (e.g. the Coxeter-Tits
transformationΦbI or the Coxeter transformation ΦI of a poset I). A
Φ-orbit Φ− Orb(v) ={Φk(v)}
k∈Z of a vector v∈Zn will be visualised as
an infinite graph:
. . . Φ(v) v Φ−1(v) Φ−2(v) . . .
Definition 2.11. Let Φ :Zn→Znbe a non-trivial group automorphism
(e.g. the Coxeter-Tits transformationΦbIor the Coxeter transformation ΦI
of a posetI). We say that the vectorsu, v1, . . . , vs, w∈Znform a Φ-mesh
starting fromu and terminating atw, if the following two conditions are satisfied:
(i) u= Φ(w) and u+w= Ps
i=1 vi,
(ii) the vectors v1, . . . , vs are pairwise different, lie in pairwise different
A Φ-mesh we visualise as the following triangular quiver:
v1
v2
$
$
Φ(w) =u
B
B
9
9
w
vs
@
@
(2.12)
Definition 2.13. Let n > 2, let Φ : Zn → Zn be a non-trivial group
automorphism and let R be a Φ-invariant subset of Zn (e.g. R = Rb I
if Φ = ΦbI or R = RI if Φ = ΦI). We say that R admits a geometry
of Φ-mesh quiver if there exists a quiver R~ = (R~0, ~R1) with R~0 = 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 verticesw and Φ(w) is a Φ-mesh of the form (2.12), and if
v1′
v2′
#
#
Φ(w) =u
C
C
:
:
w
vs′′
@
@
is a Φ-mesh, then s′=sandv1 =v1′, . . . , vs=vs′, up to permutation of
the set {1, . . . , s}.
Definition 2.14. Let Γ(R,Φ) be a Φ-mesh quiver inZn as in Definition
2.13. Aslicein 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 andI′ defined by the following
Hasse quivers:
1
2
A
A
3
O
O
4,
^
^ 1
2
O
O
4
@
@
3
^
respectively. Note that the setRbI⊆Z4 of Tits roots of I consists of 24
vectors. One easily see that the setRbI admits the following geometry of b
ΦI-mesh quiver Γ(RbI,ΦbI) (we identify the vectors in frames):
1010 1101 0010
b10b10
b1b10b1
00b10
1000 @ @ 2111 @ @ 1111 @ @ b1000 @ @
b2b1b1b1
@ @
b1b1b1b1
@ @ 1000 1100 @ @ 1011 @ @ 0100 @ @
b1b100
@
@
b10b1b1
@
@
0b100
@ @ 1001 G G 1110 G G 0001 G G
b100b1
G
G
b1b1b10
G
G
000b1
G
G
Moreover the setRbI′ of Tits roots ofI′ consists of 24 vectors and admits
the following geometry of ΦbI′-mesh quiver Γ(RbI′,ΦbI′) (we identify the
vectors in frames):
1101 0010
0b101
b100b1
00b10
010b1
1100 @ @ 1011 @ @
0b111
@ @
b1b100
@ @
b10b1b1
@ @
01b1b1
@ @ 1100 0101 @ @ 0001 @ @
0b110
@
@
b10b10
@
@
000b1
@
@
01b10
@ @ 1000 G G 0100 G G
1b100
G
G
b1000
G
G
0b100
G
G
b11b1b1
G
G
1000
G
G
Here we setba=−a, fora∈N.
In the algorithm presented in Section 3 first we look for a slice canditate Σ in Γ(RbI,ΦbI). Then the remaining part of Γ(RbI,ΦbI) is easy to compute.
In Γ(RbI,ΦbI) presented in Example 2.15 the quiver 1010 1000 & & 8 8 1100 1001
is a slice. Applying definition of a ΦbI-mesh we can construct now the b
ΦI-mesh translation quiver Γ(RbI,ΦbI) by knitting ΦbI-meshes as follows: 1010 $ $ a 1000 & & 8 8 b > > e @ @ 1100 : : c > > 1001 D D d G G
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 ΦbI(a) = (1010), ΦbI(b) = (1000), ΦbI(c) = (1100), ΦbI(d) =
(1001), andΦbI(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,Pb,Radd,Raddcomp, k)7→Γ := Γ(b RbI,ΦbI)
presented below, calleda horizontal mesh algorithm, associates to an arbi-trary posetI, with initial dataPb,Radd,Raddcomp, k, a ΦbI-mesh translation
quiver Γ(ReI,ΦbI) such that bΓ defines a ΦbI-mesh root system structure
Γ(RbI,ΦbI) on the setRbI of Tits roots of I, in case whenI 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,Pb,Radd,Raddcomp, k), where
• I = (I,) is a poset such that I ={1, . . . , n}, • Pb={pb1, . . . ,pbn} is the set of Tits-projective vectors,
• Rad =d {rb1, . . . ,rbn} is the set of Tits-radical vectors,
• Raddcomp = {br11, . . . ,rb1k1, . . . ,br1n, . . . ,brnkn}, where rb j
i are defined by
formula 2.7, • k∈N.
Output: The quiver Γ = Γ(b RbI,ΦbI).
S t e p 1 . Inductively, we construct the following data:
• ordered lists Lb[i], for any i= 1, . . . , n; • quiversGbi= (Gbi0,Gbi1), for i= 0,1,2, . . .; • quiversΓbi = (Γbi
0,Γbi1), for i= 0,1,2, . . .;
• sets Pb0 ⊆Pb1⊆. . .⊆Pbk⊆Pb ={pb1, . . . ,pnb };
in the following way.
S t e p 1 . 2 . Let
b
P0=Gb00 ={pbi∈Pb ; i∈max I} and Γb00 =Γb01 =Gb01 =∅.
S t e p 1 . 3 . We put
b
C1={pbi ; bri 6= 0 andrbji ∈Gb00 for all j= 1, . . . , ki}, b
P1:=Gb10 :=Gb00∪Cb1 and Γb10 =Γb11=∅ b
G11={brij →pbi; for allpbi∈Cb1 and allj = 1, . . . , ki}.
S t e p 1 . 4 . Assume that, for i= 0, . . . , m−1,m >2, data Gbi,Γbi, Pb
i
are constructed. We set
b
Pm′ ={pbi ∈P \b Pbm−1 ; rbi 6= 0 and brji ∈Gb0m−1 for all j = 1, . . . , ki}
and
b
Pm=Pbm′ ∪Pbm−1.
We define
b
Cm =Pbm′ ∪ {z=−x+ X x→y
y; y∈Cbm−1},
b
Gm0 =Gbm−0 1∪Cbm
and
b
Gm1 ={brji →pbi; for allpbi ∈Cbm and allj= 1, . . . , ki}∪ ∪{y→z; for ally such thatz=−x+ X
x→y y}.
Moreover, if Pbm 6= Pb,z = −x+Px→yy and x ∈ Lb[i], then we add z
at the end of the listLb[i] and delete the first element of the listLb[i].If
b
Pm6=Pb, then we set Γbm0 =Γbm1 =∅; otherwise we set b
Γm0 =Γbm−0 1∪Cbm
and
b
Γm1 =Γbm−1 1∪ {y→z; for ally→z∈Gb1msuch that y, z∈Γb0m−1∪Γbm0 }.
Moreover, ifPbm=Pb,z=−x+Px→yy and x∈Lb[i], then we add z at
the end of the listLb[i].
Remark 3.2. In this algorithm the set Pb of Tits-projective and the set
d
Rad of Tits-radical can be replaced by the setP 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 Lb[i] by L[i], Γbk by Γk etc.).
We illustrate Algorithm 3.1 by the following example.
Example 3.3. Consider the following poset
1 2
@
@
3
^
^
4
^
^ @@
Note that
b
P ={pb1= (1,0,0,0),pb2 = (1,1,0,0),pb3 = (1,0,1,0),pb4 = (1,0,0,1)}, d
Rad ={rb2= (1,0,0,0),rb3 = (1,0,0,0),rb4 = (1,1,1,0)}
and Raddcomp = {br12 = br2,br31 = br3,br41 = br4}. We set k = 5. Applying
Algorithm 3.1 to (I,Pb,Radd,Raddcomp, k) we get
1100
%
%
0010
%
%
00b11 1000
9
9
1110
%
%
9
9
0001
9
9
%
%
1001
9
9
b1000 1010
B
B
0100
B
B
0b101
Indeed:
m=0: Pb0 = Gb00 = {pb1 = (1,0,0,0)}; Γb00 = Γb01 = Gb01 = ∅; Lb[1] =
[pb1],Lb[2] = [pb2],Lb[3] = [pb3],Lb[4] = [pb4].
m=1: Cb1 = {pb2 = (1,1,0,0),pb3 = (1,0,1,0)}, Pb1 = Gb10 = {pb1,pb2,pb3}, b
G11 = {(pb1,pb2),(pb1,pb3)}, Γb10 = Γb11 = ∅. Lb[1] = [pb1],Lb[2] = [pb2], b
L[3] = [pb3],Lb[4] = [pb4].
m=2: Pb2′ =∅,Pb2 =Pb1,Cb2 ={pb2+pb3−pb1 = (1,1,1,0)},Gb2
0=Gb10∪Cb2, b
G21 = {(pb2,(1,1,1,0)),(pb3,(1,1,1,0))}, Γb20 = Γb21 = ∅. b
m=3: Pb3′ ={pb4 = (1,0,0,1)}, Pb3 ={pb1,pb2,pb3,pb4}, Cb3 = {pb4,(0,1,0,0),
(0,0,1,0)}, Gb30 = Gb20 ∪ Cb3, Gb31 = {((1,1,1,0),pb4),
((1,1,1,0),(0,1,0,0)),((1,1,1,0),(0,0,1,0))}, Γb3
0 = Cb3,Γb31 = ∅. b
L[1] = [(1,1,1,0)],Lb[2] = [(0,0,1,0)],Lb[3] = [(0,1,0,0)],Lb[4] = [pb4].
m=4: Pb′
4 = ∅, Pb4 = Pb, Cb4 = {(0,0,0,1)}, Gb40 = Gb03 ∪ Cb3, Gb41 =
{(pb4,(0,0,0,1)),((0,1,0,0),(0,0,0,1)),((0,0,1,0),(0,0,0,1))},Γb40
= Γb30 ∪ Cb3, Γb41 = Gb41. Lb[1] = [(1,1,1,0),(0,0,0,1)], Lb[2] =
[(0,0,1,0)],Lb[3] = [(0,1,0,0)],Lb[4] = [pb4].
m=5: Pb′
5=∅,Pb5 =Pb,Cb5 ={(0,0,b1,1),(1b,0,0,0),(0,b1,0,1)},Gb50=Gb40∪ b
C4,Gb51 ={((0,0,0,1),(0,0,b1,1)),((0,0,0,1),(b1,0,0,0)),((0,0,0,1),
(0,b1,0,1))},Γb5
0 =Γb40∪Cb4,Γb51 =Γb41∪Gb51.Lb[1] = [(1,1,1,0),(0,0,0,1)], b
L[2] = [(0,0,1,0),(0,0,b1,1)],Lb[3] = [(0,1,0,0),(0,b1,0,1)], Lb[4] = [pb4,(b1,0,0,0)].
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 ={r11=r1, r12 =r2, r13 =r3, r14=r4}. We setk= 5 and get 1100
&
&
0010
&
&
0101
1000
8
8
1110
%
%
8
8
0111
8
8
%
%
1111
9
9
b1000 1010
B
B
0100
B
B
0011
Indeed:
m=0: P0 = G00 = {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 = G10 = {p1, p2, p3}, G11 = {(p1, p2),(p1, p3)}, Γ10 = Γ11 = ∅. L[1] = [p
1], L[2] = [p2], L[3] = [p3], L[4] = [p4].
m=2: P′2 =∅,P2 =P1,C2 ={p
2+p3−p1 = (1,1,1,0)}, G 2 0 =G
1 0∪ C2, G21 = {(p2,(1,1,1,0)),(p3,(1,1,1,0))}, Γ20 = Γ21 = ∅.
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)}, G30 = G20 ∪ C3, G31 = {((1,1,1,0), p4),
m=4: P′4=∅,P4 =P,C4 ={(0,1,1,1)},G4
0=G 3 0∪ C3,
G41={(p4,(0,1,1,1)),((0,1,0,0),(0,1,1,1)),((0,0,1,0),(0,1,1,1))}, Γ40 = Γ30 ∪ C3, Γ41 = 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),(b1,0,0,0),(0,0,1,1)},
G50=G40∪ C4, G15 ={((0,1,1,1),(0,1,0,1)),((0,1,1,1),(b1,0,0,0)),
((0,1,1,1),(0,0,1,1))}, Γ50 = Γ40∪ C4, Γ51 = 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,(b1,0,0,0)].
4. Correctness of Algorithm 3.1
Following [28, 29], we define a group isomorphism
σI0 :ZI→ZI (4.1)
by the formula σI0(x) =x·CI0, where CI0 is the reduced incidence matrix
CI0 =
"
CI− 0
0 E
#
, (4.2)
where E is the identity matrix. By [29, Proposition 3.13],σI0 gives Z -equivalence ofqIb and qI, i.e.qI(σI0(x)) =qb(x).
Lemma 4.3. For any poset I and for all i∈I, we have (σ0I)−1(p
i) =pbi
and (σI0)−1(ri) = bri, where pbi, rbi, pi and ri are projective and radical
vectors defined in Section 2.1, and σI0:ZI →ZI is the isomorphism(4.1).
Proof. The proof is straightforward.
Theorem 4.4. Assume that I is a connected positive poset. Let Γb be the
quiver constructed by Algorithm 3.1 with input (I,Pb,Radd,Raddcomp, k)
withk large enough(e.g.k=|RbI|). The following conditions are satisfied.
(a) Pb=SkPbk, in particular there exists m such that Pbm =Pb.
(b) The sequence Γb0 ⊆Γb1 ⊆. . . stabilizes.
(c) RbI =SmΓbm0 .
(d) Lb[1], . . . ,Lb[n] are theΦbI-orbits in RbI of the Coxeter-Tits
transfor-mation ΦbI.
(e) The ΦbI-mesh translation quiver Γ = Γ(b RbI,ΦbI) defines aΦbI-mesh
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 =CI−1 satisfies the
non-cycle condition defined in [14], see Remark 2.3. Therefore, [14, Theo-rem 4.13] and [14, Section 5] yield:
(¯a) P =SkPk, in particular there exists m such thatPm=P.
(¯b) The sequence Γ0 ⊆Γ1 ⊆. . . stabilizes. (¯c) RI =SmΓm0 .
(¯d) L[1], . . . , L[n] are the ΦI-orbits inRI of the Coxeter transformation
ΦI.
(¯e) The ΦI-mesh translation quiver Γ = Γ(RI,ΦI) defines a ΦI-mesh
root system structure Γ(RI,ΦI) on the setRI of Euler roots ofI.
By Lemma 4.3, we have (σ0I)−1(pi) =pbi and (σI0)−1(ri) =rbi, see (4.1).
It is easy to verify that the automorphism (σ0i)−1 sendsP,Rad,Radcomp
toPb,Radd,Raddcomp, respectively. From [29, Proposition 3.13], it follows
thatΦbI = (σI0)−1◦ΦI◦σ0I. Now, applying the linearity ofσ0I, it is easy
to deduce that the conditions (¯a)-(¯e) imply the conditions (a)-(e), and the theorem follows.
Remark 4.5. It follows from the proof of Theorem 4.4 that the ΦbI
-mesh quiver Γ(RI,ΦbI) is the image of Γ(RI,ΦI) via the automorphism
(σI0)−1 :ZI →ZI (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 RbI. The Coxeter polynomial
coxI(t) is equal to the Coxeter polynomialcoxDI(t) of the Dynkin diagram DI and the Coxeter numbercI 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 aZ-invertible matrixB∈MI(Z) such that CoxDI =B·CoxI·B−1, where CoxDI is the Coxeter matrix
associated with the simply laced Dynkin diagramDI. Moreover by [29, Proposition 3.13], we haveCoxb I =CI0·CoxI·(CI0)−1. Now it is easy to
deduce that coxI(t) = coxDI(t) andcI=cDI.
Example 4.7. Consider the poset I given by the Hasse quiver
By applying Algorithm 3.1 to (I,P,Rad,Radcomp, k= 6) we get the b
ΦI-mesh quiver Γ(RI,ΦI):
1111 b1000
0b100
00b10
000b1
1111 1110 A A 0111 A A
b1b100
A
A
0b1b10
A
A
00b1b1
A A 1110 A A 1100 A A 0110 A A 0011 A A
b1b1b10
A
A
0b1b1b1
A A 1100 A A 1000 A A 0100 A A 0010 A A 0001 A A
b1b1b1b1
A
A
1000
A
A
where vectors in frames lying in the same orbit are identified.
Moreover, by applying Algorithm 3.1 to (I,Pb,Radd,Raddcomp, k= 6)
we get the ΦbI-mesh quiver Γ(RbI,ΦbI):
1001 b1000
0b100
01b10
001b1
1001 1010 A A 0001 A A
b1b100
A
A
00b10
A
A
010b1
A A 1010 A A 1100 A A 0010 A A
0b101
A
A
b10b10
A
A
000b1
A A 1100 A A 1000 A A 0100 A A
0b110
A
A
00b11
A
A
b100b1
A
A
1000
A
A
Note that the ΦbI-mesh quiver Γ(RI,ΦI) is isomorphic with the ΦbI
-mesh quiver Γ(RbI,ΦbI) via the authomorphism σI0:Z4 →Z4 (4.1).
Example 4.9. Consider the posetI given by the Hasse quiver
1 2 @ @ 3 ^ ^ 4 O
O 77
5
O
O
(4.10)
By applying Algorithm 3.1 we get the ΦbI-mesh quiver Γ(RI,ΦI):
00100 11111 b10000 00010
b1b1b1b10
00b10b1
0b10b10
11000 00100 11110 00101 01010
b1b1000
00b100
b1b1b1b1b1
10000
000b10
11110 11100 C C I I 11211 C C I I 01111 C C I I b10010 C C I I
b1b1b100
C C I I
b1b1b2b1b1
C C I I
0b1b1b1b1
C C I I
100b10
C C I I 11100 C C I I 11101 C C 01110 C C 00111 C C
b10b100
C
C
b1b1b10b1
C
C
0b1b1b10
C
C
00b1b1b1
C C 10100 C C 11101 10101 C C 01000 C C 00110 C C 00001 C C
b10b10b1
C
C
0b1000
C
C
00b1b10
C
C
0000b1
C
C
10101
C
and theΦbI-mesh quiver Γ(RbI,ΦbI): 00100
10b111
b10000
0b1b110
b100b10
0000b1
001b10
11000 00100 10010 00001
00b110
b1b1000
00b100
b101b1b1
10000
011b10
10010 11100 C C I I 10011 C C I I
00b111
C C I I
b1b1b110
C C I I
b1b1b100
C C I I
b100b1b1
C C I I
001b1b1
C C I I
111b10
C C I I 11100 C C I I 11001 C C 00010 C C
0b1b111
C
C
b10b100
C
C
b1b100b1
C
C
000b10
C
C
011b1b1
C C 10100 C C 11001 10001 C C 01000 C C
0b1010
C
C
00b101
C
C
b1000b1
C
C
0b1000
C
C
010b10
C
C
0010b1
C
C
10001
C
C
Note that theΦbI-mesh quiver Γ(RI,ΦI) is isomorphic with the ΦbI
-mesh quiver Γ(RbI,ΦbI) via the authomorphismσ0I :Z5→Z5 (4.1).
Example 4.11. Consider the poset I given by the Hasse quiver
1oo 3
y
y
5
o
o oo 6
t
t
2oo 4
By applying Algorithm 3.1 to (I,P,Rad,Radcomp, k= 24) we get the b
ΦI-mesh quiver Γ(RI,ΦI): 111111 ! ! 001000 ! ! 000010 ! ! 000101 ! !
0b10b100
!
!
b10b1000
!
!
0b1b10b10
112111 = = 001010 = = 000111 = =
0b10001
=
=
b1b1b1b100
=
=
b1b1b20b10
= = 101010 ! ! 011111 ! !
0b10000
!
!
000011
!
!
b1b1b10b10
!
!
0b1b1b100
! ! 112110 F F = = ! ! 112121 F F = = ! ! 001111 F F = = ! !
0b10011
F F = = ! !
b1b1b1001
F F = = ! !
b1b2b2b1b10
F F = = ! !
b1b1b2b1b10
J J D D 011110 = = ! ! 101111 = = ! ! 001011 = = ! !
b1b1b1000
=
=
!
!
0b1b1001
=
=
!
!
b1b1b1b1b10
= = ! ! 011010 = = 000100 = = 101011 = = b100000 = =
0b1b1000
=
=
000001
=
=
b1b1b1b1b1b1
D
D
000b100
!
!
b10b10b1b1
! ! 100000 ! ! 011000 ! !
00000b1
!
!
111111
!
!
0b1b1b1b10
=
=
b10b1b1b1b1
=
=
00b10b1b1
= = 111000 = =
01100b1
= = 111110 = = 112111
b10b10b10
!
!
0b1b1b1b1b1
!
!
010000
!
!
0000b1b1
! ! 111010 ! ! 011100 ! ! 101010 J J D D
b1b1b2b1b2b1
F F = = ! !
00b1b1b1b1
F F = = ! !
0100b1b1
F F = = ! !
11100b1
F F = = ! ! 122110 F F = = ! ! 112110 F F = = ! !
b1b1b2b1b1b1
=
=
!
!
00b10b10
=
=
!
!
000b1b1b1
=
=
!
!
01000b1
= = ! ! 111100 = = ! ! 112010 = = ! ! 011110 D D
00b1000
=
=
0000b10
=
=
000b10b1
= = 010100 = = 101000 = = 011010 = =
Moreover, by applying Algorithm 3.1 to (I,Pb,Radd,Raddcomp, k= 24)
we get the ΦbI-mesh quiver Γ(RbI,ΦbI): 110001 ! ! 001000 ! !
00b1010
!
!
0000b11
!
!
0b10b100
!
!
b10b1000
!
!
0b100b10
111001 = = 000010 = =
00b1001
=
=
0b10b1b11
=
=
b1b1b1b100
=
=
b1b1b10b10
= = 100010 ! ! 010001 ! !
0b10000
!
!
00b1b101
!
!
b1b100b10
!
!
0b1b1b100
! ! 111110 F F = = ! ! 110011 F F = = ! ! 000001 F F = = ! !
0b1b1b101
F F = = ! !
b1b1b1b1b11
F F = = ! !
b1b2b1b1b10
F F = = ! !
b1b1b1b1b10
D D J J 010110 = = ! ! 100001 = = ! !
000b101
=
=
!
!
b1b1b1000
=
=
!
!
0b1b1b1b11
=
=
!
!
b1b10b1b10
= = ! ! 010010 = = 000100 = =
100b101
=
=
b100000
=
=
0b1b1000
=
=
000b1b11
=
=
b1b1000b1
D
D
000b100
!
!
b10010b1
! ! 100000 ! ! 011000 ! !
00011b1
!
!
110001
!
!
0b10b1b10
=
=
b10000b1
=
=
00010b1
= = 111000 = =
01111b1
= = 110110 = = 111001
b1000b10
!
!
0b1000b1
!
!
010000
!
!
00110b1
! ! 110010 ! ! 011100 ! ! 100010 J J D D
b1b100b1b1
F F = = ! !
00000b1
F F = = ! !
01110b1
F F = = ! !
11111b1
F F = = ! ! 121110 F F = = ! ! 111110 F F = = ! !
b1b1b100b1
=
=
!
!
0000b10
=
=
!
!
00100b1
=
=
!
!
01011b1
= = ! ! 111100 = = ! ! 111010 = = ! ! 010110 D D
00b1000
=
=
0010b10
=
=
00001b1
= = 010100 = = 101000 = = 010010 = =
Note that the ΦbI-mesh quiver Γ(RI,ΦI) is isomorphic with the ΦbI
-mesh quiver Γ(RbI,ΦbI) via the authomorphism σI0:Z6 →Z6 (4.1).
Example 4.12. Consider the poset I given by the Hasse quiver
1oo 3
x x 5 o o x x 6 o o
2oo 4
By applying Algorithm 3.1 to (I,P,Rad,Radcomp, k = 24) we get
theΦbI-mesh quiver Γ(RI,ΦI): 111111 ! ! 001000 ! ! 000110 ! ! 101011 ! ! b100000 ! !
0b1b1000
!
!
000b100
112111 = = 001110 = = 101121 = = 001011 = =
b1b1b1000
=
=
0b1b1b100
= = 011110 ! ! 101111 ! ! 001010 ! ! 000111 ! !
0b10b100
!
!
b10b1000
! ! 112110 F F = = ! ! 112221 F F = = ! ! 102121 F F = = ! ! 001121 F F = = ! !
0b10011
F F = = ! !
b1b1b1b100
F F = = ! !
b1b1b2b1b10
J J D D 101110 = = ! ! 112121 = = ! ! 001111 = = ! !
0b10010
= = ! ! 000011 = = ! !
b1b1b1b1b10
= = ! ! 000100 = = 101010 = = 011111 = =
0b10000
= = 000010 = = 000001 = =
b1b1b1b1b1b1
D
b10b10b10
!
!
0b1b1b1b1b1
!
!
010000
!
!
0000b10
!
!
00000b1
!
!
111111
!
!
b10b1b1b10
=
=
b1b1b2b1b2b1
=
=
00b1b1b1b1
=
=
0100b10
=
=
0000b1b1
= = 111110 = = 112111
0b1b1b1b10
!
!
b10b1b1b1b1
!
!
00b10b10
!
!
000b1b1b1
! ! 010100 ! ! 101000 ! ! 011110 J J D D
b1b1b2b2b2b1
F F = = ! !
b10b2b1b2b1
F F = = ! !
00b1b1b2b1
F F = = ! !
0100b1b1
F F = = ! ! 111100 F F = = ! ! 112110 F F = = ! !
b1b1b2b1b1b1
=
=
!
!
00b1b1b10
=
=
!
!
b10b1b1b2b1
=
=
!
!
00b10b1b1
= = ! ! 111000 = = ! ! 011100 = = ! ! 101110 D D
00b1000
=
=
000b1b10
=
=
b10b10b1b1
= = 100000 = = 011000 = = 000100 = =
where vectors in frames lying in the same orbit are identified.
Moreover, by applying Algorithm 3.1 to (I,Pb,Radd,Raddcomp, k= 24)
we get theΦbI-mesh quiver Γ(RbI,ΦbI):
110001 ! ! 001000 ! !
00b1010
"
"
100b101
!
!
b100000
!
!
0b1b1000
!
!
000b100
111001 = = 000010 = =
10b1b111
<
<
000b101
=
=
b1b1b1000
=
=
0b1b1b100
= = 010010 ! ! 100001 ! !
000b1100
"
"
00b1001
!
!
0b10b100
!
!
b10b1000
! ! 111010 F F = = ! ! 110011 F F = = ! !
100b111
E E < < " "
00b1b111
F F = = ! !
0b1b1b101
F F = = ! !
b1b1b1b100
F F = = ! !
b1b1b10b10
D D J J 100010 = = ! !
110b111
= = ! ! 000001 < < " "
0b1b1b110
=
=
!
!
00b1b101
=
=
!
!
b1b100b10
= = ! ! 000100 = =
100b110
=
=
010001
<
<
0b10000
=
=
00b1b110
=
=
0000b11
=
=
b1b1000b1
D
D
b1001b10
!
!
0b1000b1
!
!
010000
!
!
0011b10
!
!
00001b1
!
!
110001
!
!
b1000b10
=
=
b1b101b1b1
=
=
00000b1
=
=
0111b10
=
=
00110b1
= = b110010 = = 111001
0b100b10
!
!
b10000b1
!
!
0001b10
!
!
00100b1
! ! 010100 ! ! 101000 ! ! 010010 J J D D
b1b100b1b1
F F = = ! !
b1001b1b1
F F = = ! !
0011b1b1
F F = = ! !
01110b1
F F = = ! ! 111100 F F = = ! ! 111010 F F = = ! !
b1b1b100b1
=
=
!
!
0000b10
=
=
!
!
b1011b1b1
=
=
!
!
00010b1
= = ! ! 111000 = = ! ! 011100 = = ! ! 100010 D D
00b1000
=
=
0010b10
=
=
b10010b1
= = 100000 = = 011000 = = 000100 = =
Note that theΦbI-mesh quiver Γ(RI,ΦI) is isomorphic with the ΦbI
-mesh quiver Γ(RbI,ΦbI) via the authomorphismσ0I :Z6→Z6 (4.1). References
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C o n tac t i n f o r m at i o n
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):[email protected], [email protected], [email protected], [email protected]
Web-page(s):www.mat.umk.pl/∼justus, www.mat.umk.pl/∼pmalicki, www.mat.umk.pl/∼lielow
