We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.
Citation: Shuguang Li. Efficient algorithms for scheduling equal-length jobs with processing set restrictions on uniform parallel batch machines[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 10731-10740. doi: 10.3934/mbe.2022502
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We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.
Path algebras are very important in representation theory and related mathematical fields, it is interesting to study their counterparts in higher representation theory [21,20]. The
It is well known that the path algebras of acyclic quivers are classified as finite, tame and wild representation types according to their quivers. This classification has a great influence in representation theory. When Iyama and his coauthors study
Regarding
McKay quiver for a finite subgroup of a general linear group is introduced in [27], which connects representation theory, singularity theory and many other mathematical fields. McKay quiver is also very interesting in studying higher representation theory of algebras [20,15]. Over the algebraically closed field of characteristic zero, the preprojective algebras of path algebras of tame type are Morita equivalent to the skew group algebras of finite subgroups of
We also describe the quivers and relations for the tame
The paper is organized as follows. In Section 2, concepts and results needed in this paper are recalled. We recall the constructions of the
In this paper, we assume that
Recall that a bound quiver
Let
A bound quiver
ρ⊥=⋃i,j∈Q0ρ⊥i,j. | (1) |
The quadratic dual quiver of
To define and study
Recall that a bound quiver
Let
The
With an
(Z|n−1Q)0={(u,t)|u∈Q0,t∈Z} |
and arrow set
(Z|n−1Q)1=Z×Q1∪Z×Q1,M, |
with
Z×Q1={(α,t):(i,t)⟶(i′,t)|α:i⟶i′∈Q1,t∈Z}, |
and
Z×Q1,M={(βp,t):(i′,t)⟶(i,t+1)|p∈M,s(p)=i,t(p)=i′}. |
The relation set of
Zρ={∑sas(αs,t)(α′s,t)|∑sasαsα′s∈ρ,t∈Z}, |
ZρM={(βp′,t)(βp,t+1)|βp′βp∈ρM,t∈Z}, |
and
Zρ0={∑s′as′(βp′s′,t)(α′s′,t+1)+∑sbs(αs,t)(βps,t)|∑s′as′βp′s′α′s′+∑sbsαsβps∈ρ0,t∈Z}, |
when the returning arrow quiver
Recall a complete
Given a finite stable
Z⋄˜Q0={(i,n)|i∈Q0,n∈Z}; |
and the arrow set
Z⋄˜Q1={(α,n):(i,n)→(j,n+1)|α:i→j∈Q1,n∈Z}. |
If
ρZ⋄˜Q={ζ[m]|ζ∈˜ρ,m∈Z} | (2) |
here
A connected quiver
Clearly, a nicely graded quiver is acyclic.
Proposition 2.1. 1. Let
2. All the connected components of
3. Each connected component of
Proof. The first and the second assertions follow from Proposition 4.3 and Proposition 4.5 of [12], respectively. The last follows from the definition of
An
Proposition 2.2. Assume that
Proof. Let
ϕ(i,t)=(i,t(n+1)+d(i)−d(i0)). |
It is easy to see that
By definition, an
Proposition 2.3. If
Let
Z⋄Q0[m,l]={(i,t)|m≤t≤l}. |
ρZ⋄˜Q[m,l]={x|x∈ρZ⋄˜Q,s(x),t(x)∈Z⋄Q0[m,l]}. | (3) |
By Proposition 6.2 of [12], we get the following for any
Proposition 2.4.
The complete
Starting with an acyclic
We have the following picture to illustrate the relationship among these quivers.
![]() |
(4) |
The quadratic dual quiver
Recall that a graded algebra
⋯⟶Ptft⟶⋯⟶P1f1⟶P0f0⟶˜Λ0⟶0, | (5) |
such that
Let
The following Proposition justifies the name
Proposition 2.5. Let
Starting with a quadratic acyclic
With the quivers related to
We also associate algebras to the bound quivers
Taking the usual path grading on
So we can get criteria for
Proposition 2.6. Let
Write
![]() |
(6) |
The left triangle is induced by the picture (4) depicting quivers. The left vertically up arrow indicate taking a
Now we assume that
For
At(˜Λ)=(dimke1˜Λte1dimke2˜Λte1⋯dimkem˜Λte1dimke1˜Λte2dimke2˜Λte2⋯dimkem˜Λte2⋯⋯⋯⋯dimke1˜Λtemdimke2˜Λtem⋯dimkem˜Λtem). |
Let
The Loewy matrix
L(˜Λ)=(A1(˜Λ)−E0⋯00A2(˜Λ)0−E⋯00⋅⋅⋅⋯⋅⋅An(˜Λ)00⋯0−EAn+1(˜Λ)00⋯00) | (7) |
with size
Let
l.dimM=(dimMh⋮dimMh+n), | (8) |
where
Assume that
⟶Ps⟶⋯⟶P1→P0⟶M→0 |
such that
l.dimΩsM=Ls(˜Λ)l.dimM | (9) |
by Proposition 1.1 of [18].
Write
V0=(E0⋮0)m(n+1)×m. |
The following Proposition follows from (9) and the definition of the
Proposition 2.7. Assume that
Now assume that
We can restate the Theorems 2.4 and 2.5 of [18] as follows.
Proposition 2.8.
Then by Theorem 3.1 of [30],
The following is part (a) of Proposition 2.9 of [18].
Proposition 2.9. Let
Let
By applying Proposition 2.9, the following is proved as Theorem 2.10 in [18].
Proposition 2.10. If
Recall that for a graded algebra
GKdimΓ=¯limm→∞logmm⨁t=1dimkΓt. | (10) |
By using Koszul duality, the Gelfand-Kirilov dimension of the quadratic dual
The following is Theorem 3.2 in [18]. Let
Theorem 2.11. If
If
The following is Theorem 3.1 in [18].
Theorem 2.12. If
By picture (6), we see an
To classify the
We call an algebra
Lemma 3.1. A stable
Proof. For stable
Now assume that
Clearly, being weakly periodic is a special case of complexity
We have the following characterization of the periodicity and of the complexities for stable
Theorem 3.2. Let
1. The algebra
2. The algebra
3. The algebra
Proof. Assume that
Now assume that
If
Now assume that
Let
Theorem 3.3. Let
1.
2.
3.
Proof. If
If
Combine Theorems 3.2 and 3.3, we get the following.
Theorem 3.4. Let
1.
2.
3.
By Theorem 2.12, we also have the following.
Theorem 3.5. If
Assume that
ΔνΛ=ΔνΓ!,op≃Π(Γ)!,op, |
where
When
Now we define that an
As an immediate consequence of Theorem 3.4, we get a classification of
Theorem 3.6. An
For an
Theorem 3.7. 1.
2.
3.
Proof. This follows from the above Theorems 3.3 and 3.4.
When
Proposition 3.8. If
It is natural to ask if the converse of Proposition 3.8 is true?
By Proposition 3.8, if the
In this section, we assume that
McKay quiver was introduced in [27]. Let
V⊗Si=⨁jSai,jj, 1≤i≤l, |
here
We recall some results on the McKay quivers of Abelian groups and on the relationship between McKay quivers of same group in
Let
G=G(r1,⋯,rm)=Cr1×⋯×Crm |
is a direct product of
(ξi1r1,ξi2r2,⋯,ξimrm)⟶diag(ξi1r1,ξi2r2,⋯,ξimrm), |
for
Proposition 4.1. The McKay quiver
˜Q0(r1,⋯,rm)=Z/r1Z×⋯×Z/rmZ | (11) |
and the arrow set
˜Q1(r1,⋯,rm)={αi,t:i→i+et|i∈Z/r1Z×⋯×Z/rmZ,1≤t≤m}∪{αi,m+1:i→i−e|i∈Z/r1Z×⋯×Z/rmZ}. | (12) |
Proof. We prove by induction on
The Abelian subgroup in
Assume Proposition holds for
Embed the group
(ξi1r1,⋯,ξihrh)⟶diag(ξi1r1,⋯,ξihrh,ξih+1rh+1), |
then group
G(r1,⋯,rh,rh+1)∩SL(h+1,C)=G(r1,⋯,rh), |
and we have
G(r1,⋯,rh,rh+1)=G(r1,⋯,rh)×C′rh+1, |
where
G(r1,⋯,rh,rh+1)/G(r1,⋯,rh,rh+1)∩SL(h+1,C)≃(ξrh+1). |
By Theorem 1.2 of [11], the McKay quiver of
αi,h+1:(i(h),t)→(i(h)−e(h),t+1) |
from one copy
˜Q′1(r1,⋯,rh,rh+1)={αi,t:i→i+e(h+1)t|i∈Z/r1Z×⋯×Z/rh+1Z,1≤t≤h+1} |
as the arrow set.
Now embed
(ξi1r1,⋯,ξih+1rh+1)⟶diag(ξi1r1,⋯,ξih+1rh+1,ξ−i1r1⋯ξ−ih+1rh+1). |
Since Nakayama permutation
This shows that Proposition holds for
Note that the Nakayama permutation for the subgroup of a special linear group is trivial. As a direct consequence of Proposition 3.1 of [11], we also have the following Proposition to describe the McKay quiver of finite group
Proposition 4.2. Let
Let
Theorem 4.3. Let
Let
Note that
Proposition 4.4. Let
Starting with a finite group
Let
Let
{(i,ˉm)|(i,m)∈Z|n−1Q(G),ˉm∈Z/(n+1)Z}. |
For a path
Proposition 4.5.
Proof. For any
⟶˜Ptft⟶˜Pt−1ft−1⟶⋯˜P1f1⟶˜P0⟶˜Λ0(G)ei⟶0 | (13) |
for the simple
Clearly (13) is exact if and only if
(d1,⋯,dht)=(d′1,⋯,d′ht+1)Ct+1. |
Now for any
⋯⟶˜Pt[¯m]˜ft[¯m]⟶˜Pt−1[¯m]˜ft−1[¯m]⟶⋯˜P0[¯m]˜f1[¯m]⟶˜P0⟶Δ(G)ei,¯m⟶0. | (14) |
By comparing the matrices defining the sequences (13) and (14), we see that (13) is exact if and only if (14) is exact. That is, (13) is a projective resolution of simple
Since
By comparing (13) and (14), the simple modules
We call
It is interesting to know if the converse of Proposition 4.5 is true, that is, for an indecomposable nicely graded tame
For the three pairs of quadratic duals of algebras:
For an AS-regular algebra
Theorem 4.6. The following categories are equivalent as triangulate categories:
1. the bounded derived category
2. the bounded derived category
3. the bounded derived category
4. the stable category
5. the stable category
6. the stable category
If the lengths of the oriented circles in
(7) the bounded derived category
(8) the stable category
Proof. By Theorem 4.14 of [28], we have that
By Theorem 1.1 of [7], we have that
By Corollary 1.2 of [7], we have that
On the other hand, we have
By Lemma Ⅱ.2.4 of [19],
Similarly,
When the lengths of the oriented circles in
We remark that the equivalence of (1) and (2) can be regarded as a McKay quiver version of Beilinson correspondence and the equivalence of (1) and (4) can be regarded as a McKay quiver version of Berstein-Gelfand-Gelfand correspondence [5,4]. So we have the following analog to (6), for the equivalences of the triangulate categories in Theorem 4.6:
![]() |
(15) |
In the classical representation theory, we take a slice from the translation quiver and view the path algebra as
Now we consider the case of
Assume that
Since
L(˜Λ)=(M(˜Q)−E0M′(˜Q)0−EE00). | (16) |
This is exactly the Loewy matrix of
Proposition 5.1. Let
1. If there is an arrow from
2. If there is only one arrow from
Proof. The proposition follows directly from
Therefore the number of arrows from
We also have the following Proposition.
Proposition 5.2. If
Proof. Due to that there is no arrow from
Now we determine the relations for the McKay quiver of finite Abelian subgroup of
Let
Let
![]() |
For each vertex
z(γ,i,ci)=ciβi+e1αi+αi+xe2βi,z(β,i,bi)=biαi−eγi+γi+e1αi, |
and
z(α,i,ai)=aiβi−eγi+γi+e2βi, |
for
˜ρcomm(s,r,C(a,b,c))={z(α,i,ai),z(β,i,bi),z(γ,i,ci)|i∈˜Q0}˜ρzero(s,r)={αi+e1αi,βi+e2βi,γi−eγi|i∈˜Q0}, |
and let
˜ρ(s,r,C(a,b,c))=˜ρcomm(s,r,C(a,b,c))∪˜ρzero(s,r). | (17) |
Proposition 5.3. If the quotient algebra
Proof. Assume that
Consider the square with vertices
ciβi+e1αi+c′iαi+e2βi∈I, |
by (2) of Proposition 5.1. Since
z(γ,i,ci)=z(γ,i,ci,1)∈I. |
Similarly, there are
z(β,i,bi)=biαi−eγi+γi+e1αi∈I, |
and
z(α,i,ai)=aiβi−eγi+γi+e2βi∈I. |
For each
αi+e1αi,βi+e2βi,γi−eγi∈I, |
by Proposition 5.2.
So
Let
ei˜Λ2=kβi−e2αi−e+kγi+eβi+be1+kγi+eαi+be2, |
and
˜Λ3ei=kγi+eβi+e1αi, |
by computing directly using the relations in
dimkei˜Λ3ei≤1=dimkei˜Λ3(G(s,r))ei, |
and
dimkei˜Λ2ei′{≤1,for i′=i−e,i+e1,i+e2;=0,otherwise. |
This implies that
dimkei˜Λ2ei′≤dimkei˜Λ2ei′, |
for any
So
ei′˜Λtei=ei′˜Λt(G(s,r))ei |
for all
This proves
For the quadratic dual quiver
Proposition 5.4. If the quotient algebra
˜ρ⊥(s,r,C(a,b,c))={z(α,i,−a−1i),z(β,i,−b−1i),z(γ,i,−c−1i)|i∈˜Q0}. | (18) |
Proof. Let
Now construct the quiver
Recall by (2), the relation set for
z(γ,(i,t),ci)=ciβi+e1,t+αi+e2,t+1βi,t,z(β,(i,t),bi)=biαi−e,t+1γi,t+γi+e1,t+1αi,t, |
and
z(α,(i,t),ai)=aiβi−e,t+1γi,t+γi+e2,t+1βi,t, |
for
˜ρcomm(s,r,C(a,b,c))[t]={z(α,(i,t),ai),z(β,(i,t),bi),z(γ,(i,t),ci)|i∈˜Q0}˜ρzero(s,r)[t]={αi+e1,t+1αi,t,βi+e2,t+1βi,t,γi−e,t+1γi,t|i∈˜Q0}, |
and let
ρZ⋄˜Q(s,r)=⋃t∈Z(˜ρcomm(s,r,C(a,b,c))[t]∪˜ρzero(s,r)[t]). | (19) |
By taking a connected component
![]() |
Here we denote by
Let
ρ(s,r)={x∈ρZ⋄˜Q(s,r)|e[0,2]xe[0,2]=x}=˜ρcomm(s,r,C(a,b,c))[0]∪˜ρzero(s,r)[0]. |
Since any sequence of relations in
ρ(s,r)={αi+e1,1αi,0,βi+e2,1βi,0,γi−e1,1γi,0|i∈Z/sZ×Z/rZ}∪{αi+e2,1βi,0−βi+e1,1αi,0,αi−e,1γi,0−γi+e1,1αi,0,αi−e1,1γi,0−γi+e1,1αi,0|i∈Z/sZ×Z/rZ}. | (20) |
Proposition 5.5. If the quotient algebra
The quadratic dual quiver
ρ⊥(s,r)={αi+e2,d+1βi,d1=βi+e1,d+1αi,d,αi−e,1γi,0+γi+e1,1αi,0,αi−e1,1γi,0+γi+e1,1αi,0|i∈Z/sZ×Z/rZ}. | (21) |
Proposition 5.6. If the quotient algebra
So the relations for the
Let
![]() |
![]() |
The vertices for
We have immediate the following on the arrows of these McKay quivers.
Lemma 5.7. Let
1. There is a loop at each vertex of
2. There is at most one arrow from
3. There is an arrow
4. There are at most
Let
Lemma 5.8. If
αj,hαi,j,βj,hβi,j,αj,hβi,j,βj,hαi,j∈I(Ξ) |
if such paths exist.
Proof. If
αj,hαi,j∈I(Ξ). |
Similarly we have
βj,hβi,j,αj,hβi,j,βj,hαi,j∈I(Ξ) |
if such paths exist.
Write
˜ρ11(Ξ)={αj,hαi,j|i,h,j∈˜Q0(Ξ),i≠h},˜ρ12(Ξ)={αj,hβi,j|i,h,j∈˜Q0(Ξ),i≠h},˜ρ21(Ξ)={αj,hβi,j|i,h,j∈˜Q0(Ξ),i≠h},˜ρ22(Ξ)={βj,hβi,j|i,h,j∈˜Q0(Ξ),i≠h}. |
Take
˜ρp(Ξ)=˜ρ11(Ξ)∪˜ρ12(Ξ)∪˜ρ21(Ξ)∪˜ρ22(Ξ). | (22) |
As a corollary of Lemma 5.8, we get the following.
Proposition 5.9.
Let
˜ρa(Ξ,Ca)={αi,jγi−γjαi,j,βj,iγj−γiβj,i|αi,j∈˜Q1,ai,j∈Ca,i<j}. | (23) |
Proposition 5.10. By choosing the representatives of the arrows suitably, we have a set
˜ρa(Ξ,Ca)⊆I(Ξ). |
Proof. By Lemma 5.7, there is an arrow
βj,iγj−bj,iγiβj,i∈I(Ξ), |
for some
αi,jγi−a′i,jγjαi,j∈I(Ξ), |
for some
Starting from
βj,iγj−γiβj,i∈I(Ξ) |
for all arrows
This proves that by choosing the representatives of the arrows suitably, we have
Write
Lemma 5.11. For each arrow
Write
˜ρa⊥(Ξ,Ca)={ai,jαi,jγi+γjαi,j,βj,iγj+γiβj,i|αi,j∈˜Q1,ai,j∈Ca,i<j}. | (24) |
Now consider
Fix
μi,j={αi,ji<j,βi,ji>j, and ζj,i={βj,ii<j,αj,ii>j. | (25) |
Then
Consider the following cases.
Lemma 5.12. Assume that there is only one arrow
1. If
2. We have
{γ2i−ciζj,iμi,j,ciγ2i+ζj,iμi,j} | (26) |
is a orthogonal basis for
Proof. Apply Proposition 5.1 for the arrow
The second assertion follows from direct computation.
Lemma 5.13. Assume that there are exactly two arrows
1. There is a
2. If
3. We have
{biζj1,iμi,j1+ζj2,iμi,j2,ζj1,iμi,j1−b′iζj2,iμi,j2,γ2i}and{biζj1,iμi,j1−ζj2,iμi,j2,ciζj1,iμi,j1−γ2i,ζj1,iμi,j1+biζj2,iμi,j2+ciγ2i} | (27) |
are orthogonal bases for
Proof. By Proposition 5.1, we have that the images of
If
The rest follows from direct computations.
Lemma 5.14. Assume that there are three arrows
1. There are
{b_i \zeta_{j_1,i}\mu_{i,j_1 } + \zeta_{j_2,i}\mu_{i,j_2 }, b'_i \zeta_{j_1,i}\mu_{i,j_1 } + \zeta_{j_3,i}\mu_{i,j_3 }\in I(\Xi) .} |
2. If
3. We have
\begin{equation} \label{{orthbasiii}}\begin{array}{l}{ \{b_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_2,i}\mu_{i,j_2 }, b'_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_3,i}\mu_{i,j_3}, \\ \quad \gamma^2_i - c_i\zeta_{j_1,i}\mu_{i,j_1}, c_i \gamma^2_i + \zeta_{j_1,i}\mu_{i,j_1} + b_i \zeta_{j_2,i}\mu_{i,j_2} +b'_i \zeta_{j_3,i}\mu_{i,j_3} \}\\ \mathit{\mbox{and}}\\ \{ \gamma^2, b_i \zeta_{j_1,i}\mu_{i,j_1} - \zeta_{j_2,i}\mu_{i,j_2}, b'_i \zeta_{j_1,i}\mu_{i,j_1} - \zeta_{j_3,i}\mu_{i,j_3},\\ \quad \zeta_{j_1,i}\mu_{i,j_1} + b_i \zeta_{j_2,i}\mu_{i,j_2} +b'_i \zeta_{j_3,i}\mu_{i,j_3}, \gamma^2 \} }\end{array} \end{equation} | (28) |
are orthogonal bases of
Proof. The lemma follows from Proposition 5.1, similar to above two lemmas.
Denote by
\begin{equation} C_i = \left\{ \begin{array}{ll}\{ c_i\} \subset k^* & i\in \widetilde{Q}_{01}(\Xi), \\ \{ c_i, b_i\} \subset k^*& i\in \widetilde{Q}_{02}(\Xi), \\ \{ c_i, b_i,b'_i\} \subset k^*&i\in \widetilde{Q}_{03}(\Xi), \end{array}\right. \end{equation} | (29) |
and set
\begin{equation} C_i' = \left\{ \begin{array}{ll}\emptyset & i\in \widetilde{Q}_{01}(\Xi), \\ \{ b_i\} \subset k^*& i\in \widetilde{Q}_{02}(\Xi), \\ \{ b_i,b'_i\} \subset k^*& i\in \widetilde{Q}_{03}(\Xi). \end{array}\right. \end{equation} | (30) |
Let
\begin{equation} U_i(C_i) = \left\{ \begin{array}{ll}\{ c_i \gamma^2_i +\zeta_{j,i}\mu_{i,j}\} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \zeta_{j_1,i}\mu_{i,j_1 } - b_i \zeta_{j_2,i}\mu_{i,j_2 }, \gamma^2_i - c_i\zeta_{j_1,i}\mu_{i,j_1}\} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ b_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_2,i}\mu_{i,j_2 }, b'_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_3,i}\mu_{i,j_3 }, \\ \quad \gamma^2_i - c_i\zeta_{j_1,i}\mu_{i,j_1} \} & i\in \widetilde{Q}_{03}(\Xi). \end{array}\right. \end{equation} | (31) |
and let
\begin{equation}U^-_i(C'_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i}\mu_{i,j} \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \gamma^2, \zeta_{j_1,i}\mu_{i,j_1 } - b_i\zeta_{j_2,i}\mu_{i,j_2 } \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \gamma_i^2, b_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_2,i}\mu_{i,j_2 }, b'_i \zeta_{j_1,i}\mu_{i,j_1 } - \zeta_{j_3,i}\mu_{i,j_3 } \} & i\in \widetilde{Q}_{03}(\Xi). \end{array}\right. \end{equation} | (32) |
Lemma 5.15. A basis of the orthogonal subspace of
\begin{equation} U_i^{\perp}(C_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i}\mu_{i,j} + c_i \gamma^2_i \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \zeta_{j_1,i}\mu_{i,j_1 } + b_i^{-1}\zeta_{j_2,i}\mu_{i,j_2 }+ c^{-1}_i \gamma_i^2 \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i}\mu_{i,j_1 } + b^{-1}_i\zeta_{j_2,i}\mu_{i,j_2 }+ {b'}^{-1}_i \zeta_{j_3,i}\mu_{i,j_3 } +c_i \gamma^2_i \} & i\in \widetilde{Q}_{03}(\Xi). \end{array}\right. \end{equation} | (33) |
and let
\begin{equation} U^{-,\perp}_i(C'_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i}\mu_{i,j} \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ b_i \zeta_{j_1,i} \mu_{i,j_1 } + \zeta_{j_2,i}\mu_{i,j_2 } \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i}\mu_{i,j_1 } + b^{-1}_i \zeta_{j_2,i}\mu_{i,j_2 }+ {b'}^{-1}_i \zeta_{j_3,i}\mu_{i,j_3 } \} & i\in \widetilde{Q}_{03}(\Xi). \end{array}\right.\end{equation} | (34) |
Proof. This follow immediately from (2) of Lemma 5.12, (3) of Lemma 5.13 and (3) of Lemma 5.14.
Fix
\begin{equation} \label{{defCJ}}{C_J = C_a \cup \bigcup\limits_{i\in Q_0\setminus J} C_{i}\cup \bigcup\limits_{i\in J} C'_i,} \end{equation} | (35) |
for
\begin{equation} \ \tilde{\rho}(\Xi,J,C_J) = \tilde{\rho}_p(\Xi) \cup \tilde{\rho}_a(\Xi,C_a) \cup \bigcup\limits_{i\in \widetilde{Q}_0\setminus J} U_i (C_i)\cup \bigcup\limits_{i\in J} U^-_i (C'_i). \end{equation} | (36) |
By Lemma 5.11 and Lemma 5.15, we have the following.
Proposition 5.16.
\begin{equation} \tilde{\rho}^{\perp}(\Xi,J,C_J) = { \tilde{\rho}_a}^\perp(\Xi,C_a) \cup \bigcup\limits_{i\in \widetilde{Q}_0\setminus J} U^{\perp}_i (C_i) \cup \bigcup\limits_{i\in J} U^{-,\perp}_i (C_i). \end{equation} | (37) |
Proposition 5.17. Let
Proof. Take
Let
It is also immediate that
By Lemma 5.16, a quadratic dual relation of
Proposition 5.18. Let
Now construct the nicely-graded quiver
\begin{array}{lll}\rho_{ {\mathbb Z}_{ \diamond} \widetilde{Q}(\Xi)} & = & \rho_{ {\mathbb Z}_{ \diamond} \widetilde{Q}(\Xi)}(J,C_J)\\ & = & \{z[m]| z\in \tilde{\rho}(\Xi,J,C_J),t\in {\mathbb Z}\}\end{array} |
for some parameter set
\Lambda(\Xi,J,C) \simeq k {\mathbb Z}_{ \diamond} \widetilde{Q}(\Xi)/(\rho_{ {\mathbb Z}_{ \diamond} \widetilde{Q}(\Xi)}(J,C_J)) , |
if
By taking the complete
![]() |
![]() |
They are all nicely-graded quivers. We get
\begin{equation} \label{{relnpg}}{\rho(\Xi,J,C) = \{z[0]| z\in \tilde{\rho}(\Xi, J, C)\}.} \end{equation} | (38) |
For any parameter set
Write
\mu_{i,j,t} = \left\{\begin{array}{ll} \alpha_{i,j,t} & i < j, t = 0,1,\\ \beta_{i,j,t} & i > j, t = 0,1, \end{array}\right. \mbox{ and } \zeta_{j,i} = \left\{\begin{array}{ll} \beta_{j,i,t} & i < j,t = 0,1,\\ \alpha_{j,i,t} & i > j, t = 0,1. \end{array}\right. |
Let
\begin{equation} U_i(C_i) = \left\{ \begin{array}{ll}\{ \gamma_{i,1} \gamma_{i,0} +\zeta_{j,i,1}\mu_{i,j,0}\} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } - \zeta_{j_2,i,1}\mu_{i,j_2,0 }, \gamma_{i,1} \gamma_{i,0} - \zeta_{j_1,i,1}\mu_{i,j_1,0}\} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } - \zeta_{j_2,i,1}\mu_{i,j_2,0 }, \\ \quad \zeta_{j_1,i,1}\mu_{i,j_1,0 } - \zeta_{j_3,i,1}\mu_{i,j_3,0 }, \gamma_{i,1} \gamma_{i,0} - \zeta_{j_1,i,1}\mu_{i,j_1,0} \} & i\in \widetilde{Q}_{03}(\Xi), \end{array}\right. \end{equation} | (39) |
and let
\begin{equation} U^-_i(C'_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i,1}\mu_{i,j,0} \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \gamma^2, \zeta_{j_1,i,1}\mu_{i,j_1,0 } - \zeta_{j_2,i,1}\mu_{i,j_2,0 } \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } + \zeta_{j_2,i,1}\mu_{i,j_2,0 }+ \zeta_{j_3,i,1}\mu_{i,j_3,0 } \} & i\in \widetilde{Q}_{03}(\Xi), \end{array}\right. \end{equation} | (40) |
\begin{equation} U_i^{\perp}(C_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i,1}\mu_{i,j,0} + \gamma_{i,1} \gamma_{i,0} \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } + \zeta_{j_2,i,1}\mu_{i,j_2,0 }+ \gamma_{i,1} \gamma_{i,0} \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } + b^{-1}_i\zeta_{j_2,i,1}\mu_{i,j_2,0 }+ \\ \quad \zeta_{j_3,i,1}\mu_{i,j_3,0 } + \gamma_{i,1} \gamma_{i,0} \} & i\in \widetilde{Q}_{03}(\Xi), \end{array}\right. \end{equation} | (41) |
and let
\begin{equation} U^{-,\perp}_i(C'_i) = \left\{ \begin{array}{ll}\{ \zeta_{j,i,1}\mu_{i,j,0} \} & i\in \widetilde{Q}_{01}(\Xi), \\ \{ \zeta_{j_1,i,1} \mu_{i,j_1,0 } + \zeta_{j_2,i,1}\mu_{i,j_2,0 } \} & i\in \widetilde{Q}_{02}(\Xi), \\ \{ \zeta_{j_1,i,1}\mu_{i,j_1,0 } + \zeta_{j_2,i,1}\mu_{i,j_2,0 }+ \zeta_{j_3,i,1}\mu_{i,j_3,0 } \} & i\in \widetilde{Q}_{03}(\Xi). \end{array}\right. \end{equation} | (42) |
Take
\begin{array}{rl}\rho_p(\Xi) = & \{ \alpha_{i,j,1} \alpha_{j,h,0} | i,h ,j \in \widetilde{Q}_0(\Xi), i\neq h \} \cup \{ \beta_{i,j,1} \beta_{j,h,0} | i,h ,j \in \widetilde{Q}_0(\Xi), i\neq h \} \\ &\cup \{ \beta_{i,j,1} \alpha_{j,h,0}| i,h ,j \in \widetilde{Q}_0(\Xi), i\neq h \} \\ &\cup \{ \beta_{i,j,1} \alpha_{j,h,0} | i,h ,j \in \widetilde{Q}_0(\Xi), i\neq h\} ,\end{array} |
and
\rho_a(\Xi) = \{ \alpha_{i,j,1} \gamma_{i,0} - \gamma_{j,1} \alpha_{i,j,0}, \beta_{j,i,1} \gamma_{j,0} - \gamma_{i,1} \beta_{j,i,0} | \alpha_{i,j}\in \widetilde{Q}_1, i < j \}. |
Write
{\rho_a}^\perp(\Xi) = \{ \alpha_{i,j,1} \gamma_{i,0} + \gamma_{j,1} \alpha_{i,j,0}, \beta_{j,i,1} \gamma_{j,0} + \gamma_{i,1} \beta_{j,i,0} | \alpha_{i,j}\in \widetilde{Q}_1, i < j\}. |
They are subsets of the space
Take a subset
\begin{equation} \rho(\Xi,J) = \rho_p(\Xi) \cup \rho_a(\Xi) \cup \bigcup\limits_{i\in \widetilde{Q}_0\setminus J} U_{i,0} \cup \bigcup\limits_{i\in J} U^-_{i,0}, \end{equation} | (43) |
and
\begin{equation} \rho^{\perp}(\Xi,J) = {\rho_a}^\perp(\Xi) \cup \bigcup\limits_{i\in \widetilde{Q}_0\setminus J} U^{\perp}_{i,0} \cup \bigcup\limits_{i\in J} U^{-,\perp}_{i,0}. \end{equation} | (44) |
We have the following descriptions of the relation sets for the
Proposition 5.19. Let
Proposition 5.20. Let
1. By constructing
2. The complete
3. Though we need the field
We would like to thank the referees for reading the manuscript carefully and for suggestions and comments on revising and improving the paper. They also thank the referee for bring [24] to their attention.
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