git-equivalence and semi-stable subcategories of quiver...
TRANSCRIPT
![Page 1: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/1.jpg)
GIT-Equivalence and Semi-Stable Subcategoriesof Quiver Representations
Valerie GrangerJoint work with Calin Chindris
November 21, 2016
![Page 2: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/2.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 3: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/3.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0
V = a representation of the quiver Q V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 4: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/4.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 5: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/5.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i
V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 6: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/6.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a.
dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 7: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/7.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 8: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/8.jpg)
Notation
Q = (Q0,Q1, t,h) is a quiver
K = algebraically closed field of characteristic 0 V = a representation of the quiver Q
V (i) is the K -vector space at vertex i V (a) is the K -linear map along arrow a. dimV = the dimension vector of V .
rep(Q) is the category of finite dimensional quiverrepresentations.
![Page 9: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/9.jpg)
Euler Inner Product and Semi-stability
Given a quiver Q with vertex set Qo and arrow set Q1, we definethe Euler inner product of two vectors α and β in ZQ0 to be
⟨α,β⟩ = ∑i∈Qo
α(i)β(i) − ∑a∈Q1
α(ta)β(ha)
From now on, assume that Q is a connected acyclic quiver.
Let α ∈ QQ0 .A representation V ∈ rep(Q) is said to be ⟨α,−⟩-semi-stable if:
⟨α,dimV ⟩ = 0 and ⟨α,dimV ′⟩ ≤ 0
for all subrepresentations V ′ ≤ V .
Likewise, it is ⟨α,−⟩-stable if the inequality is strict for proper,non-trivial subrepresentations.
![Page 10: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/10.jpg)
Euler Inner Product and Semi-stability
Given a quiver Q with vertex set Qo and arrow set Q1, we definethe Euler inner product of two vectors α and β in ZQ0 to be
⟨α,β⟩ = ∑i∈Qo
α(i)β(i) − ∑a∈Q1
α(ta)β(ha)
From now on, assume that Q is a connected acyclic quiver.
Let α ∈ QQ0 .A representation V ∈ rep(Q) is said to be ⟨α,−⟩-semi-stable if:
⟨α,dimV ⟩ = 0 and ⟨α,dimV ′⟩ ≤ 0
for all subrepresentations V ′ ≤ V .
Likewise, it is ⟨α,−⟩-stable if the inequality is strict for proper,non-trivial subrepresentations.
![Page 11: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/11.jpg)
Euler Inner Product and Semi-stability
Given a quiver Q with vertex set Qo and arrow set Q1, we definethe Euler inner product of two vectors α and β in ZQ0 to be
⟨α,β⟩ = ∑i∈Qo
α(i)β(i) − ∑a∈Q1
α(ta)β(ha)
From now on, assume that Q is a connected acyclic quiver.
Let α ∈ QQ0 .A representation V ∈ rep(Q) is said to be ⟨α,−⟩-semi-stable if:
⟨α,dimV ⟩ = 0 and ⟨α,dimV ′⟩ ≤ 0
for all subrepresentations V ′ ≤ V .
Likewise, it is ⟨α,−⟩-stable if the inequality is strict for proper,non-trivial subrepresentations.
![Page 12: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/12.jpg)
Euler Inner Product and Semi-stability
Given a quiver Q with vertex set Qo and arrow set Q1, we definethe Euler inner product of two vectors α and β in ZQ0 to be
⟨α,β⟩ = ∑i∈Qo
α(i)β(i) − ∑a∈Q1
α(ta)β(ha)
From now on, assume that Q is a connected acyclic quiver.
Let α ∈ QQ0 .A representation V ∈ rep(Q) is said to be ⟨α,−⟩-semi-stable if:
⟨α,dimV ⟩ = 0 and ⟨α,dimV ′⟩ ≤ 0
for all subrepresentations V ′ ≤ V .
Likewise, it is ⟨α,−⟩-stable if the inequality is strict for proper,non-trivial subrepresentations.
![Page 13: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/13.jpg)
Euler Inner Product and Semi-stability
Given a quiver Q with vertex set Qo and arrow set Q1, we definethe Euler inner product of two vectors α and β in ZQ0 to be
⟨α,β⟩ = ∑i∈Qo
α(i)β(i) − ∑a∈Q1
α(ta)β(ha)
From now on, assume that Q is a connected acyclic quiver.
Let α ∈ QQ0 .A representation V ∈ rep(Q) is said to be ⟨α,−⟩-semi-stable if:
⟨α,dimV ⟩ = 0 and ⟨α,dimV ′⟩ ≤ 0
for all subrepresentations V ′ ≤ V .
Likewise, it is ⟨α,−⟩-stable if the inequality is strict for proper,non-trivial subrepresentations.
![Page 14: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/14.jpg)
Schur Representations and Generic Dimension Vectors
Recall that a representation V is called Schur ifHomQ(V ,V ) = K .
We say a dimension vector β is a Schur root if there exists aβ-dimensional Schur representation.
We say that β′ β if every β-dimensional representation has asubrepresentation of dimension β′.
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Schur Representations and Generic Dimension Vectors
Recall that a representation V is called Schur ifHomQ(V ,V ) = K .
We say a dimension vector β is a Schur root if there exists aβ-dimensional Schur representation.
We say that β′ β if every β-dimensional representation has asubrepresentation of dimension β′.
![Page 16: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/16.jpg)
Schur Representations and Generic Dimension Vectors
Recall that a representation V is called Schur ifHomQ(V ,V ) = K .
We say a dimension vector β is a Schur root if there exists aβ-dimensional Schur representation.
We say that β′ β if every β-dimensional representation has asubrepresentation of dimension β′.
![Page 17: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/17.jpg)
Semi-Stability
rep(Q)ss⟨α,−⟩ is the full subcategory of rep(Q) whose objects are
⟨α,−⟩-semi-stable.
We say that β is ⟨α,−⟩-(semi)-stable if there exists aβ-dimensional, ⟨α,−⟩-(semi)-stable representation. This isequivalent to saying
⟨α,β⟩ = 0 and ⟨α,β′⟩ ≤ 0 for all β′ β
And respectively, β is ⟨α,−⟩-stable if the second inequality is strictfor β′ ≠ 0, β.
![Page 18: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/18.jpg)
Semi-Stability
rep(Q)ss⟨α,−⟩ is the full subcategory of rep(Q) whose objects are
⟨α,−⟩-semi-stable.
We say that β is ⟨α,−⟩-(semi)-stable if there exists aβ-dimensional, ⟨α,−⟩-(semi)-stable representation. This isequivalent to saying
⟨α,β⟩ = 0 and ⟨α,β′⟩ ≤ 0 for all β′ β
And respectively, β is ⟨α,−⟩-stable if the second inequality is strictfor β′ ≠ 0, β.
![Page 19: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/19.jpg)
Semi-Stability
rep(Q)ss⟨α,−⟩ is the full subcategory of rep(Q) whose objects are
⟨α,−⟩-semi-stable.
We say that β is ⟨α,−⟩-(semi)-stable if there exists aβ-dimensional, ⟨α,−⟩-(semi)-stable representation. This isequivalent to saying
⟨α,β⟩ = 0 and ⟨α,β′⟩ ≤ 0 for all β′ β
And respectively, β is ⟨α,−⟩-stable if the second inequality is strictfor β′ ≠ 0, β.
![Page 20: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/20.jpg)
Semi-Stability
rep(Q)ss⟨α,−⟩ is the full subcategory of rep(Q) whose objects are
⟨α,−⟩-semi-stable.
We say that β is ⟨α,−⟩-(semi)-stable if there exists aβ-dimensional, ⟨α,−⟩-(semi)-stable representation. This isequivalent to saying
⟨α,β⟩ = 0 and ⟨α,β′⟩ ≤ 0 for all β′ β
And respectively, β is ⟨α,−⟩-stable if the second inequality is strictfor β′ ≠ 0, β.
![Page 21: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/21.jpg)
The cone of effective weights
The cone of effective weights for a dimension vector β:
D(β) = α ∈ QQo ∣⟨α,β⟩ = 0, ⟨α,β′⟩ ≤ 0, β′ β
Theorem (Schofield)
β is a Schur root if and only ifD(β) = α ∈ QQ0 ∣⟨α,β⟩ = 0, ⟨α,β′⟩ < 0 ∀ β′ β,β ≠ 0, β isnon-empty if and only if β is ⟨β,−⟩ − ⟨−, β⟩-stable.
![Page 22: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/22.jpg)
The cone of effective weights
The cone of effective weights for a dimension vector β:
D(β) = α ∈ QQo ∣⟨α,β⟩ = 0, ⟨α,β′⟩ ≤ 0, β′ β
Theorem (Schofield)
β is a Schur root if and only ifD(β) = α ∈ QQ0 ∣⟨α,β⟩ = 0, ⟨α,β′⟩ < 0 ∀ β′ β,β ≠ 0, β isnon-empty if and only if β is ⟨β,−⟩ − ⟨−, β⟩-stable.
![Page 23: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/23.jpg)
Big Question
Two rational vectors α1, α2 ∈ QQ0 , are said to be GIT-equivalent(or ss-equivalent) if:
rep(Q)ss⟨α1,−⟩ = rep(Q)ss⟨α2,−⟩
Main Question: Find necessary and sufficient conditions for α1 andα2 to be GIT-equivalent.
Colin Ingalls, Charles Paquette, and Hugh Thomas gave acharacterization in the case that Q is tame, which was published in2015. Their work was motivated by studying what subcategories ofrep(Q) arise as semi-stable-subcategories, with an eye towardsforming a lattice of subcategories.
![Page 24: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/24.jpg)
Big Question
Two rational vectors α1, α2 ∈ QQ0 , are said to be GIT-equivalent(or ss-equivalent) if:
rep(Q)ss⟨α1,−⟩ = rep(Q)ss⟨α2,−⟩
Main Question: Find necessary and sufficient conditions for α1 andα2 to be GIT-equivalent.
Colin Ingalls, Charles Paquette, and Hugh Thomas gave acharacterization in the case that Q is tame, which was published in2015. Their work was motivated by studying what subcategories ofrep(Q) arise as semi-stable-subcategories, with an eye towardsforming a lattice of subcategories.
![Page 25: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/25.jpg)
Big Question
Two rational vectors α1, α2 ∈ QQ0 , are said to be GIT-equivalent(or ss-equivalent) if:
rep(Q)ss⟨α1,−⟩ = rep(Q)ss⟨α2,−⟩
Main Question: Find necessary and sufficient conditions for α1 andα2 to be GIT-equivalent.
Colin Ingalls, Charles Paquette, and Hugh Thomas gave acharacterization in the case that Q is tame, which was published in2015. Their work was motivated by studying what subcategories ofrep(Q) arise as semi-stable-subcategories, with an eye towardsforming a lattice of subcategories.
![Page 26: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/26.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 27: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/27.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 28: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/28.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 29: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/29.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives
Remaining indecomposables occur in tubes Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 30: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/30.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 31: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/31.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many)
Finitely many non-homogeneous tubes
![Page 32: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/32.jpg)
A tiny bit of AR Theory for tame path algebras
We can build the Auslander-Reiten quiver, Γ, of the path algebraKQ.
Each indecomposable KQ-module corresponds to a vertex in Γ
All projective indecomposables lie in the same connectedcomponent, and all indecomposables in that component(called preprojectives) are exceptional (i.e., their dimensionvectors are real Schur roots)
Similarly for injectives/preinjectives Remaining indecomposables occur in tubes
Homogeneous tubes (infinitely many) Finitely many non-homogeneous tubes
![Page 33: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/33.jpg)
Non-Homogeneous tubes
Now for example, a rank 3 tube looks like:
β11β12β13β11
β12β11
β13β12
β11β13
β11β12β13
β12β13β11
β13β11β12
β11β12β13
⋮ ↑ Not Schur
↓ Schur
τ−τ−τ−
τ−τ−
τ−τ−τ− δ-dimensional, Schur
![Page 34: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/34.jpg)
Previous work on the tame case
IPT did the following:
Label the non-homogeneous regular tubes in the A-R quiver1, . . . ,N, and let the period of the i th tube be ri .
Let βi ,j be the j th quasi-simple root from the i th tube, where1 ≤ j ≤ ri .
Set I to be the multi-index (a1, . . . , aN), where 1 ≤ ai ≤ ri , andR to be the set of all permissible such multi-indices.
Define the cone CI to be the rational convex polyhedral conegenerated by δ, together with βi ,j , except for βiai .
![Page 35: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/35.jpg)
Previous work on the tame case
IPT did the following:
Label the non-homogeneous regular tubes in the A-R quiver1, . . . ,N, and let the period of the i th tube be ri .
Let βi ,j be the j th quasi-simple root from the i th tube, where1 ≤ j ≤ ri .
Set I to be the multi-index (a1, . . . , aN), where 1 ≤ ai ≤ ri , andR to be the set of all permissible such multi-indices.
Define the cone CI to be the rational convex polyhedral conegenerated by δ, together with βi ,j , except for βiai .
![Page 36: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/36.jpg)
Previous work on the tame case
IPT did the following:
Label the non-homogeneous regular tubes in the A-R quiver1, . . . ,N, and let the period of the i th tube be ri .
Let βi ,j be the j th quasi-simple root from the i th tube, where1 ≤ j ≤ ri .
Set I to be the multi-index (a1, . . . , aN), where 1 ≤ ai ≤ ri , andR to be the set of all permissible such multi-indices.
Define the cone CI to be the rational convex polyhedral conegenerated by δ, together with βi ,j , except for βiai .
![Page 37: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/37.jpg)
Previous work on the tame case
IPT did the following:
Label the non-homogeneous regular tubes in the A-R quiver1, . . . ,N, and let the period of the i th tube be ri .
Let βi ,j be the j th quasi-simple root from the i th tube, where1 ≤ j ≤ ri .
Set I to be the multi-index (a1, . . . , aN), where 1 ≤ ai ≤ ri , andR to be the set of all permissible such multi-indices.
Define the cone CI to be the rational convex polyhedral conegenerated by δ, together with βi ,j , except for βiai .
![Page 38: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/38.jpg)
Previous work on the tame case
Define J = CII∈R ∪ D(β)β, where β is a real Schur root. SetJα = C ∈ J ∣α ∈ C.
Theorem (Ingalls, Paquette, Thomas)
For α1, α2 ∈ ZQ0 , we have that α1 and α2 are GIT equivalent if andonly if Jα1 = Jα2 .
![Page 39: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/39.jpg)
Previous work on the tame case
Define J = CII∈R ∪ D(β)β, where β is a real Schur root. SetJα = C ∈ J ∣α ∈ C.
Theorem (Ingalls, Paquette, Thomas)
For α1, α2 ∈ ZQ0 , we have that α1 and α2 are GIT equivalent if andonly if Jα1 = Jα2 .
![Page 40: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/40.jpg)
The GIT-cone
Let Q be any acyclic quiver.
The semi-stable locus of α with respect to β is:
rep(Q, β)ss⟨α,−⟩The GIT-cone of α with respect to β:
C(β)α = α′ ∈ D(β)∣ rep(Q, β)ss⟨α,−⟩ ⊆ rep(Q, β)ss⟨α′,−⟩
This consists of all effective weights “weaker” than α.The GIT-fan associated to (Q, β) is:
F(β) = C(β)α∣α ∈ D(β) ∪ 0
![Page 41: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/41.jpg)
The GIT-cone
Let Q be any acyclic quiver.
The semi-stable locus of α with respect to β is:
rep(Q, β)ss⟨α,−⟩
The GIT-cone of α with respect to β:
C(β)α = α′ ∈ D(β)∣ rep(Q, β)ss⟨α,−⟩ ⊆ rep(Q, β)ss⟨α′,−⟩
This consists of all effective weights “weaker” than α.The GIT-fan associated to (Q, β) is:
F(β) = C(β)α∣α ∈ D(β) ∪ 0
![Page 42: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/42.jpg)
The GIT-cone
Let Q be any acyclic quiver.
The semi-stable locus of α with respect to β is:
rep(Q, β)ss⟨α,−⟩The GIT-cone of α with respect to β:
C(β)α = α′ ∈ D(β)∣ rep(Q, β)ss⟨α,−⟩ ⊆ rep(Q, β)ss⟨α′,−⟩
This consists of all effective weights “weaker” than α.The GIT-fan associated to (Q, β) is:
F(β) = C(β)α∣α ∈ D(β) ∪ 0
![Page 43: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/43.jpg)
The GIT-cone
Let Q be any acyclic quiver.
The semi-stable locus of α with respect to β is:
rep(Q, β)ss⟨α,−⟩The GIT-cone of α with respect to β:
C(β)α = α′ ∈ D(β)∣ rep(Q, β)ss⟨α,−⟩ ⊆ rep(Q, β)ss⟨α′,−⟩
This consists of all effective weights “weaker” than α.
The GIT-fan associated to (Q, β) is:
F(β) = C(β)α∣α ∈ D(β) ∪ 0
![Page 44: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/44.jpg)
The GIT-cone
Let Q be any acyclic quiver.
The semi-stable locus of α with respect to β is:
rep(Q, β)ss⟨α,−⟩The GIT-cone of α with respect to β:
C(β)α = α′ ∈ D(β)∣ rep(Q, β)ss⟨α,−⟩ ⊆ rep(Q, β)ss⟨α′,−⟩
This consists of all effective weights “weaker” than α.The GIT-fan associated to (Q, β) is:
F(β) = C(β)α∣α ∈ D(β) ∪ 0
![Page 45: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/45.jpg)
A few remarks about fans
A fan is finite a collection of (rational convex polyhedral) conessatisfying some additional properties. It is said to be pure ofdimension n if all cones that are maximal with respect to inclusionare of dimension n.
Theorem
F(β) is a finite fan cover of D(β), and if β is a Schur root, thenF(β) is a pure fan of dimension ∣Q0∣ − 1.
A very useful property of pure fans for our result is the following:
(Keicher, 2012) Let Σ ⊆ Qn be a pure n-dimensional fan withconvex support ∣Σ∣, and let τ ∈ Σ be such that τ ∩ ∣Σ∣ ≠ ∅. Thenτ is the intersection over all σ ∈ Σ(m) satisfying τ ⪯ σ.
![Page 46: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/46.jpg)
A few remarks about fans
A fan is finite a collection of (rational convex polyhedral) conessatisfying some additional properties. It is said to be pure ofdimension n if all cones that are maximal with respect to inclusionare of dimension n.
Theorem
F(β) is a finite fan cover of D(β), and if β is a Schur root, thenF(β) is a pure fan of dimension ∣Q0∣ − 1.
A very useful property of pure fans for our result is the following:
(Keicher, 2012) Let Σ ⊆ Qn be a pure n-dimensional fan withconvex support ∣Σ∣, and let τ ∈ Σ be such that τ ∩ ∣Σ∣ ≠ ∅. Thenτ is the intersection over all σ ∈ Σ(m) satisfying τ ⪯ σ.
![Page 47: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/47.jpg)
A few remarks about fans
A fan is finite a collection of (rational convex polyhedral) conessatisfying some additional properties. It is said to be pure ofdimension n if all cones that are maximal with respect to inclusionare of dimension n.
Theorem
F(β) is a finite fan cover of D(β), and if β is a Schur root, thenF(β) is a pure fan of dimension ∣Q0∣ − 1.
A very useful property of pure fans for our result is the following:
(Keicher, 2012) Let Σ ⊆ Qn be a pure n-dimensional fan withconvex support ∣Σ∣, and let τ ∈ Σ be such that τ ∩ ∣Σ∣ ≠ ∅. Thenτ is the intersection over all σ ∈ Σ(m) satisfying τ ⪯ σ.
![Page 48: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/48.jpg)
A few remarks about fans
A fan is finite a collection of (rational convex polyhedral) conessatisfying some additional properties. It is said to be pure ofdimension n if all cones that are maximal with respect to inclusionare of dimension n.
Theorem
F(β) is a finite fan cover of D(β), and if β is a Schur root, thenF(β) is a pure fan of dimension ∣Q0∣ − 1.
A very useful property of pure fans for our result is the following:
(Keicher, 2012) Let Σ ⊆ Qn be a pure n-dimensional fan withconvex support ∣Σ∣, and let τ ∈ Σ be such that τ ∩ ∣Σ∣ ≠ ∅. Thenτ is the intersection over all σ ∈ Σ(m) satisfying τ ⪯ σ.
![Page 49: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/49.jpg)
Main Theorem
Set
I = C(β)α∣β is a Schur root and C(β)α is maximal
Iα = C ∈ I ∣α ∈ C
Theorem (Theorem 1)
Let Q be a connected, acyclic quiver. For α1, α2 ∈ QQ0 , α1 ∼GIT α2
if and only if Iα1 = Iα2
That is, we have a collection of cones parametrized by Schur rootswhich characterizes GIT-equivalence classes.
![Page 50: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/50.jpg)
Main Theorem
Set
I = C(β)α∣β is a Schur root and C(β)α is maximal
Iα = C ∈ I ∣α ∈ C
Theorem (Theorem 1)
Let Q be a connected, acyclic quiver. For α1, α2 ∈ QQ0 , α1 ∼GIT α2
if and only if Iα1 = Iα2
That is, we have a collection of cones parametrized by Schur rootswhich characterizes GIT-equivalence classes.
![Page 51: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/51.jpg)
Main Theorem
Set
I = C(β)α∣β is a Schur root and C(β)α is maximal
Iα = C ∈ I ∣α ∈ C
Theorem (Theorem 1)
Let Q be a connected, acyclic quiver. For α1, α2 ∈ QQ0 , α1 ∼GIT α2
if and only if Iα1 = Iα2
That is, we have a collection of cones parametrized by Schur rootswhich characterizes GIT-equivalence classes.
![Page 52: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/52.jpg)
Idea of Proof
Assume Iα1 = Iα2
If β is ⟨α1,−⟩-stable, then α1 ∈ D(β), and of course α1 ∈ C(β)α1 .We can apply Keicher’s result to conclude that C(β)α1 is anintersection of all maximal cones of which it is a face.By the assumption, any such maximal cone contains α2 as well.So, α2 ∈ C(β)α1 . Similarly, α1 ∈ C(β)α2 .Now, if β is arbitrary, use a JH-filtration to break it into a sum of⟨α1,−⟩-stable factors.
![Page 53: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/53.jpg)
Idea of Proof
Assume Iα1 = Iα2
If β is ⟨α1,−⟩-stable, then α1 ∈ D(β), and of course α1 ∈ C(β)α1 .
We can apply Keicher’s result to conclude that C(β)α1 is anintersection of all maximal cones of which it is a face.By the assumption, any such maximal cone contains α2 as well.So, α2 ∈ C(β)α1 . Similarly, α1 ∈ C(β)α2 .Now, if β is arbitrary, use a JH-filtration to break it into a sum of⟨α1,−⟩-stable factors.
![Page 54: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/54.jpg)
Idea of Proof
Assume Iα1 = Iα2
If β is ⟨α1,−⟩-stable, then α1 ∈ D(β), and of course α1 ∈ C(β)α1 .We can apply Keicher’s result to conclude that C(β)α1 is anintersection of all maximal cones of which it is a face.
By the assumption, any such maximal cone contains α2 as well.So, α2 ∈ C(β)α1 . Similarly, α1 ∈ C(β)α2 .Now, if β is arbitrary, use a JH-filtration to break it into a sum of⟨α1,−⟩-stable factors.
![Page 55: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/55.jpg)
Idea of Proof
Assume Iα1 = Iα2
If β is ⟨α1,−⟩-stable, then α1 ∈ D(β), and of course α1 ∈ C(β)α1 .We can apply Keicher’s result to conclude that C(β)α1 is anintersection of all maximal cones of which it is a face.By the assumption, any such maximal cone contains α2 as well.So, α2 ∈ C(β)α1 . Similarly, α1 ∈ C(β)α2 .
Now, if β is arbitrary, use a JH-filtration to break it into a sum of⟨α1,−⟩-stable factors.
![Page 56: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/56.jpg)
Idea of Proof
Assume Iα1 = Iα2
If β is ⟨α1,−⟩-stable, then α1 ∈ D(β), and of course α1 ∈ C(β)α1 .We can apply Keicher’s result to conclude that C(β)α1 is anintersection of all maximal cones of which it is a face.By the assumption, any such maximal cone contains α2 as well.So, α2 ∈ C(β)α1 . Similarly, α1 ∈ C(β)α2 .Now, if β is arbitrary, use a JH-filtration to break it into a sum of⟨α1,−⟩-stable factors.
![Page 57: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/57.jpg)
The tame case
If Q is tame, the collection of cones I is exactly the collection Jdefined by IPT.
Precisely, CI is a maximal GIT-cone, namely C(δ)αIwhere
αI = δ +∑j≠ai ∑Ni=1 βij
Main ingredients in proof:
Realize CI as the orbit cone of a representation: Ω(ZI ), whereZI is a direct sum of Zi , where Zi is the unique δ dimensionalrepresentation with regular socle of dimension βiai .
Show that the Zi ’s and the homogeneous δ-dimensionalrepresentations are the only δ-dimensional representationswhich are polystable with respect to the weightαI = δ +∑j≠ai ∑
Ni=1 βij
Invoke a result that C(β)α = ⋂Ω(W ) (Chindris, “On GITFans for Quivers”)
![Page 58: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/58.jpg)
The tame case
If Q is tame, the collection of cones I is exactly the collection Jdefined by IPT.Precisely, CI is a maximal GIT-cone, namely C(δ)αI
whereαI = δ +∑j≠ai ∑
Ni=1 βij
Main ingredients in proof:
Realize CI as the orbit cone of a representation: Ω(ZI ), whereZI is a direct sum of Zi , where Zi is the unique δ dimensionalrepresentation with regular socle of dimension βiai .
Show that the Zi ’s and the homogeneous δ-dimensionalrepresentations are the only δ-dimensional representationswhich are polystable with respect to the weightαI = δ +∑j≠ai ∑
Ni=1 βij
Invoke a result that C(β)α = ⋂Ω(W ) (Chindris, “On GITFans for Quivers”)
![Page 59: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/59.jpg)
The tame case
If Q is tame, the collection of cones I is exactly the collection Jdefined by IPT.Precisely, CI is a maximal GIT-cone, namely C(δ)αI
whereαI = δ +∑j≠ai ∑
Ni=1 βij
Main ingredients in proof:
Realize CI as the orbit cone of a representation: Ω(ZI ), whereZI is a direct sum of Zi , where Zi is the unique δ dimensionalrepresentation with regular socle of dimension βiai .
Show that the Zi ’s and the homogeneous δ-dimensionalrepresentations are the only δ-dimensional representationswhich are polystable with respect to the weightαI = δ +∑j≠ai ∑
Ni=1 βij
Invoke a result that C(β)α = ⋂Ω(W ) (Chindris, “On GITFans for Quivers”)
![Page 60: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/60.jpg)
The tame case
If Q is tame, the collection of cones I is exactly the collection Jdefined by IPT.Precisely, CI is a maximal GIT-cone, namely C(δ)αI
whereαI = δ +∑j≠ai ∑
Ni=1 βij
Main ingredients in proof:
Realize CI as the orbit cone of a representation: Ω(ZI ), whereZI is a direct sum of Zi , where Zi is the unique δ dimensionalrepresentation with regular socle of dimension βiai .
Show that the Zi ’s and the homogeneous δ-dimensionalrepresentations are the only δ-dimensional representationswhich are polystable with respect to the weightαI = δ +∑j≠ai ∑
Ni=1 βij
Invoke a result that C(β)α = ⋂Ω(W ) (Chindris, “On GITFans for Quivers”)
![Page 61: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/61.jpg)
The tame case
If Q is tame, the collection of cones I is exactly the collection Jdefined by IPT.Precisely, CI is a maximal GIT-cone, namely C(δ)αI
whereαI = δ +∑j≠ai ∑
Ni=1 βij
Main ingredients in proof:
Realize CI as the orbit cone of a representation: Ω(ZI ), whereZI is a direct sum of Zi , where Zi is the unique δ dimensionalrepresentation with regular socle of dimension βiai .
Show that the Zi ’s and the homogeneous δ-dimensionalrepresentations are the only δ-dimensional representationswhich are polystable with respect to the weightαI = δ +∑j≠ai ∑
Ni=1 βij
Invoke a result that C(β)α = ⋂Ω(W ) (Chindris, “On GITFans for Quivers”)
![Page 62: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/62.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 63: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/63.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0
Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 64: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/64.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.
In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 65: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/65.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.
D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 66: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/66.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)
D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 67: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/67.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)
For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 68: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/68.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).
Lastly, D(δ) is generated by δ.
![Page 69: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/69.jpg)
Example
Let Q = A1:
1 2
Real Roots: (n,n + 1) and (n + 1,n) for n ≥ 0Isotropic Roots: (n,n) for n ≥ 1.In particular, δ = (1,1) is the unique isotropic Schur root.D((0,1)) is generated by (1,2) and (−1,−2)D((1,0)) is generated by (0,1) and (0,−1)For n ≥ 1, D((n,n + 1)) is generated by (n + 1,n + 2), andD((n + 1,n)) is generated by (n,n − 1).Lastly, D(δ) is generated by δ.
![Page 70: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/70.jpg)
Example
Rays extendingthrough latticepoints of y = x + 1and y = x − 1
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Example
Rays extendingthrough latticepoints of y = x + 1and y = x − 1
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Example
Two weights α1
and α2 are GITequivalent if andonly if they are onthe same collectionof rays.
![Page 73: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/73.jpg)
Example
In this case, sincethe intersection ofany two rays is(0,0), we have thatα1, α2 areGIT-equivalent ifthey are
both = (0,0) both in the
same ray, i.e.,α1 = λα2 forsome λ ∈ Q
![Page 74: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/74.jpg)
Further Questions
How can we get our hands on these maximal GIT-cones ofSchur roots for wild quivers?
Would a similar result hold, using similar techniques, forquivers with relations?
![Page 75: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/75.jpg)
Example
Let Q = A2:
0
1 2
We want to give an idea of the cones in I. Recall that
I = C(δ)αII∈R ∪ D(β)β
where the union is over all real Schur roots β.
![Page 76: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/76.jpg)
Example
Let Q = A2:
0
1 2
We want to give an idea of the cones in I. Recall that
I = C(δ)αII∈R ∪ D(β)β
where the union is over all real Schur roots β.
![Page 77: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/77.jpg)
Example
Starting with the dimension vectors of the projective and injectiveindecomposables, and applying the A-R translate, we get infinitelymany real Schur roots:
dimP0 = (1,0,1)τ−Ð→ (2,2,3)
τ−Ð→ (4,3,4)τ−Ð→ (5,5,6)
τ−Ð→ ⋯dimP1 = (1,1,2)
τ−Ð→ (3,2,3)τ−Ð→ (4,4,5)
τ−Ð→ (6,5,6)τ−Ð→ ⋯
dimP2 = (0,0,1)τ−Ð→ (2,1,2)
τ−Ð→ (3,3,4)τ−Ð→ (5,4,5)
τ−Ð→ ⋯dimI0 = (1,1,0)
τÐ→ (2,3,2)τÐ→ (4,4,3)
τÐ→ (5,6,5)τ−Ð→ ⋯
dimI1 = (0,1,0)τÐ→ (2,2,1)
τÐ→ (3,4,3)τÐ→ (5,5,4)
τ−Ð→ ⋯dimI2 = (1,2,1)
τÐ→ (3,3,2)τÐ→ (4,5,4)
τÐ→ (6,6,5)τ−Ð→ ⋯
Each one of these real Schur roots will correspond to a D(β) ∈ I.
![Page 78: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/78.jpg)
Example
Starting with the dimension vectors of the projective and injectiveindecomposables, and applying the A-R translate, we get infinitelymany real Schur roots:
dimP0 = (1,0,1)τ−Ð→ (2,2,3)
τ−Ð→ (4,3,4)τ−Ð→ (5,5,6)
τ−Ð→ ⋯dimP1 = (1,1,2)
τ−Ð→ (3,2,3)τ−Ð→ (4,4,5)
τ−Ð→ (6,5,6)τ−Ð→ ⋯
dimP2 = (0,0,1)τ−Ð→ (2,1,2)
τ−Ð→ (3,3,4)τ−Ð→ (5,4,5)
τ−Ð→ ⋯dimI0 = (1,1,0)
τÐ→ (2,3,2)τÐ→ (4,4,3)
τÐ→ (5,6,5)τ−Ð→ ⋯
dimI1 = (0,1,0)τÐ→ (2,2,1)
τÐ→ (3,4,3)τÐ→ (5,5,4)
τ−Ð→ ⋯dimI2 = (1,2,1)
τÐ→ (3,3,2)τÐ→ (4,5,4)
τÐ→ (6,6,5)τ−Ð→ ⋯
Each one of these real Schur roots will correspond to a D(β) ∈ I.
![Page 79: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/79.jpg)
Example
Starting with the dimension vectors of the projective and injectiveindecomposables, and applying the A-R translate, we get infinitelymany real Schur roots:
dimP0 = (1,0,1)τ−Ð→ (2,2,3)
τ−Ð→ (4,3,4)τ−Ð→ (5,5,6)
τ−Ð→ ⋯dimP1 = (1,1,2)
τ−Ð→ (3,2,3)τ−Ð→ (4,4,5)
τ−Ð→ (6,5,6)τ−Ð→ ⋯
dimP2 = (0,0,1)τ−Ð→ (2,1,2)
τ−Ð→ (3,3,4)τ−Ð→ (5,4,5)
τ−Ð→ ⋯dimI0 = (1,1,0)
τÐ→ (2,3,2)τÐ→ (4,4,3)
τÐ→ (5,6,5)τ−Ð→ ⋯
dimI1 = (0,1,0)τÐ→ (2,2,1)
τÐ→ (3,4,3)τÐ→ (5,5,4)
τ−Ð→ ⋯dimI2 = (1,2,1)
τÐ→ (3,3,2)τÐ→ (4,5,4)
τÐ→ (6,6,5)τ−Ð→ ⋯
Each one of these real Schur roots will correspond to a D(β) ∈ I.
![Page 80: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/80.jpg)
Example
For example, if we takeβ = (0,0,1), we have D(β) isgenerated by−dimP0 = (−1,0,−1),−dimP1 = (−1,−1,−2) and(1,0,1), which is −⟨−, β⟩-stable.
Thus, D(β) looks like:
![Page 81: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/81.jpg)
Example
For example, if we takeβ = (0,0,1), we have D(β) isgenerated by−dimP0 = (−1,0,−1),−dimP1 = (−1,−1,−2) and(1,0,1), which is −⟨−, β⟩-stable.
Thus, D(β) looks like:
![Page 82: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/82.jpg)
Example
If we take β = (1,1,2), which issincere, we have D(β) isgenerated by (0,1,1) and(2,1,2) which are both−⟨−, β⟩-stable.
Thus, D(β) looks like:
![Page 83: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/83.jpg)
Example
If we take β = (1,1,2), which issincere, we have D(β) isgenerated by (0,1,1) and(2,1,2) which are both−⟨−, β⟩-stable.
Thus, D(β) looks like:
![Page 84: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/84.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).
If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 + z2 − xy − xz − yz = 1
β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 85: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/85.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0
⟨β,β⟩ = x2 + y2 + z2 − xy − xz − yz = 1β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 86: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/86.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 + z2 − xy − xz − yz = 1
β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 87: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/87.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 − 2xy = 1
β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 88: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/88.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 − 2xy = 1β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 89: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/89.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 − 2xy = 1β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.
Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 90: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/90.jpg)
Example
Now, turning to the regular representations, we have δ = (1,1,1).If β = (x , y , z) is quasi-simple, it must satisfy:
⎧⎪⎪⎪⎨⎪⎪⎪⎩
⟨δ, β⟩ = y − z = 0⟨β,β⟩ = x2 + y2 − 2xy = 1β ≤ δ, i.e., x ≤ 1, y ≤ 1, z ≤ 1
So, the only quasi-simples are (0,1,1) and (1,0,0). That is, wehave a single non-homogeneous tube in the regular component ofthe A-R quiver, and it has period 2.Now, β11 = (0,1,1) and β12 = (1,0,0) are themselves real Schurroots, and so D(β11) and D(β12) are in I.
![Page 91: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/91.jpg)
Example
Lastly, we need the C(δ)αI’s.
For I = (1), we haveαI = δ + β12 = (2,1,1) andC(δ)αI
is generated, as a cone,by (1,1,1) and (1,0,0).
For I = (2), we haveαI = δ + β11 = (1,2,2) andC(δ)αI
is generated, as a cone,by (1,1,1) and (0,1,1).
[Animation with many of the cones from I]
![Page 92: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/92.jpg)
Example
Lastly, we need the C(δ)αI’s.
For I = (1), we haveαI = δ + β12 = (2,1,1) andC(δ)αI
is generated, as a cone,by (1,1,1) and (1,0,0).
For I = (2), we haveαI = δ + β11 = (1,2,2) andC(δ)αI
is generated, as a cone,by (1,1,1) and (0,1,1).
[Animation with many of the cones from I]
![Page 93: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/93.jpg)
Example
Lastly, we need the C(δ)αI’s.
For I = (1), we haveαI = δ + β12 = (2,1,1) andC(δ)αI
is generated, as a cone,by (1,1,1) and (1,0,0).
For I = (2), we haveαI = δ + β11 = (1,2,2) andC(δ)αI
is generated, as a cone,by (1,1,1) and (0,1,1).
[Animation with many of the cones from I]
![Page 94: GIT-Equivalence and Semi-Stable Subcategories of Quiver …homepage.math.uiowa.edu/~fbleher/CGMRT2016/Slides/... · 2016. 11. 29. · GIT-Equivalence and Semi-Stable Subcategories](https://reader034.vdocuments.us/reader034/viewer/2022051902/5ff21c4fab241510265e9ef9/html5/thumbnails/94.jpg)
Example
Lastly, we need the C(δ)αI’s.
For I = (1), we haveαI = δ + β12 = (2,1,1) andC(δ)αI
is generated, as a cone,by (1,1,1) and (1,0,0).
For I = (2), we haveαI = δ + β11 = (1,2,2) andC(δ)αI
is generated, as a cone,by (1,1,1) and (0,1,1).
[Animation with many of the cones from I]