international conference on representations of algebras · 2012-08-23 · international conference...
TRANSCRIPT
International Conference on Representations of Algebras
Bielefeld, August 3-17, 2012
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Markus Schmidmeier (Florida Atlantic University):
ADE Posets
Contents
ADE posetsDilworth’s TheoremNoether’s TheoremRelated categories
SymmetriesReflection at the centerRotationShift
InvariantsSlopeDriftCurvature
Summary
ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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ADE Posets
Dilworth’s Theorem. For P a finite poset,
max{
|A| : A ⊆ P antichain}
= min{
|C| : C partition of P by chains}
Recall: A poset of width at most 2 has finite representation type.
In this talk, we consider posets of width at most 3 with lots ofsymmetries.
P(n) :
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Poset representations
Let P be one of the above posets.
Poset representations: By repK P we denote the category of allsystems
(
V∗, (Vx)x∈P)
which satisfy:
◮ V∗ is a finite dimensional K -vector space,
◮ Vx ⊂ V∗ is a subspace for each x ∈ P,
◮ Vx ⊂ Vy holds whenever x < y in P, and
◮ there exist x , y ∈ P with Vx = 0 and Vy = V∗ (finitesupport).
Poset representations
Let P be one of the above posets.
Poset representations: By repK P we denote the category of allsystems
(
V∗, (Vx)x∈P)
which satisfy:
◮ V∗ is a finite dimensional K -vector space,
◮ Vx ⊂ V∗ is a subspace for each x ∈ P,
◮ Vx ⊂ Vy holds whenever x < y in P, and
◮ there exist x , y ∈ P with Vx = 0 and Vy = V∗ (finitesupport).
A chain of categories: The relations for P(n + 1) are satisfied inP(n), hence the categories repK P(n), n ∈ N, form a chain:
repK P(1) ⊂ repK P(2) ⊂ repK P(3) ⊂ · · ·
Symmetries of the poset
We consider three symmetry operations of the posets, picturedhere for the poset P(3), and the endofunctors which they induceon its category of representations.
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“rotation”
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“reflection”
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“shift”
AimNoether’s Theorem: Any symmetry of the action of a physical
system has a corresponding conservation law.
AimNoether’s Theorem: Any symmetry of the action of a physical
system has a corresponding conservation law.
◮ Symmetries of the posets emphasize the role of operations:
reflection dualityrotation AR-translation τ
shift graded shift
AimNoether’s Theorem: Any symmetry of the action of a physical
system has a corresponding conservation law.
◮ Symmetries of the posets emphasize the role of operations:
reflection dualityrotation AR-translation τ
shift graded shift
◮ In turn, the operations on repP(n) motivate invariants:
graded shift slope σ
duality drift δ = d σ
d n
AR-translation curvature κ = d σ
d τ
AimNoether’s Theorem: Any symmetry of the action of a physical
system has a corresponding conservation law.
◮ Symmetries of the posets emphasize the role of operations:
reflection dualityrotation AR-translation τ
shift graded shift
◮ In turn, the operations on repP(n) motivate invariants:
graded shift slope σ
duality drift δ = d σ
d n
AR-translation curvature κ = d σ
d τ
◮ Using the invariants we study the chain of categories:
repK P(1) ⊂ repK P(2) ⊂ repK P(3) ⊂ · · ·
Related categoriesThe category S(n) of invariant subspaces (C.M. Ringel, D. Simson,P. Zhang e.a.) occurs as a factor (M. Kleiner):
Proposition. The functor G : repK P(n) → S(n) given by the
picture
V ′′1
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V ′2
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Related categoriesThe category S(n) of invariant subspaces (C.M. Ringel, D. Simson,P. Zhang e.a.) occurs as a factor (M. Kleiner):
Proposition. The functor G : repK P(n) → S(n) given by the
picture
V ′′1
V ′′2
V ′2
V ′3
V3
V4
V∗
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is full and has kernel the ideal Z consisting of all maps which factor
through a sum of projective poset representations of type P ′′i .
Related categoriesThe category S(n) of invariant subspaces (C.M. Ringel, D. Simson,P. Zhang e.a.) occurs as a factor (M. Kleiner):
Proposition. The functor G : repK P(n) → S(n) given by the
picture
V ′′1
V ′′2
V ′2
V ′3
V3
V4
V∗
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is full and has kernel the ideal Z consisting of all maps which factor
through a sum of projective poset representations of type P ′′i .
Corollary. The stable category repKP(n) is equivalent to the
stable category of vector bundles over weighted projective linesX(2, 3, n) (X.-W. Chen, D. Kussin, H. Lenzing, H. Meltzer).Moreover, repK P(n) is equivalent to the category of gradedlattices over tiled orders (W. Rump).
The reflectionEach of the posets P(n) is symmetric with respect to reflection atone of the centers. On the module category, this operation inducesthe reflection duality:
V1′′
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The reflectionEach of the posets P(n) is symmetric with respect to reflection atone of the centers. On the module category, this operation inducesthe reflection duality:
V1′′
V2′′
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V3′
V3
V4
V∗
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Lemma. The reflection duality D : repK P(n) → repK P(n)preserves the ideal Z and induces the duality on the quotient
S(n) = repK P(n)/Z given by
D(
U ⊂ V)
=(
D(V /U) ⊂ DV)
.
The rotationThe rotation of the poset gives rise to a autoequivalence R oforder three on its category of representations.
We pick the orientation such that V 7→ V ′ 7→ V ′′ 7→ V .
The rotationThe rotation of the poset gives rise to a autoequivalence R oforder three on its category of representations.
We pick the orientation such that V 7→ V ′ 7→ V ′′ 7→ V .
Remark: For a representation M = (M ′i ⊂ Mi )i∈Z ∈ S(n), the
corresponding poset representation encodes the subspaces M ′i , the
ambient spaces Mi and the factor spaces Mi/M′i in the following
subposets of P(n).
poset P
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subspace
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4.....................................................
The rotationThe rotation of the poset gives rise to a autoequivalence R oforder three on its category of representations.
We pick the orientation such that V 7→ V ′ 7→ V ′′ 7→ V .
Remark: For a representation M = (M ′i ⊂ Mi )i∈Z ∈ S(n), the
corresponding poset representation encodes the subspaces M ′i , the
ambient spaces Mi and the factor spaces Mi/M′i in the following
subposets of P(n).
poset P
•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
subspace
•
•
•
• ◦
◦....................................
....................................
1′′
2′′
2′
3′.....................................................
ambient space
•
•
•
•◦
◦....................................
....................................
1′′
2′′
3
4..........................................
...........................
.............
factor space
•
•
•
•◦
◦ ....................................
....................................
2′
3′
3
4.....................................................
Proposition. For the unbounded posets of discrete representation
type, the rotation is just the square of the Auslander-Reiten
translation, up to a shift.
R = τ2[
6− n
3
]
The shiftThe shift in the poset in vertical direction gives rise to the gradedshift on the category of representations.
The position with respect to the shift is measured by the slope.
Roughly, the slope is just the barycenter of the generators in thealigned poset, where generators for multiple columns are averagedout.
The shiftThe shift in the poset in vertical direction gives rise to the gradedshift on the category of representations.
The position with respect to the shift is measured by the slope.
Roughly, the slope is just the barycenter of the generators in thealigned poset, where generators for multiple columns are averagedout.
Slope formula: σ(V ) =1
g
∑
x∈P(n)
µ(x)σx(V ) for V ∈ repK P(n),
where g is the number of generators,
µ(x) =
y + n3 if x = y ′′
y if x = y ′
y − n3 if x = y
and
σx(V ) = dim Vx
Vx+1− 1
2
(
dim Vx+1+V ′
x
Vx+1+ dim
V−
x+1+Vx
V−
x+1
)
−16
(
dim Vx+1+V ′′
x
Vx+1+ dim
V−
x+1+V ′
x
V−
x+1
+ dimV=x+1+Vx
V=x+1
)
.
ExampleThe category repk P(3) for P(3) =
•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
has the following
partial Auslander-Reiten quiver:
1000000
1100000
1010000
1001000
2111000
1011000
1101000
1110000
1011010
1111000
1101001
1110100
1111010
1111001
1111100
2222111
1111101
1111110
1111011
1111111
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
....................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
....................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
..............
..............
..............
..............
..............
..............
ExampleThe category repk P(3) for P(3) =
•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
has the following
partial Auslander-Reiten quiver:
1000000
1100000
1010000
1001000
2111000
1011000
1101000
1110000
1011010
1111000
1101001
1110100
1111010
1111001
1111100
2222111
1111101
1111110
1111011
1111111
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
....................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
....................................................................
............................................. ................
.......................................................
.........................................................................
.............................................................
....................................... ................
.........................................................................
............................................. ................
.......................................................
.........................................................................
..............
..............
..............
..............
..............
..............
1 114 11
2 134 2 21
4 212 23
4 3
...................
........................
.........
...
.........
...................
........................
.........
...
.........
........
........
.........
...
.........
...................
........................
.........
...
.........
...................
........................
◮ The slope increases with the graded shift.
◮ It is invariant under rotation.
◮ There is constant d such that σ(DX ) = d − σ(X ) holds.
The drift
For X ∈ repK P(n0), we consider the sequence σn(X ) of slopeswhen X is considered an object in repK P(n) for n ≥ n0.
Four Examples: Consider the poset P(3) =•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
.
dimX σn(X ) for n > 3 σ3(X ) drift
1
1
1
1
1
0 0©©© 1
2
(
1 + n3 + 1
2(3 + 3− n3 ))
= 2 + n12 21
4112
The drift
For X ∈ repK P(n0), we consider the sequence σn(X ) of slopeswhen X is considered an object in repK P(n) for n ≥ n0.
Four Examples: Consider the poset P(3) =•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
.
dimX σn(X ) for n > 3 σ3(X ) drift
1
1
1
1
1
0 0©©© 1
2
(
1 + n3 + 1
2(3 + 3− n3 ))
= 2 + n12 21
4112
1
1 1
1
1
00©©© 1
2
(
3− n3 + 1
2(2 +n3 + 2)
)
= 212 − n
12 214 − 1
12
The drift
For X ∈ repK P(n0), we consider the sequence σn(X ) of slopeswhen X is considered an object in repK P(n) for n ≥ n0.
Four Examples: Consider the poset P(3) =•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
.
dimX σn(X ) for n > 3 σ3(X ) drift
1
1
1
1
1
0 0©©© 1
2
(
1 + n3 + 1
2(3 + 3− n3 ))
= 2 + n12 21
4112
1
1 1
1
1
00©©© 1
2
(
3− n3 + 1
2(2 +n3 + 2)
)
= 212 − n
12 214 − 1
12
1 1
1
1
1
0 0©©© 1
3
(
2 + 4− n3 + 1 + n
3
)
= 213 21
4 0
The drift
For X ∈ repK P(n0), we consider the sequence σn(X ) of slopeswhen X is considered an object in repK P(n) for n ≥ n0.
Four Examples: Consider the poset P(3) =•
•
•
•
•
•
..................................................
..................................................
..................................................
1′′
2′′
2′
3′
3
4
...................................................................................................................
..........................................................................
..........................................................................
.
dimX σn(X ) for n > 3 σ3(X ) drift
1
1
1
1
1
0 0©©© 1
2
(
1 + n3 + 1
2(3 + 3− n3 ))
= 2 + n12 21
4112
1
1 1
1
1
00©©© 1
2
(
3− n3 + 1
2(2 +n3 + 2)
)
= 212 − n
12 214 − 1
12
1 1
1
1
1
0 0©©© 1
3
(
2 + 4− n3 + 1 + n
3
)
= 213 21
4 0
1 1 1
2 2 2
2
©©©
©©© 1
4
(
12(3 + 3− n
3 ) +12(2 +
n3 + 2) + 4 + 1
)
= 212 21
2 0
The curvatureTheorem. If X → Y is an irreducible morphism in repK P(n) then
σ(Y ) = σ(X ) +6− n
12
The curvatureTheorem. If X → Y is an irreducible morphism in repK P(n) then
σ(Y ) = σ(X ) +6− n
12
Definition: The curvature in repK P(n) is defined as
κ(n) =d σ
d τ=
6− n
6
n 1 2 3 4 5 6 7 8 9 · · · 12
Γ ∅ ZA2 ZD4 ZE6 ZE8 E8..................................................................................... ZA∞ · · ·
[
1]
− τ32ϕ τ2ρ τ3ϕ τ6 − τ−6
κ(n) ∅ 23
12
13
16 0 −1
6 −13 −1
2 · · · −1
Summary
Symmetry properties of the posets P(n) lead to invariants for thestudy of repK P(n).
◮ For each object, the slope determines its position withinrepK P(n).
◮ The drift is the change of the slope as the object moves alongthe chain
repK P(1) ⊂ repK P(2) ⊂ repK P(3) ⊂ · · ·
◮ The curvature is positive, zero or negative, depending onwhether repK P(n) is of discrete, tame or wild representationtype.
Summary
Symmetry properties of the posets P(n) lead to invariants for thestudy of repK P(n).
◮ For each object, the slope determines its position withinrepK P(n).
◮ The drift is the change of the slope as the object moves alongthe chain
repK P(1) ⊂ repK P(2) ⊂ repK P(3) ⊂ · · ·
◮ The curvature is positive, zero or negative, depending onwhether repK P(n) is of discrete, tame or wild representationtype.
Thank You!