Fractals and modular representations of SL
2Or: All I know about SL2
Daniel Tubbenhauer
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Joint with Lousie Sutton, Paul Wedrich, Jieru Zhu
February 2021
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 1 / 8
Question. What can we say about finite-dimensional modules ofSL2...
• ...in the context of the representation theory of classical groups? The modules and their structure.
• ...in the context of the representation theory of Hopf algebras? Fusion rules i.e. tensor products rules.
• ...in the context of categories? Morphisms of representations and their structure.
The most amazing things happen if the characteristic of the underlying fieldK=K ofSL2=SL2(K) is finite, and we will see fractals,e.g.
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Spoiler: What will be the take away?
Well, in some sense modular (charp<∞) representation theory so much harder than classical one (char∞a.k.a. char 0)
because secretly we are doing fractal geometry. In my toy exampleSL2we can do everything explicitly.
Question. What can we say about finite-dimensional modules ofSL2...
• ...in the context of the representation theory of classical groups? The modules and their structure.
• ...in the context of the representation theory of Hopf algebras? Fusion rules i.e. tensor products rules.
• ...in the context of categories? Morphisms of representations and their structure.
The most amazing things happen if the characteristic of the underlying fieldK=K ofSL2=SL2(K) is finite, and we will see fractals,e.g.
1 100 200 300 400 486
1
100
200
300
400
486
1 100 200 300 400 486
1
100
200
300
400
486
Spoiler: What will be the take away?
Well, in some sense modular (charp<∞) representation theory so much harder than classical one (char∞a.k.a. char 0)
because secretly we are doing fractal geometry.
In my toy exampleSL2we can do everything explicitly.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 2 / 8
Weyl ∼1923. TheSL2(dual)Weyl modules ∆(v−1).
∆(1−1)
∆(2−1)
∆(3−1)
∆(4−1)
∆(5−1)
∆(6−1)
∆(7−1)
X0Y0
X1Y0 X0Y1
X2Y0 X1Y1 X0Y2
X3Y0 X2Y1 X1Y2 X0Y3
X4Y0 X3Y1 X2Y2 X1Y3 X0Y4
X5Y0 X4Y1 X3Y2 X2Y3 X1Y4 X0Y5
X6Y0 X5Y1 X4Y2 X3Y3 X2Y4 X1Y5 X0Y6
a bc d
7→matrix who’s rows are expansions of (aX +cY)v−i(bX+dY)i−1.
Example∆(7−1) =KX6Y0⊕ · · · ⊕KX0Y6.
(a bc d) acts as
The rows are expansions of (aX+cY)7−i(bX +dY)i−1. Binomials!
Example∆(7−1), characteristic0. No common eigensystem⇒∆(7−1) simple.
Example∆(7−1), characteristic2.
(a bc d) acts as
(0,0,0,1,0,0,0) is a common eigenvector, so we found a submodule. When is∆(v−1)simple?
∆(v−1) is simple
⇔
v−1 w−1
6= 0 for allw ≤v
⇔(Lucas’s theorem) v = [ar,0,...,0]p.
General. Weyl ∆(λ) and dual Weyl∇(λ)
are easy a.k.a. standard; are parameterized by dominant integral weights;
are highest weight modules; are defined overZ; have the classical Weyl characters;
form a basis of the Grothendieck group unitriangular w.r.t. simples; satisfy (a version of) Schur’s lemma dimKExti(∆(λ),∆(µ)) = ∆i,0∆λ,µ;
are simple generically;
have a root-binomial-criterion to determine whether they are simple (Jantzen’s thesis∼1973). Lucas∼1878.
“Binomials mod p are the product of binomials of the p-adic digits”:
a b
=Qr
i=0 ai bi
modp, wherea= [ar,...,a0]p=Pr
i=0ai pi etc.
Weyl ∼1923. TheSL2(dual)Weyl modules ∆(v−1).
∆(1−1)
∆(2−1)
∆(3−1)
∆(4−1)
∆(5−1)
∆(6−1)
∆(7−1)
X0Y0
X1Y0 X0Y1
X2Y0 X1Y1 X0Y2
X3Y0 X2Y1 X1Y2 X0Y3
X4Y0 X3Y1 X2Y2 X1Y3 X0Y4
X5Y0 X4Y1 X3Y2 X2Y3 X1Y4 X0Y5
X6Y0 X5Y1 X4Y2 X3Y3 X2Y4 X1Y5 X0Y6
a bc d
7→matrix who’s rows are expansions of (aX +cY)v−i(bX+dY)i−1.
The simples
Example∆(7−1) =KX6Y0⊕ · · · ⊕KX0Y6.
(a bc d) acts as
The rows are expansions of (aX+cY)7−i(bX +dY)i−1. Binomials!
Example∆(7−1), characteristic0. No common eigensystem⇒∆(7−1) simple.
Example∆(7−1), characteristic2.
(a bc d) acts as
(0,0,0,1,0,0,0) is a common eigenvector, so we found a submodule. When is∆(v−1)simple?
∆(v−1) is simple
⇔
v−1 w−1
6= 0 for allw ≤v
⇔(Lucas’s theorem) v = [ar,0,...,0]p.
General. Weyl ∆(λ) and dual Weyl∇(λ)
are easy a.k.a. standard; are parameterized by dominant integral weights;
are highest weight modules; are defined overZ; have the classical Weyl characters;
form a basis of the Grothendieck group unitriangular w.r.t. simples; satisfy (a version of) Schur’s lemma dimKExti(∆(λ),∆(µ)) = ∆i,0∆λ,µ;
are simple generically;
have a root-binomial-criterion to determine whether they are simple (Jantzen’s thesis∼1973). Lucas∼1878.
“Binomials mod p are the product of binomials of the p-adic digits”:
a b
=Qr
i=0 ai bi
modp, wherea= [ar,...,a0]p=Pr
i=0ai pi etc.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 3 / 8
Weyl ∼1923. TheSL2(dual)Weyl modules ∆(v−1).
∆(1−1)
∆(2−1)
∆(3−1)
∆(4−1)
∆(5−1)
∆(6−1)
∆(7−1)
X0Y0
X1Y0 X0Y1
X2Y0 X1Y1 X0Y2
X3Y0 X2Y1 X1Y2 X0Y3
X4Y0 X3Y1 X2Y2 X1Y3 X0Y4
X5Y0 X4Y1 X3Y2 X2Y3 X1Y4 X0Y5
X6Y0 X5Y1 X4Y2 X3Y3 X2Y4 X1Y5 X0Y6
a bc d
7→matrix who’s rows are expansions of (aX +cY)v−i(bX+dY)i−1. Example∆(7−1) =KX6Y0⊕ · · · ⊕KX0Y6.
(a bc d) acts as
The rows are expansions of (aX+cY)7−i(bX +dY)i−1. Binomials!
Example∆(7−1), characteristic0.
No common eigensystem⇒∆(7−1) simple.
Example∆(7−1), characteristic2.
(a bc d) acts as
(0,0,0,1,0,0,0) is a common eigenvector, so we found a submodule.
When is∆(v−1)simple?
∆(v−1) is simple
⇔
v−1 w−1
6= 0 for allw ≤v
⇔(Lucas’s theorem) v = [ar,0,...,0]p.
General. Weyl ∆(λ) and dual Weyl∇(λ)
are easy a.k.a. standard; are parameterized by dominant integral weights;
are highest weight modules; are defined overZ; have the classical Weyl characters;
form a basis of the Grothendieck group unitriangular w.r.t. simples; satisfy (a version of) Schur’s lemma dimKExti(∆(λ),∆(µ)) = ∆i,0∆λ,µ;
are simple generically;
have a root-binomial-criterion to determine whether they are simple (Jantzen’s thesis∼1973). Lucas∼1878.
“Binomials mod p are the product of binomials of the p-adic digits”:
a b
=Qr
i=0 ai bi
modp, wherea= [ar,...,a0]p=Pr
i=0ai pi etc.
Weyl ∼1923. TheSL2(dual)Weyl modules ∆(v−1).
∆(1−1)
∆(2−1)
∆(3−1)
∆(4−1)
∆(5−1)
∆(6−1)
∆(7−1)
X0Y0
X1Y0 X0Y1
X2Y0 X1Y1 X0Y2
X3Y0 X2Y1 X1Y2 X0Y3
X4Y0 X3Y1 X2Y2 X1Y3 X0Y4
X5Y0 X4Y1 X3Y2 X2Y3 X1Y4 X0Y5
X6Y0 X5Y1 X4Y2 X3Y3 X2Y4 X1Y5 X0Y6
a bc d
7→matrix who’s rows are expansions of (aX +cY)v−i(bX+dY)i−1.
The simples
Example∆(7−1) =KX6Y0⊕ · · · ⊕KX0Y6.
(a bc d) acts as
The rows are expansions of (aX+cY)7−i(bX +dY)i−1. Binomials!
Example∆(7−1), characteristic0. No common eigensystem⇒∆(7−1) simple.
Example∆(7−1), characteristic2.
(a bc d) acts as
(0,0,0,1,0,0,0) is a common eigenvector, so we found a submodule.
When is∆(v−1)simple?
∆(v−1) is simple
⇔
v−1 w−1
6= 0 for allw ≤v
⇔(Lucas’s theorem) v = [ar,0,...,0]p.
General.
Weyl ∆(λ) and dual Weyl∇(λ) are easy a.k.a. standard;
are parameterized by dominant integral weights;
are highest weight modules;
are defined overZ;
have the classical Weyl characters;
form a basis of the Grothendieck group unitriangular w.r.t. simples;
satisfy (a version of) Schur’s lemma dimKExti(∆(λ),∆(µ)) = ∆i,0∆λ,µ; are simple generically;
have a root-binomial-criterion to determine whether they are simple (Jantzen’s thesis∼1973).
Lucas∼1878.
“Binomials mod p are the product of binomials of the p-adic digits”:
a b
=Qr
i=0 ai bi
modp, wherea= [ar,...,a0]p=Pr
i=0ai pi etc.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 3 / 8
Ringel, Donkin∼1991. The indecomposable SL2 tilting modulesT(v−1) are the indecomposable summands of ∆(1)⊗i ∼= (K2)⊗i
. Tilting modulesT(v−1)
• are those modules with a ∆(w−1)- and a∇(w−1)-filtration;
• are parameterized by dominant integral weights;
• are highest weight modules;
• T(v−1) : ∆(w−1)
determines
∆(v−1) :L(w−1)
;
• form a basis of the Grothendieck group unitriangular w.r.t. simples;
• satisfy (a version of) Schur’s lemma dimKHom T(v−1),T(w−1) P =
x<min(v,w) T(v−1) : ∆(x−1)
T(w−1) : ∆(x−1)
;
• are simple generically;
• have a root-binomial-criterion to determine whether they are simple.
Slogan. Indecomposable tilting modules are akin to indecomposable projectives.
Warning: SL2has finite-dimensional projectives if and only ifchar(K) = 0.
General.
These facts hold in general, and the first bullet point is the general definition.
How many Weyl factors doesT(v−1)have?
# Weyl factors ofT(v−1) is 2k where k= max{νp v−1
w−1
,w ≤v}. (Order of vanishing of wv−−11 .) determined by (Lucas’s theorem)
non-zeronon-leadingdigits ofv = [ar,ar−1,...,a0]p. ExampleT(220540−1)forp= 11?
v = 220540 = [1,4,0,7,7,1]11;
Maximal vanishing forw= 75594 = [0,5,1,8,8,2]11;
v−1 w−1
= (HUGE) = [...,6= 0,0,0,0,0]11.
⇒T(220540−1) has 24Weyl factors.
Which Weyl factors doesT(v−1)have a.k.a. the negative digits game? Weyl factors ofT(v−1) are
∆([ar,±ar−1,...,±a0]p−1) wherev= [ar,...,a0]p.
ExampleT(220540−1)forp= 11? v = 220540 = [1,4,0,7,7,1]11; has Weyl factors [1,±4,0,±7,±7,±1]11; e.g. ∆(218690 = [1,4,0,−7,−7,−1]11−1) appears. The tilting-Cartan matrix a.k.a. T(v−1) : ∆(w−1)
?
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150
201
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50
100
150
201
This is characteristic 3.
Ringel, Donkin∼1991. The indecomposable SL2 tilting modulesT(v−1) are the indecomposable summands of ∆(1)⊗i ∼= (K2)⊗i
. Tilting modulesT(v−1)
• are those modules with a ∆(w−1)- and a∇(w−1)-filtration;
• are parameterized by dominant integral weights;
• are highest weight modules;
• T(v−1) : ∆(w−1)
determines
∆(v−1) :L(w−1)
;
• form a basis of the Grothendieck group unitriangular w.r.t. simples;
• satisfy (a version of) Schur’s lemma dimKHom T(v−1),T(w−1) P =
x<min(v,w) T(v−1) : ∆(x−1)
T(w−1) : ∆(x−1)
;
• are simple generically;
• have a root-binomial-criterion to determine whether they are simple.
Slogan. Indecomposable tilting modules are akin to indecomposable projectives.
Warning: SL2has finite-dimensional projectives if and only ifchar(K) = 0.
General. These facts hold in general, and
the first bullet point is the general definition.
How many Weyl factors doesT(v−1)have?
# Weyl factors ofT(v−1) is 2k where k= max{νp v−1
w−1
,w≤v}. (Order of vanishing of wv−−11 .) determined by (Lucas’s theorem)
non-zeronon-leadingdigits ofv = [ar,ar−1,...,a0]p. ExampleT(220540−1)forp= 11?
v= 220540 = [1,4,0,7,7,1]11;
Maximal vanishing forw= 75594 = [0,5,1,8,8,2]11;
v−1 w−1
= (HUGE) = [...,6= 0,0,0,0,0]11.
⇒T(220540−1) has 24Weyl factors.
Which Weyl factors doesT(v−1)have a.k.a. the negative digits game? Weyl factors ofT(v−1) are
∆([ar,±ar−1,...,±a0]p−1) wherev= [ar,...,a0]p.
ExampleT(220540−1)forp= 11? v = 220540 = [1,4,0,7,7,1]11; has Weyl factors [1,±4,0,±7,±7,±1]11; e.g. ∆(218690 = [1,4,0,−7,−7,−1]11−1) appears. The tilting-Cartan matrix a.k.a. T(v−1) : ∆(w−1)
?
1 50 100 150 201
1
50
100
150
201
1 50 100 150 201
1
50
100
150
201
This is characteristic 3.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 4 / 8
Ringel, Donkin∼1991. The indecomposable SL2 tilting modulesT(v−1) are the indecomposable summands of ∆(1)⊗i ∼= (K2)⊗i
. Tilting modulesT(v−1)
• are those modules with a ∆(w−1)- and a∇(w−1)-filtration;
• are parameterized by dominant integral weights;
• are highest weight modules;
• T(v−1) : ∆(w−1)
determines
∆(v−1) :L(w−1)
;
• form a basis of the Grothendieck group unitriangular w.r.t. simples;
• satisfy (a version of) Schur’s lemma dimKHom T(v−1),T(w−1) P =
x<min(v,w) T(v−1) : ∆(x−1)
T(w−1) : ∆(x−1)
;
• are simple generically;
• have a root-binomial-criterion to determine whether they are simple.
Slogan. Indecomposable tilting modules are akin to indecomposable projectives.
Warning: SL2has finite-dimensional projectives if and only ifchar(K) = 0.
General. These facts hold in general, and
the first bullet point is the general definition.
How many Weyl factors doesT(v−1)have?
# Weyl factors ofT(v−1) is 2k where k= max{νp v−1
w−1
,w ≤v}. (Order of vanishing of wv−−11 .) determined by (Lucas’s theorem)
non-zeronon-leadingdigits ofv = [ar,ar−1,...,a0]p. ExampleT(220540−1)forp= 11?
v = 220540 = [1,4,0,7,7,1]11;
Maximal vanishing forw= 75594 = [0,5,1,8,8,2]11;
v−1 w−1
= (HUGE) = [...,6= 0,0,0,0,0]11.
⇒T(220540−1) has 24Weyl factors.
Which Weyl factors doesT(v−1)have a.k.a. the negative digits game?
Weyl factors ofT(v−1) are
∆([ar,±ar−1,...,±a0]p−1) wherev= [ar,...,a0]p.
ExampleT(220540−1)forp= 11?
v= 220540 = [1,4,0,7,7,1]11; has Weyl factors [1,±4,0,±7,±7,±1]11; e.g. ∆(218690 = [1,4,0,−7,−7,−1]11−1) appears.
The tilting-Cartan matrix a.k.a. T(v−1) : ∆(w−1)
?
1 50 100 150 201
1
50
100
150
201
1 50 100 150 201
1
50
100
150
201
This is characteristic 3.
Ringel, Donkin∼1991. The indecomposable SL2 tilting modulesT(v−1) are the indecomposable summands of ∆(1)⊗i ∼= (K2)⊗i
. Tilting modulesT(v−1)
• are those modules with a ∆(w−1)- and a∇(w−1)-filtration;
• are parameterized by dominant integral weights;
• are highest weight modules;
• T(v−1) : ∆(w−1)
determines
∆(v−1) :L(w−1)
;
• form a basis of the Grothendieck group unitriangular w.r.t. simples;
• satisfy (a version of) Schur’s lemma dimKHom T(v−1),T(w−1) P =
x<min(v,w) T(v−1) : ∆(x−1)
T(w−1) : ∆(x−1)
;
• are simple generically;
• have a root-binomial-criterion to determine whether they are simple.
Slogan. Indecomposable tilting modules are akin to indecomposable projectives.
Warning: SL2has finite-dimensional projectives if and only ifchar(K) = 0.
General. These facts hold in general, and
the first bullet point is the general definition.
How many Weyl factors doesT(v−1)have?
# Weyl factors ofT(v−1) is 2k where k= max{νp v−1
w−1
,w ≤v}. (Order of vanishing of wv−−11 .) determined by (Lucas’s theorem)
non-zeronon-leadingdigits ofv = [ar,ar−1,...,a0]p. ExampleT(220540−1)forp= 11?
v = 220540 = [1,4,0,7,7,1]11;
Maximal vanishing forw= 75594 = [0,5,1,8,8,2]11;
v−1 w−1
= (HUGE) = [...,6= 0,0,0,0,0]11.
⇒T(220540−1) has 24Weyl factors.
Which Weyl factors doesT(v−1)have a.k.a. the negative digits game? Weyl factors ofT(v−1) are
∆([ar,±ar−1,...,±a0]p−1) wherev= [ar,...,a0]p.
ExampleT(220540−1)forp= 11? v = 220540 = [1,4,0,7,7,1]11; has Weyl factors [1,±4,0,±7,±7,±1]11; e.g. ∆(218690 = [1,4,0,−7,−7,−1]11−1) appears.
The tilting-Cartan matrix a.k.a. T(v−1) : ∆(w−1)
?
1 50 100 150 201
1
50
100
150
201
1 50 100 150 201
1
50
100
150
201
This is characteristic 3.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 4 / 8
Tilting modules form a braided monoidal category Tilt.
Simple⊗simple6=simple, Weyl⊗Weyl6=Weyl, but tilting⊗tilting=tilting.
The Grothendieck algebra [Tilt] ofTiltis a commutative algebra with basis [T(v−1)]. So what I would like to answer on the object level,i.e. for [Tilt]:
• What are the fusion rules? Answer
• Find theNv,wx ∈N[0] inT(v−1)⊗T(v−1)∼=L
xNv,wx T(x−1).
B For [Tilt] this means finding the structure constants.
• What are the thick⊗-ideals? Answer
B For [Tilt] this means finding the ideals.
General.
These facts hold in general, and
tilting modules form the “nicest possible” monoidal subcategory.
Tilting modules form a braided monoidal category Tilt.
Simple⊗simple6=simple, Weyl⊗Weyl6=Weyl, but tilting⊗tilting=tilting.
The Grothendieck algebra [Tilt] ofTiltis a commutative algebra with basis [T(v−1)]. So what I would like to answer on the object level,i.e. for [Tilt]:
• What are the fusion rules? Answer
• Find theNv,wx ∈N[0] inT(v−1)⊗T(v−1)∼=L
xNv,wx T(x−1).
B For [Tilt] this means finding the structure constants.
• What are the thick⊗-ideals? Answer
B For [Tilt] this means finding the ideals.
General. These facts hold in general, and
tilting modules form the “nicest possible” monoidal subcategory.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 5 / 8
The morphism. There exists a K-algebra Zp defined as a (very explicit) quotient of the path algebra of an infinite, fractal-like quiver. LetpMod-Zp denote the category of finitely-generated, projective (right-)modules forZp. There is an equivalence of additive,K-linear categories
F:Tilt−→∼= pMod-Zp,
sending indecomposable tilting modules to indecomposable projectives.
v T Δ
012345678910 11 12 13 14 15 161718 19 20 21 22 23 24 252627 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0123
1 4 0
56 4 7 3
89 7 10
6 11
5 12 10 6 4 13 9 7 3 14 2 15 13 3 1 16 12 4 0
1718 16 19 15 20 14 21 19 15 13 22 18 16 12 23 11 24 22 12 10 25 21 13 9
2627 25 28 24 29 23 30 28 24 22 31 27 25 21 32 20 33 31 21 19 34 30 22 18 35 17 36 34 18 16 37 33 19 15 38 32 20 14 39 37 33 31 21 19 15 13 40 36 34 30 22 18 16 12 41 29 23 11 42 40 30 28 24 22 12 10 43 39 31 27 25 21 13 9 44
8 45 43 9 7 46 42 10 6 47 41 11 5 48 46 42 40 12 10 6 4 49 45 43 39 13 9 7 3 50 38 14 2 51 49 39 37 15 13 3 1 52 48 40 36 16 12 4 0 12345678910 11 12 13 14 15 16 171819 20 21 22 23 24 25 262728 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
.
Figure:My favorite rainbow: The full subquiver containing the first 53 vertices of the quiver underlyingZ3.
Example, generation0, i.e. up top. In this case the quiver has no edges.
Continuing this periodically gives a quiver forTiltfor charp=∞. (This is the semisimple case: the quiver has to be boring.)
Example, generation1, i.e. up top2.
In this case the quiver is a bunch of type A graphs. The algebra is a zigzag algebra, with arrows acting on the 0th digit.
Continuing this periodically gives a quiver forTilt
for the quantum group at a complex root of unity (due to Andersen∼2014). Example, generation2, i.e. up top3.
In this case every connected component
of the quiver is a bunch of type A graphs glued together in a matrix-grid. Each row and column is a zigzag algebra, with arrows acting on the 0th digit or 1digit,
and there are “squares commute” relations.
Continuing this periodically gives a quiver for projectiveG2T-modules (due to Andersen∼2019).
In general, Zp is basically a bunch of zigzag algebras (there are scalars and a lower-order-error term, but never mind) glued together in a fractal-way, according to the digits ofv = [ar,...,a0]p.
The morphism. There exists a K-algebra Zp defined as a (very explicit) quotient of the path algebra of an infinite, fractal-like quiver. LetpMod-Zp denote the category of finitely-generated, projective (right-)modules forZp. There is an equivalence of additive,K-linear categories
F:Tilt−→∼= pMod-Zp,
sending indecomposable tilting modules to indecomposable projectives.
v T Δ
012345678910 11 12 13 14 15 161718 19 20 21 22 23 24 252627 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0123
1 4 0
56 4 7 3
89 7 10
6 11
5 12 10 6 4 13 9 7 3 14 2 15 13 3 1 16 12 4 0
1718 16 19 15 20 14 21 19 15 13 22 18 16 12 23 11 24 22 12 10 25 21 13 9
2627 25 28 24 29 23 30 28 24 22 31 27 25 21 32 20 33 31 21 19 34 30 22 18 35 17 36 34 18 16 37 33 19 15 38 32 20 14 39 37 33 31 21 19 15 13 40 36 34 30 22 18 16 12 41 29 23 11 42 40 30 28 24 22 12 10 43 39 31 27 25 21 13 9 44
8 45 43 9 7 46 42 10 6 47 41 11 5 48 46 42 40 12 10 6 4 49 45 43 39 13 9 7 3 50 38 14 2 51 49 39 37 15 13 3 1 52 48 40 36 16 12 4 0 12345678910 11 12 13 14 15 16 171819 20 21 22 23 24 25 262728 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
.
Figure:My favorite rainbow: The full subquiver containing the first 53 vertices of the quiver underlyingZ3.
Proof? Time’s up
Example, generation0, i.e. up top.
In this case the quiver has no edges.
Continuing this periodically gives a quiver forTiltfor charp=∞. (This is the semisimple case: the quiver has to be boring.)
Example, generation1, i.e. up top2.
In this case the quiver is a bunch of type A graphs. The algebra is a zigzag algebra, with arrows acting on the 0th digit.
Continuing this periodically gives a quiver forTilt
for the quantum group at a complex root of unity (due to Andersen∼2014). Example, generation2, i.e. up top3.
In this case every connected component
of the quiver is a bunch of type A graphs glued together in a matrix-grid. Each row and column is a zigzag algebra, with arrows acting on the 0th digit or 1digit,
and there are “squares commute” relations.
Continuing this periodically gives a quiver for projectiveG2T-modules (due to Andersen∼2019).
In general, Zp is basically a bunch of zigzag algebras (there are scalars and a lower-order-error term, but never mind) glued together in a fractal-way, according to the digits ofv = [ar,...,a0]p.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 6 / 8
The morphism. There exists a K-algebra Zp defined as a (very explicit) quotient of the path algebra of an infinite, fractal-like quiver. LetpMod-Zp denote the category of finitely-generated, projective (right-)modules forZp. There is an equivalence of additive,K-linear categories
F:Tilt−→∼= pMod-Zp,
sending indecomposable tilting modules to indecomposable projectives.
v T Δ
012345678910 11 12 13 14 15 161718 19 20 21 22 23 24 252627 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0123
1 4 0
56 4 7 3
89 7 10
6 11
5 12 10 6 4 13 9 7 3 14 2 15 13 3 1 16 12 4 0
1718 16 19 15 20 14 21 19 15 13 22 18 16 12 23 11 24 22 12 10 25 21 13 9
2627 25 28 24 29 23 30 28 24 22 31 27 25 21 32 20 33 31 21 19 34 30 22 18 35 17 36 34 18 16 37 33 19 15 38 32 20 14 39 37 33 31 21 19 15 13 40 36 34 30 22 18 16 12 41 29 23 11 42 40 30 28 24 22 12 10 43 39 31 27 25 21 13 9 44
8 45 43 9 7 46 42 10 6 47 41 11 5 48 46 42 40 12 10 6 4 49 45 43 39 13 9 7 3 50 38 14 2 51 49 39 37 15 13 3 1 52 48 40 36 16 12 4 0 12345678910 11 12 13 14 15 16 171819 20 21 22 23 24 25 262728 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
.
Figure:My favorite rainbow: The full subquiver containing the first 53 vertices of the quiver underlyingZ3.
Example, generation0, i.e. up top. In this case the quiver has no edges.
Continuing this periodically gives a quiver forTiltfor charp=∞. (This is the semisimple case: the quiver has to be boring.)
Example, generation1, i.e. up top2.
In this case the quiver is a bunch of type A graphs. The algebra is a zigzag algebra, with arrows acting on the 0th digit.
Continuing this periodically gives a quiver forTilt
for the quantum group at a complex root of unity (due to Andersen∼2014).
Example, generation2, i.e. up top3. In this case every connected component
of the quiver is a bunch of type A graphs glued together in a matrix-grid. Each row and column is a zigzag algebra, with arrows acting on the 0th digit or 1digit,
and there are “squares commute” relations.
Continuing this periodically gives a quiver for projectiveG2T-modules (due to Andersen∼2019).
In general, Zp is basically a bunch of zigzag algebras (there are scalars and a lower-order-error term, but never mind) glued together in a fractal-way, according to the digits ofv = [ar,...,a0]p.
The morphism. There exists a K-algebra Zp defined as a (very explicit) quotient of the path algebra of an infinite, fractal-like quiver. LetpMod-Zp denote the category of finitely-generated, projective (right-)modules forZp. There is an equivalence of additive,K-linear categories
F:Tilt−→∼= pMod-Zp,
sending indecomposable tilting modules to indecomposable projectives.
v T Δ
012345678910 11 12 13 14 15 161718 19 20 21 22 23 24 252627 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0123
1 4 0
56 4 7 3
89 7 10
6 11
5 12 10 6 4 13 9 7 3 14 2 15 13 3 1 16 12 4 0
1718 16 19 15 20 14 21 19 15 13 22 18 16 12 23 11 24 22 12 10 25 21 13 9
2627 25 28 24 29 23 30 28 24 22 31 27 25 21 32 20 33 31 21 19 34 30 22 18 35 17 36 34 18 16 37 33 19 15 38 32 20 14 39 37 33 31 21 19 15 13 40 36 34 30 22 18 16 12 41 29 23 11 42 40 30 28 24 22 12 10 43 39 31 27 25 21 13 9 44
8 45 43 9 7 46 42 10 6 47 41 11 5 48 46 42 40 12 10 6 4 49 45 43 39 13 9 7 3 50 38 14 2 51 49 39 37 15 13 3 1 52 48 40 36 16 12 4 0 12345678910 11 12 13 14 15 16 171819 20 21 22 23 24 25 262728 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
.
Figure:My favorite rainbow: The full subquiver containing the first 53 vertices of the quiver underlyingZ3.
Proof? Time’s up
Example, generation0, i.e. up top. In this case the quiver has no edges.
Continuing this periodically gives a quiver forTiltfor charp=∞. (This is the semisimple case: the quiver has to be boring.)
Example, generation1, i.e. up top2.
In this case the quiver is a bunch of type A graphs. The algebra is a zigzag algebra, with arrows acting on the 0th digit.
Continuing this periodically gives a quiver forTilt
for the quantum group at a complex root of unity (due to Andersen∼2014).
Example, generation2, i.e. up top3. In this case every connected component
of the quiver is a bunch of type A graphs glued together in a matrix-grid.
Each row and column is a zigzag algebra, with arrows acting on the 0th digit or 1digit, and there are “squares commute” relations.
Continuing this periodically gives a quiver for projectiveG2T-modules (due to Andersen∼2019).
In general, Zp is basically a bunch of zigzag algebras (there are scalars and a lower-order-error term, but never mind) glued together in a fractal-way, according to the digits ofv = [ar,...,a0]p.
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 6 / 8
The morphism. There exists a K-algebra Zp defined as a (very explicit) quotient of the path algebra of an infinite, fractal-like quiver. LetpMod-Zp denote the category of finitely-generated, projective (right-)modules forZp. There is an equivalence of additive,K-linear categories
F:Tilt−→∼= pMod-Zp,
sending indecomposable tilting modules to indecomposable projectives.
v T Δ
012345678910 11 12 13 14 15 161718 19 20 21 22 23 24 252627 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 0123
1 4 0
56 4 7 3
89 7 10
6 11
5 12 10 6 4 13 9 7 3 14 2 15 13 3 1 16 12 4 0
1718 16 19 15 20 14 21 19 15 13 22 18 16 12 23 11 24 22 12 10 25 21 13 9
2627 25 28 24 29 23 30 28 24 22 31 27 25 21 32 20 33 31 21 19 34 30 22 18 35 17 36 34 18 16 37 33 19 15 38 32 20 14 39 37 33 31 21 19 15 13 40 36 34 30 22 18 16 12 41 29 23 11 42 40 30 28 24 22 12 10 43 39 31 27 25 21 13 9 44
8 45 43 9 7 46 42 10 6 47 41 11 5 48 46 42 40 12 10 6 4 49 45 43 39 13 9 7 3 50 38 14 2 51 49 39 37 15 13 3 1 52 48 40 36 16 12 4 0 12345678910 11 12 13 14 15 16 171819 20 21 22 23 24 25 262728 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
.
Figure:My favorite rainbow: The full subquiver containing the first 53 vertices of the quiver underlyingZ3.
Example, generation0, i.e. up top. In this case the quiver has no edges.
Continuing this periodically gives a quiver forTiltfor charp=∞. (This is the semisimple case: the quiver has to be boring.)
Example, generation1, i.e. up top2.
In this case the quiver is a bunch of type A graphs. The algebra is a zigzag algebra, with arrows acting on the 0th digit.
Continuing this periodically gives a quiver forTilt
for the quantum group at a complex root of unity (due to Andersen∼2014). Example, generation2, i.e. up top3.
In this case every connected component
of the quiver is a bunch of type A graphs glued together in a matrix-grid. Each row and column is a zigzag algebra, with arrows acting on the 0th digit or 1digit,
and there are “squares commute” relations.
Continuing this periodically gives a quiver for projectiveG2T-modules (due to Andersen∼2019).
In general, Zp is basically a bunch of zigzag algebras (there are scalars and a lower-order-error term, but never mind) glued together in a fractal-way, according to the digits ofv = [ar,...,a0]p.
The whole story generalizes to Lusztig’s quantum group over Kwithq∈Kvia:
• We needp, the characteristic ofK, andl, the order ofq2.
• Thep-l-adic expansion ofv = [ar,...,a0]p,l isv =Pr
i=0aip(r) withp(0)= 1 andp(k) =pk−1l. Here 0≤a0<l−1 and 0≤ai <p−1.
B Example. ForK=F7 andq= 2∈F7, we havep= 7 andl= 3.
B Example. 68 = [68]p,∞= [66,2]∞,3= [1,2,5]7,7= [3,1,2]7,3
• Repeat everything I told you for these expansions.
Here is the tilting-Cartan matrix in mixed characteristicp= 5 andl= 2:
1 20 40 60 80 101
1
20
40
60
80
101
1 20 40 60 80 101
1
20
40
60
80
101, 1 100 200 300 401
1
100
200
300
401
1 100 200 300 401
1
100
200
300
401
Daniel Tubbenhauer Fractals and modular representations of SL2 February 2021 7 / 8