an example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/cordoba.pdf · example:...
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An example of higher representation theory
Geordie WilliamsonMax Planck Institute, Bonn
Quantum 2016,Cordoba, February 2016.
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First steps in representation theory.
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We owe the term group(e) to Galois (1832).
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H Ă G is a subgroup
Letter to Auguste Chevalier in 1832
written on the eve of Galois’ death
notion of a normal subgroup
notion of a simple group
notion of a soluble group
main theorem of Galois theory
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Mathematicians were studying group theory for 60 years beforethey began studying representations of finite groups.
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The first character table ever published. Here G is the alternatinggroup on 4 letters, or equivalently the symmetries of the
tetrahedron.
Frobenius, Uber Gruppencharaktere, S’ber. Akad. Wiss. Berlin, 1896.
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Now G “ S5, the symmetric group on 5 letters of order 120:
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Conway, Curtis, Norton, Parker, Wilson, Atlas of finite groups. Maximal subgroups and ordinary
characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, 1985.
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However around 1900 other mathematicians took some convincingat to the utility of representation theory...
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– Burnside, Theory of groups of finite order, 1897.(One year after Frobenius’ definition of the character.)
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– Burnside, Theory of groups of finite order, Second edition, 1911.(15 years after Frobenius’ definition of the character table.)
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Representation theory is useful because symmetry is everywhereand linear algebra is powerful!
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Categories can have symmetry too!
Caution: What “linear” means is more subtle.
Usually it means to study categories in which one has operations like direct sums, limits and colimits, kernels . . .
(Using these operations one can try to “categorify linear algebra” by taking sums, cones etc. If we are lucky Ben
Elias will have more to say about this.)
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Categories can have symmetry too!
Caution: What “linear” means is more subtle.
Usually it means to study categories in which one has operations like direct sums, limits and colimits, kernels . . .
(Using these operations one can try to “categorify linear algebra” by taking sums, cones etc. If we are lucky Ben
Elias will have more to say about this.)
![Page 15: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/15.jpg)
Example: Given a variety X one can think about CohpX q orDbpCohX q as a linearisation of X .
Example: Given a finite group G its “C-linear shadow” is thecharacter table (essentially by semi-simplicity).
The subtle homological algebra of kG if kG is not semi-simplemeans that Rep kG or DbpRep kG q is better thought of as its
k-linear shadow.
Amusing: Under this analogy difficult conjectures about derived equivalence (e.g. Broue conjecture) are higher
categorical versions of questions like “can two groups have the same character table”?
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Example: Given a variety X one can think about CohpX q orDbpCohX q as a linearisation of X .
Example: Given a finite group G its “C-linear shadow” is thecharacter table (essentially by semi-simplicity).
The subtle homological algebra of kG if kG is not semi-simplemeans that Rep kG or DbpRep kG q is better thought of as its
k-linear shadow.
Amusing: Under this analogy difficult conjectures about derived equivalence (e.g. Broue conjecture) are higher
categorical versions of questions like “can two groups have the same character table”?
![Page 17: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/17.jpg)
Example: Given a variety X one can think about CohpX q orDbpCohX q as a linearisation of X .
Example: Given a finite group G its “C-linear shadow” is thecharacter table (essentially by semi-simplicity).
The subtle homological algebra of kG if kG is not semi-simplemeans that Rep kG or DbpRep kG q is better thought of as its
k-linear shadow.
Amusing: Under this analogy difficult conjectures about derived equivalence (e.g. Broue conjecture) are higher
categorical versions of questions like “can two groups have the same character table”?
![Page 18: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/18.jpg)
Example: Given a variety X one can think about CohpX q orDbpCohX q as a linearisation of X .
Example: Given a finite group G its “C-linear shadow” is thecharacter table (essentially by semi-simplicity).
The subtle homological algebra of kG if kG is not semi-simplemeans that Rep kG or DbpRep kG q is better thought of as its
k-linear shadow.
Amusing: Under this analogy difficult conjectures about derived equivalence (e.g. Broue conjecture) are higher
categorical versions of questions like “can two groups have the same character table”?
![Page 19: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/19.jpg)
In classical representation theory we ask:
What are the basic linear symmetries?
Partial answer: Groups, Lie algebras, quantum groups, Heckealgebras, . . .
(Note that here there are objects (e.g. compact Lie groups) whose representation theory is particularly appealing.
We have both a general theory and a rich base of examples.)
In higher representation theory we ask:
What are the basic categorical symmetries?
I would suggest that we don’t know the answer to this question. We are
witnessing the birth of a theory. We know some examples which are both
intrinsically beautiful and powerful, but are far from a general theory.
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In classical representation theory we ask:
What are the basic linear symmetries?
Partial answer: Groups, Lie algebras, quantum groups, Heckealgebras, . . .
(Note that here there are objects (e.g. compact Lie groups) whose representation theory is particularly appealing.
We have both a general theory and a rich base of examples.)
In higher representation theory we ask:
What are the basic categorical symmetries?
I would suggest that we don’t know the answer to this question. We are
witnessing the birth of a theory. We know some examples which are both
intrinsically beautiful and powerful, but are far from a general theory.
![Page 21: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/21.jpg)
In classical representation theory we ask:
What are the basic linear symmetries?
Partial answer: Groups, Lie algebras, quantum groups, Heckealgebras, . . .
(Note that here there are objects (e.g. compact Lie groups) whose representation theory is particularly appealing.
We have both a general theory and a rich base of examples.)
In higher representation theory we ask:
What are the basic categorical symmetries?
I would suggest that we don’t know the answer to this question. We are
witnessing the birth of a theory. We know some examples which are both
intrinsically beautiful and powerful, but are far from a general theory.
![Page 22: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/22.jpg)
In classical representation theory we ask:
What are the basic linear symmetries?
Partial answer: Groups, Lie algebras, quantum groups, Heckealgebras, . . .
(Note that here there are objects (e.g. compact Lie groups) whose representation theory is particularly appealing.
We have both a general theory and a rich base of examples.)
In higher representation theory we ask:
What are the basic categorical symmetries?
I would suggest that we don’t know the answer to this question. We are
witnessing the birth of a theory. We know some examples which are both
intrinsically beautiful and powerful, but are far from a general theory.
![Page 23: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/23.jpg)
R. Rouquier, 2-Kac-Moody algebras, 2008
Over the past ten years, we have advocated the idea that thereshould exist monoidal categories (or 2-categories) with an
interesting “representation theory”: we propose to call“2-representation theory” this higher version of representationtheory and to call “2-algebras” those “interesting” monoidaladditive categories. The difficulty in pinning down what is a2-algebra (or a Hopf version) should be compared with the
difficulty in defining precisely the meaning of quantum groups (orquantum algebras).
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First steps in higher representation theory.
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Given a vector space V its symmetries are EndpV q.
In representation theory we study homomorphisms AÑ EndpV qfor A an algebra and ask what they might tell us about V .
Given an (additive) category C its symmetries are EndpCq (all(additive) endofunctors of C).
In higher representation theory we study homomorphismsAÑ EndpV q for A a monoidal category and ask what such
homomorphisms might tell us about C.
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Given a vector space V its symmetries are EndpV q.
In representation theory we study homomorphisms AÑ EndpV qfor A an algebra and ask what they might tell us about V .
Given an (additive) category C its symmetries are EndpCq (all(additive) endofunctors of C).
In higher representation theory we study homomorphismsAÑ EndpV q for A a monoidal category and ask what such
homomorphisms might tell us about C.
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Given a vector space V its symmetries are EndpV q.
In representation theory we study homomorphisms AÑ EndpV qfor A an algebra and ask what they might tell us about V .
Given an (additive) category C its symmetries are EndpCq (all(additive) endofunctors of C).
In higher representation theory we study homomorphismsAÑ EndpV q for A a monoidal category and ask what such
homomorphisms might tell us about C.
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Given a vector space V its symmetries are EndpV q.
In representation theory we study homomorphisms AÑ EndpV qfor A an algebra and ask what they might tell us about V .
Given an (additive) category C its symmetries are EndpCq (all(additive) endofunctors of C).
In higher representation theory we study homomorphismsAÑ EndpV q for A a monoidal category and ask what such
homomorphisms might tell us about C.
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Thus algebras are replaced by (additive or sometimes abelian)tensor categories.
Recall: A is an additive tensor category if we have a bifunctor ofadditive categories:
pM1,M2q ÞÑ M1 bM2
together with a unit 1, associator, . . .
Examples: Vectk , RepG , G -graded vector spaces, EndpCq(endofunctors of an additive category), . . .
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A A-module is an additive category M together with a b-functor
AÑ FunpM,Mq.
What exactly this means can take a little getting used to.
As in classical representation theory it is often more useful to thinkabout an “action” of A on M.
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A A-module is an additive category M together with a b-functor
AÑ FunpM,Mq.
What exactly this means can take a little getting used to.
As in classical representation theory it is often more useful to thinkabout an “action” of A on M.
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A first example:
A :“ Rep SU2 p“ Repfd sl2pCqq
A is generated under sums and summands by nat :“ C2.
An A-module is a recipe M ÞÑ nat ¨M and a host of maps
HomApnatbm,natbnq Ñ HomMpnat
bm ¨M, natbn ¨Mq
satisfying an even larger host of identities which I will let youcontemplate.
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A first example:
A :“ Rep SU2 p“ Repfd sl2pCqq
A is generated under sums and summands by nat :“ C2.
An A-module is a recipe M ÞÑ nat ¨M and a host of maps
HomApnatbm,natbnq Ñ HomMpnat
bm ¨M, natbn ¨Mq
satisfying an even larger host of identities which I will let youcontemplate.
![Page 35: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/35.jpg)
A first example:
A :“ Rep SU2 p“ Repfd sl2pCqq
A is generated under sums and summands by nat :“ C2.
An A-module is a recipe M ÞÑ nat ¨M and a host of maps
HomApnatbm,natbnq Ñ HomMpnat
bm ¨M, natbn ¨Mq
satisfying an even larger host of identities which I will let youcontemplate.
![Page 36: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/36.jpg)
Let M be an A “ Rep SU2-module which is
1. abelian and semi-simple,
2. indecomposable as an A-module.
Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
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Let M be an A “ Rep SU2-module which is
1. abelian and semi-simple,
2. indecomposable as an A-module.
Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
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Let M be an A “ Rep SU2-module which is
1. abelian and semi-simple,
2. indecomposable as an A-module.
Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
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Let M be an A “ Rep SU2-module which is
1. abelian and semi-simple,
2. indecomposable as an A-module.
Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
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Let M be an A “ Rep SU2-module which is
1. abelian and semi-simple,
2. indecomposable as an A-module.
Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
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Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
Theorem
(Classification of representations of Rep SU2.) These are all.
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Examples:
M :“ VectC with V ¨M :“ ForpV q bM (“trivial rep”)
M :“ RepSU2 with V ¨M :“ V bM (“regular rep”)
M :“ Rep S1 with V ¨M :“ pResS1
SU2V q bM.
M :“ Rep Γ (Γ Ă SU2 finite or NSU2pS1q) with
V ¨M :“ pResΓSU2
V q bM.
Theorem
(Classification of representations of Rep SU2.) These are all.
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Remarkably, the action of Rep SU2 on the Grothendieck group ofM already determines the structure of M as an RepSU2-module!
This is an example of “rigidity” in higher representation theory.
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Remarkably, the action of Rep SU2 on the Grothendieck group ofM already determines the structure of M as an RepSU2-module!
This is an example of “rigidity” in higher representation theory.
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An example of higher representation theory(joint with Simon Riche).
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We want to apply these ideas to the modular (i.e. characteristic p)representation theory of finite and algebraic groups.
Here the questions are very difficult and we will probably neverknow a complete and satisfactory answer.
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Some motivation from characteristic 0:
Recall the famous Kazhdan-Lusztig conjecture (1979):
chpLw q “ÿ
yPW
p1q`pwq´`pyqPy ,w p1qchpMy q
(Here Lw (resp. My ) is a simple highest weight module (resp. Verma
module) for a complex semi-simple Lie algebra, and Py ,w is a
“Kazhdan-Lusztig” polynomial.)
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The Kazhdan-Lusztig conjecture has 2 distinct proofs:
1. Geometric: Apply the localization theorem for g-modules topass to differential operators (D-modules) on the flag variety,then pass through the Riemann-Hilbert correspondence toland in perverse sheaves, and using some deep geometric tools(e.g. proof of Weil conjectures) complete the proof(Kazhdan-Lusztig, Beilinson-Bernstein, Brylinsky-Kashiwara1980s). This proof uses every trick in the book!
2. Categorical: Show that translation functors give an action of“Soergel bimodules” on category O. Then theKazhdan-Lusztig conjecture follows from the calculation ofthe character of indecomposable Soergel bimodules (Soergel1990, Elias-W 2012). This proof is purely algebraic.
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The Kazhdan-Lusztig conjecture has 2 distinct proofs:
1. Geometric: Apply the localization theorem for g-modules topass to differential operators (D-modules) on the flag variety,then pass through the Riemann-Hilbert correspondence toland in perverse sheaves, and using some deep geometric tools(e.g. proof of Weil conjectures) complete the proof(Kazhdan-Lusztig, Beilinson-Bernstein, Brylinsky-Kashiwara1980s). This proof uses every trick in the book!
2. Categorical: Show that translation functors give an action of“Soergel bimodules” on category O. Then theKazhdan-Lusztig conjecture follows from the calculation ofthe character of indecomposable Soergel bimodules (Soergel1990, Elias-W 2012). This proof is purely algebraic.
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We want to apply the second approach to the representationtheory of reductive algebraic groups.
The first approach has also seen recent progress (Bezrukavnikov-Mirkovic-Rumynin) however it seems much more
likely at this stage that the second approach will yield computable character formulas.
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For the rest of the talk fix a field k and a connected reductivegroup G like GLn (where we will state a theorem later) of Sp4
(where we can draw pictures).
If k is of characteristic 0 then RepG looks “just likerepresentations of a compact Lie group”. In positive characteristicone still has a classification of simple modules via highest weight,
character theory etc. However the simple modules are usuallymuch smaller than in characteristic zero.
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For the rest of the talk fix a field k and a connected reductivegroup G like GLn (where we will state a theorem later) of Sp4
(where we can draw pictures).
If k is of characteristic 0 then RepG looks “just likerepresentations of a compact Lie group”. In positive characteristicone still has a classification of simple modules via highest weight,
character theory etc. However the simple modules are usuallymuch smaller than in characteristic zero.
![Page 54: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/54.jpg)
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Rep0‘Ă RepG the principal block.
Rep0 Ă RepG depends on p!
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The analogue of the Kazhdan-Lusztig conjecture in this setting is:
Lusztig’s character formula (1979): If x ¨ 0 is “restricted” (all digitsin fundamental weights less than p) then
chpx ¨p 0q “ÿ
y
p´1q`pyq´`pxqPw0y ,w0xp1qchp∆py ¨p 0qq.
For non-trivial reasons this gives a character formula for all simple modules.
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1. 1979: Lusztig conjecture this formula to hold for p ě 2h ´ 2(h = Coxeter number). Later Kato suggested that p ě h isreasonable.
2. 1994: Proved to hold for large p without an explicit bound byAndersen-Janzten-Soergel, Kashiwara-Tanisaki, Lusztig,Kazhdan-Lusztig.
3. 2008: Fiebig gave a new proof for p " 0 as well as an explicit(enormous) bound (e.g. at least of the order of p ą nn
2for
SLn)
4. 2013: Building on work of Soergel and joint work with Elias,He, Kontorovich and Mcnamara I showed that the Lusztigconjecture does not hold for many p which grow exponentiallyin n. (E.g. fails for p “ 470 858 183 for SL100.)
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1. 1979: Lusztig conjecture this formula to hold for p ě 2h ´ 2(h = Coxeter number). Later Kato suggested that p ě h isreasonable.
2. 1994: Proved to hold for large p without an explicit bound byAndersen-Janzten-Soergel, Kashiwara-Tanisaki, Lusztig,Kazhdan-Lusztig.
3. 2008: Fiebig gave a new proof for p " 0 as well as an explicit(enormous) bound (e.g. at least of the order of p ą nn
2for
SLn)
4. 2013: Building on work of Soergel and joint work with Elias,He, Kontorovich and Mcnamara I showed that the Lusztigconjecture does not hold for many p which grow exponentiallyin n. (E.g. fails for p “ 470 858 183 for SL100.)
![Page 60: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/60.jpg)
1. 1979: Lusztig conjecture this formula to hold for p ě 2h ´ 2(h = Coxeter number). Later Kato suggested that p ě h isreasonable.
2. 1994: Proved to hold for large p without an explicit bound byAndersen-Janzten-Soergel, Kashiwara-Tanisaki, Lusztig,Kazhdan-Lusztig.
3. 2008: Fiebig gave a new proof for p " 0 as well as an explicit(enormous) bound (e.g. at least of the order of p ą nn
2for
SLn)
4. 2013: Building on work of Soergel and joint work with Elias,He, Kontorovich and Mcnamara I showed that the Lusztigconjecture does not hold for many p which grow exponentiallyin n. (E.g. fails for p “ 470 858 183 for SL100.)
![Page 61: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/61.jpg)
1. 1979: Lusztig conjecture this formula to hold for p ě 2h ´ 2(h = Coxeter number). Later Kato suggested that p ě h isreasonable.
2. 1994: Proved to hold for large p without an explicit bound byAndersen-Janzten-Soergel, Kashiwara-Tanisaki, Lusztig,Kazhdan-Lusztig.
3. 2008: Fiebig gave a new proof for p " 0 as well as an explicit(enormous) bound (e.g. at least of the order of p ą nn
2for
SLn)
4. 2013: Building on work of Soergel and joint work with Elias,He, Kontorovich and Mcnamara I showed that the Lusztigconjecture does not hold for many p which grow exponentiallyin n. (E.g. fails for p “ 470 858 183 for SL100.)
![Page 62: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/62.jpg)
Rep0‘Ă RepG the principal block.
On Rep0 one has the action of wall-crossing functors:
“matrix coefficients of tensoring with objects in RepG”
Let W denote the affine Weyl group and S “ ts0, . . . , snu itssimple reflections. For each s P S one has a wall-crossing functor
Ξs . These generate the category of wall-crossing functors.
xΞs0 ,Ξs1 , . . . ,Ξsny ýRep0 .
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Rep0‘Ă RepG the principal block.
On Rep0 one has the action of wall-crossing functors:
“matrix coefficients of tensoring with objects in RepG”
Let W denote the affine Weyl group and S “ ts0, . . . , snu itssimple reflections. For each s P S one has a wall-crossing functor
Ξs . These generate the category of wall-crossing functors.
xΞs0 ,Ξs1 , . . . ,Ξsny ýRep0 .
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Main conjecture: This action of wall-crossing functors can beupgraded to an action of diagrammatic Soergel bimodules.
The category of diagrammatic Soergel bimodules is a fundamentalmonoidal category in representation theory.
It can be thought of as one of the promised objects which hasinteresting 2-representation theory.
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Main conjecture: This action of wall-crossing functors can beupgraded to an action of diagrammatic Soergel bimodules.
The category of diagrammatic Soergel bimodules is a fundamentalmonoidal category in representation theory.
It can be thought of as one of the promised objects which hasinteresting 2-representation theory.
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Theorem: Our conjecture holds for G “ GLn.
Consequences of the conjecture...
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Theorem: Our conjecture holds for G “ GLn.
Consequences of the conjecture...
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The category of diagrammatic Soergel bimodules is a natural homefor the canonical basis and Kazhdan-Lusztig polynomials. In fact,
because it is defined over Z we get the p-canonical basis andp-Kazhdan-Lusztig polynomials for all p.
Theorem: Assume our conjecture or G “ GLn. Then there existsimple formulas for the irreducible (if p ą 2h ´ 2) and tilting (if
p ą h) characters in terms of the p-canonical basis.
Thus the p-canonical basis controls precisely when Lusztig’sconjecture holds, and tells us what happens when it fails.
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The category of diagrammatic Soergel bimodules is a natural homefor the canonical basis and Kazhdan-Lusztig polynomials. In fact,
because it is defined over Z we get the p-canonical basis andp-Kazhdan-Lusztig polynomials for all p.
Theorem: Assume our conjecture or G “ GLn. Then there existsimple formulas for the irreducible (if p ą 2h ´ 2) and tilting (if
p ą h) characters in terms of the p-canonical basis.
Thus the p-canonical basis controls precisely when Lusztig’sconjecture holds, and tells us what happens when it fails.
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The category of diagrammatic Soergel bimodules is a natural homefor the canonical basis and Kazhdan-Lusztig polynomials. In fact,
because it is defined over Z we get the p-canonical basis andp-Kazhdan-Lusztig polynomials for all p.
Theorem: Assume our conjecture or G “ GLn. Then there existsimple formulas for the irreducible (if p ą 2h ´ 2) and tilting (if
p ą h) characters in terms of the p-canonical basis.
Thus the p-canonical basis controls precisely when Lusztig’sconjecture holds, and tells us what happens when it fails.
![Page 72: An example of higher representation theorypeople.mpim-bonn.mpg.de/geordie/Cordoba.pdf · Example: Given a nite group G its \C-linear shadow" is the character table (essentially by](https://reader034.vdocument.in/reader034/viewer/2022042812/5fabf975b855183cf4359174/html5/thumbnails/72.jpg)
Other consequences of our conjecture:
1. A complete description of Rep0 in terms of the Heckecategory, existence of a Z-grading, etc.
2. All three main conjectures in this area (Lusztig conjecture,Andersen conjecture, James conjecture) are all controlled bythe p-canonical basis. (Actually, the links to the Jamesconjecture need some other conjectures. They should followfrom work in progress by Elias-Losev.)
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Other consequences of our conjecture:
1. A complete description of Rep0 in terms of the Heckecategory, existence of a Z-grading, etc.
2. All three main conjectures in this area (Lusztig conjecture,Andersen conjecture, James conjecture) are all controlled bythe p-canonical basis. (Actually, the links to the Jamesconjecture need some other conjectures. They should followfrom work in progress by Elias-Losev.)
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Thanks!
Slides:
people.mpim-bonn.mpg.de/geordie/Cordoba.pdf
Paper with Riche (all 136 pages!):
Tilting modules and the p-canonical basis,
http://arxiv.org/abs/1512.08296