Representation Theory and Geometry

Geordie Williamson

University of Sydney http://www.maths.usyd.edu.au/u/geordie/ICM.pdf representations We obtain a representation of our group of symmetries

ρ : G Ñ GLpVq.

representations

¨ ˛ ˚ ‹ ⇝ Sym ˝ ‚Ă GLpR3q.

2 representations

¨ ˛ ˚ ‹ ⇝ Sym ˝ ‚Ă GLpR3q.

We obtain a representation of our group of symmetries

ρ : G Ñ GLpVq.

2 representations

´e1 ´ e2 e1

#˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸+ 1 0 1 ´1 ´1 0 0 ´1 ´1 1 0 1 , , , , , 0 1 0 ´1 ´1 1 1 ´1 ´1 0 1 0

3 Symmetric group Sn The problem of understanding

„ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation

is hard. The theory of representations of Sn is rich, highly-developed and useful.

Galois representations The passage(s)

tvarieties{Qu ÝÑ tGalois representationsu

is one of the most powerful tools of modern number theory.

why study representations?

4 The theory of representations of Sn is rich, highly-developed and useful.

Galois representations The passage(s)

tvarieties{Qu ÝÑ tGalois representationsu

is one of the most powerful tools of modern number theory.

why study representations?

Symmetric group Sn The problem of understanding

„ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation

is hard.

4 Galois representations The passage(s)

tvarieties{Qu ÝÑ tGalois representationsu

is one of the most powerful tools of modern number theory.

why study representations?

Symmetric group Sn The problem of understanding

„ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation

is hard. The theory of representations of Sn is rich, highly-developed and useful.

4 why study representations?

Symmetric group Sn The problem of understanding

„ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation

is hard. The theory of representations of Sn is rich, highly-developed and useful.

Galois representations The passage(s)

tvarieties{Qu ÝÑ tGalois representationsu

is one of the most powerful tools of modern number theory.

4 There are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

R3 “ L ‘ H “ trivial ‘

A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations.

simple representations

Example Consider the symmetric group S3. It acts via permutation of coordinates on R3.

5 R3 “ L ‘ H “ trivial ‘

simple representations

Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

5 A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations.

simple representations

Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

R3 “ L ‘ H “ trivial ‘

5 A representation is semi-simple if it is isomorphic to a direct sum of simple representations.

simple representations

Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

R3 “ L ‘ H “ trivial ‘

A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V.

5 simple representations

Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

R3 “ L ‘ H “ trivial ‘

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

However now L Ă H because 3 “ 0. We obtain a composition series

F3 0 Ă L Ă H Ă 3. trivial We write (“Grothendieck group”, “multiplicities”)

F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns.

trivial

jordan-hölder theorem

Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the ﬁnite ﬁeld with 3 elements.)

6 However now L Ă H because 3 “ 0. We obtain a composition series

F3 0 Ă L Ă H Ă 3. trivial We write (“Grothendieck group”, “multiplicities”)

F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns.

trivial

jordan-hölder theorem

Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the ﬁnite ﬁeld with 3 elements.) As before, there are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

6 We obtain a composition series

F3 0 Ă L Ă H Ă 3. trivial We write (“Grothendieck group”, “multiplicities”)

F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns.

trivial

jordan-hölder theorem

Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the ﬁnite ﬁeld with 3 elements.) As before, there are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

However now L Ă H because 3 “ 0.

6 trivial We write (“Grothendieck group”, “multiplicities”)

F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns.

trivial

jordan-hölder theorem

Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the ﬁnite ﬁeld with 3 elements.) As before, there are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

However now L Ă H because 3 “ 0. We obtain a composition series

F3 0 Ă L Ă H Ă 3.

6 jordan-hölder theorem

Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the ﬁnite ﬁeld with 3 elements.) As before, there are two invariant subspaces:

L “ tall coordinates equalu, H “ tcoordinates sum to zerou.

However now L Ă H because 3 “ 0. We obtain a composition series

F3 0 Ă L Ă H Ă 3. trivial We write (“Grothendieck group”, “multiplicities”)

F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns.

trivial

6 We search for a classiﬁcation (“periodic table”), character formulas (“mass”, “number of neutrons”), …

simple representations

representations Ø “matter”

␣ ( simple representations Ø “elements” , ,

semi-simple Ø “elements don’t interract”

7 simple representations

representations Ø “matter”

␣ ( simple representations Ø “elements” , ,

semi-simple Ø “elements don’t interract”

We search for a classiﬁcation (“periodic table”), character formulas (“mass”, “number of neutrons”), …

7 ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world”

‚ Inﬁnite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups…

plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897).

8 “Beyond the semi-simple world”

plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925).

8 ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups…

plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world”

‚ Inﬁnite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979).

8 ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups…

plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world”

8 Related situations: non-compact Lie groups, p-adic groups…

plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world”

8 plan of talk

“Semi-simple world”

‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world”

8 geometric structure invariant forms

symmetric, hermitian, …

representation theory geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

9 geometric structure invariant forms

symmetric, hermitian, …

geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

representation theory

9 geometric structure invariant forms

symmetric, hermitian, …

geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

representation theory geometry

9 geometric structure

geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

invariant forms

symmetric, hermitian, …

representation theory geometry

9 geometric structure

geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

invariant forms

symmetric, hermitian, …

representation theory geometry

9 geometric structure

geometry

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout.

common features

invariant forms

symmetric, hermitian, …

representation theory geometry

Related feature: (hidden) semi-simplicity

9 geometric structure

geometry

common features

invariant forms

symmetric, hermitian, …

representation theory geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive deﬁnite; signature plays an important role throughout. 9 geometry

common features

geometric structure invariant forms

symmetric, hermitian, …

representation theory geometry

Related feature: (hidden) semi-simplicity

A geometric structure on a real (resp. complex) vector space V will mean a non-degenerate symmetric (resp. Hermitian) form on V.

We do not assume that our forms are positive definite; signature plays an important role throughout. 9 the semi-simple world Observation 1: If V has a positive-definite G-invariant geometric structure, then V is semi-simple. If U Ă V is a subrepresentation, then V “ U ‘ UK. Observation 2: Any representation of G admits a positive-definite geometric structure. Take a positive-definite geometric structure x´, ý on V. Then ÿ 1 xv, wy :“ xgv, gwy G |G|

deﬁnes a positive-deﬁnite and G-invariant geometric structure.

maschke’s theorem

Maschke (1897) Any representation V of a ﬁnite group G over R or C is semi-simple.

11 Observation 2: Any representation of G admits a positive-definite geometric structure. Take a positive-definite geometric structure x´, ý on V. Then ÿ 1 xv, wy :“ xgv, gwy G |G|

deﬁnes a positive-deﬁnite and G-invariant geometric structure.

maschke’s theorem

Maschke (1897) Any representation V of a ﬁnite group G over R or C is semi-simple.

Observation 1: If V has a positive-deﬁnite G-invariant geometric structure, then V is semi-simple. If U Ă V is a subrepresentation, then V “ U ‘ UK.

11 maschke’s theorem

Maschke (1897) Any representation V of a ﬁnite group G over R or C is semi-simple.

Observation 1: If V has a positive-definite G-invariant geometric structure, then V is semi-simple. If U Ă V is a subrepresentation, then V “ U ‘ UK. Observation 2: Any representation of G admits a positive-definite geometric structure. Take a positive-definite geometric structure x´, ý on V. Then ÿ 1 xv, wy :“ xgv, gwy G |G|

deﬁnes a positive-deﬁnite and G-invariant geometric structure.

11 If V is simple and deﬁned over the complex numbers, then Schur’s lemma shows that the geometric structure is unique up to positive scalar. This is an example of “unicity of geometric structure”.

⇝

geometric structure

Example of “semi-simplicity via introduction of geometric structure”.

12 This is an example of “unicity of geometric structure”.

⇝

geometric structure

Example of “semi-simplicity via introduction of geometric structure”. If V is simple and deﬁned over the complex numbers, then Schur’s lemma shows that the geometric structure is unique up to positive scalar.

12 geometric structure

⇝

12 Weyl (1925)

Any continuous representation of a compact Lie group K is semi-simple.

Existence and uniqueness of geometric structure still holds.

weyl’s theorem

1 Consider a compact Lie group K, e.g. S or SU2 or a ﬁnite group. Weyl generalised these observations to K, with sum replaced by integral: ż

xv, wyK :“ xgv, gwydµ K

13 Existence and uniqueness of geometric structure still holds.

weyl’s theorem

1 Consider a compact Lie group K, e.g. S or SU2 or a ﬁnite group. Weyl generalised these observations to K, with sum replaced by integral: ż

xv, wyK :“ xgv, gwydµ K

Weyl (1925)

Any continuous representation of a compact Lie group K is semi-simple.

13 weyl’s theorem

1 Consider a compact Lie group K, e.g. S or SU2 or a ﬁnite group. Weyl generalised these observations to K, with sum replaced by integral: ż

xv, wyK :“ xgv, gwydµ K

Weyl (1925)

Any continuous representation of a compact Lie group K is semi-simple.

Existence and uniqueness of geometric structure still holds.

13 cartan’s letter

Élie Cartan to Hermann Weyl, 28 of March 1925:

“…the difficulty, I dare not say the impossibility, of finding a proof which does not leave the strictly infinitesimal domain shows the necessity of not sacrificing either point of view …” An algebraic (“infinitesimal”) proof took 10 years, and involves the Casimir element (arises from an invariant form called the trace form).

14 extended example: su2 and sl2 “I don’t think it is the representations themselves, but the groups. I ﬁnd

SU2, SL2,Sn etc. amazing and beautiful animals (if I have a favourite, it

is SU2), but will probably never really understand them. I might someday understand their linear shadows though...” – Quindici

su2

# ˜ ¸ ˇ + ˇ a b ˇ unit SU “ A “ ˇ AA˚ “ id, det A “ 1 “ 2 c d ˇ quaternions.

˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 LiepSU qC “ sl pCq “C ‘ C ‘ C 2 2 1 0 0 ´1 0 0 “ “ “ f h e

16 su2

# ˜ ¸ ˇ + ˇ a b ˇ unit SU “ A “ ˇ AA˚ “ id, det A “ 1 “ 2 c d ˇ quaternions.

˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 LiepSU qC “ sl pCq “C ‘ C ‘ C 2 2 1 0 0 ´1 0 0 “ “ “ f h e

“I don’t think it is the representations themselves, but the groups. I ﬁnd

SU2, SL2,Sn etc. amazing and beautiful animals (if I have a favourite, it

is SU2), but will probably never really understand them. I might someday understand their linear shadows though...” – Quindici

16 simple representations of su2

SU2 acts on its “natural representation”: ˜ ¸ ˜ ¸ 0 1 C2 “ C ‘ C “ CY ‘ CX. 1 0

For any m ě 0, SU2 acts naturally on homogenous polynomials in X, Y of degree m:

m m´1 m m´1 m Lm :“ CY ‘ CY X ‘ ¨ ¨ ¨ ‘ CY X ‘ CX .

The Lm for m ě 0 are all irreducible representations of SU2. “spherical harmonics”, “quantum mechanics”.

17 5 4 3 2 1 e 0 C C C C C C

1 2 3 4 5 f

´5 ´3 ´1 1 3 5 h

Action on Lm (here m “ 5):

Y5 Y4X1 Y3X2 Y2X3 YX4 X5

action of the lie algebra

Differentiate to get representation of the (complexiﬁed) Lie algebra

LiepSU2qC “ sl2pCq ˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

18 5 4 3 2 1 e 0

1 2 3 4 5 f

´5 ´3 ´1 1 3 5 h

action of the lie algebra

Differentiate to get representation of the (complexiﬁed) Lie algebra

LiepSU2qC “ sl2pCq ˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

Action on Lm (here m “ 5):

CY5 CY4X1 CY3X2 CY2X3 CYX4 CX5

18 5 4 3 2 1 e 0 C C C C C C

1 2 3 4 5 f

´5 ´3 ´1 1 3 5 h

action of the lie algebra

Differentiate to get representation of the (complexiﬁed) Lie algebra

LiepSU2qC “ sl2pCq ˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

Action on Lm (here m “ 5):

Y5 Y4X1 Y3X2 Y2X3 YX4 X5

18 5 4 3 2 1 e 0 C C C C C C

1 2 3 4 5 f

action of the lie algebra

Differentiate to get representation of the (complexiﬁed) Lie algebra

LiepSU2qC “ sl2pCq ˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

Action on Lm (here m “ 5):

Y5 Y4X1 Y3X2 Y2X3 YX4 X5

´5 ´3 ´1 1 3 5 h

18 C C C C C C

action of the lie algebra

Differentiate to get representation of the (complexiﬁed) Lie algebra

LiepSU2qC “ sl2pCq ˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

Action on Lm (here m “ 5): 5 4 3 2 1 e 0 Y5 Y4X1 Y3X2 Y2X3 YX4 X5

1 2 3 4 5 f

´5 ´3 ´1 1 3 5 h

18 In the algebraic theory an important role is played by inﬁnite-dimensional representations called Verma modules. The study of Verma modules has led to important advances in representation theory beyond the semi-simple world.

For sl2pCq they depend on a parameter λ P C.

As vector spaces: à C ∆λ “ vi iPZě0

verma modules

For SU2 and sl2pCq we can do everything “by hand”. This is no longer possible for more complicated groups.

19 For sl2pCq they depend on a parameter λ P C.

As vector spaces: à C ∆λ “ vi iPZě0

verma modules

For SU2 and sl2pCq we can do everything “by hand”. This is no longer possible for more complicated groups. In the algebraic theory an important role is played by inﬁnite-dimensional representations called Verma modules. The study of Verma modules has led to important advances in representation theory beyond the semi-simple world.

19 As vector spaces: à C ∆λ “ vi iPZě0

verma modules

For sl2pCq they depend on a parameter λ P C.

19 verma modules

For sl2pCq they depend on a parameter λ P C.

As vector spaces: à C ∆λ “ vi iPZě0

19 verma modules

˜ ¸ ˜ ¸ ˜ ¸ 0 0 1 0 0 1 sl pCq “C ‘ C ‘ C 2 1 0 0 ´1 0 0 “ “ “ f h e

The Verma module ∆λ determined by λ P C:

λ ´ 5 λ ´ 4 λ ´ 3 λ ´ 2 λ ´ 1 λ e 0 ... Cv5 Cv4 Cv3 Cv2 Cv1 Cv0

6 5 4 3 2 1 f

λ ´ 10 λ ´ 8 λ ´ 6 λ ´ 4 λ ´ 2 λ h

20 structure of verma modules

λ ´ 5 λ ´ 4 λ ´ 3 λ ´ 2 λ ´ 1 λ e 0 ... Cv5 Cv4 Cv3 Cv2 Cv1 Cv0

6 5 4 3 2 1 f

λ ´ 10 λ ´ 8 λ ´ 6 λ ´ 4 λ ´ 2 λ h

λ ‰ 0, 1, 2, 3,... : ∆λ is simple, call it Lλ. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ? ?

21 structure of verma modules

Example λ “ 2

´3 ´2 ´1 0 1 2 e 0 ... ‚ ‚ ‚ ‚ ‚ ‚

6 5 4 3 2 1 f

´8 ´6 ´4 ´2 0 2 h

We have a subrepresentation isomorphic to ∆´4, and 2

? ∆2{∆´4 – L2 ´4

(L2 is our simple finite-dimensional representation from earlier.) 22 (a) A single family of representations (Verma modules) produces all simple finite-dimensional representations. (b) We get new infinite-dimensional simple representations. (c) The structure of Verma modules varies (subtly) based on the parameter.

Summary:

structure of verma modules

λ ‰ 0, 1, 2, 3,... : ∆λ is simple and inﬁnite-dimensional. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ´λ ´ 2

23 (a) A single family of representations (Verma modules) produces all simple ﬁnite-dimensional representations. (b) We get new inﬁnite-dimensional simple representations. (c) The structure of Verma modules varies (subtly) based on the parameter.

structure of verma modules

λ ‰ 0, 1, 2, 3,... : ∆λ is simple and inﬁnite-dimensional. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ´λ ´ 2

Summary:

23 (b) We get new inﬁnite-dimensional simple representations. (c) The structure of Verma modules varies (subtly) based on the parameter.

structure of verma modules

λ ‰ 0, 1, 2, 3,... : ∆λ is simple and inﬁnite-dimensional. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ´λ ´ 2

Summary: (a) A single family of representations (Verma modules) produces all simple ﬁnite-dimensional representations.

23 (c) The structure of Verma modules varies (subtly) based on the parameter.

structure of verma modules

λ ‰ 0, 1, 2, 3,... : ∆λ is simple and inﬁnite-dimensional. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ´λ ´ 2

Summary: (a) A single family of representations (Verma modules) produces all simple ﬁnite-dimensional representations. (b) We get new inﬁnite-dimensional simple representations.

23 structure of verma modules

λ ‰ 0, 1, 2, 3,... : ∆λ is simple and inﬁnite-dimensional. λ

λ “ 0, 1, 2, 3,... : ∆λ is not simple. ´λ ´ 2

Summary: (a) A single family of representations (Verma modules) produces all simple ﬁnite-dimensional representations. (b) We get new inﬁnite-dimensional simple representations. (c) The structure of Verma modules varies (subtly) based on the parameter.

23 kazhdan-lusztig conjecture Motivation We think of the ﬁnite group W as being the skeleton of g. We try to answer questions about g in terms of W.

the weyl group

g is a complex semi-simple Lie algebra. h Ă g a Cartan subalgebra. W the Weyl group, which acts on h as a reﬂection group.

Example

g “ slnpCq “ t n ˆ n matrices X | trX “ 0u. h “ diagonal matrices Ă slnpCq W “ Sn acting on h via permutations.

25 the weyl group

g is a complex semi-simple Lie algebra. h Ă g a Cartan subalgebra. W the Weyl group, which acts on h as a reﬂection group.

Example

g “ slnpCq “ t n ˆ n matrices X | trX “ 0u. h “ diagonal matrices Ă slnpCq W “ Sn acting on h via permutations.

Motivation We think of the ﬁnite group W as being the skeleton of g. We try to answer questions about g in terms of W.

25 ∆λ has unique simple quotient ∆λ ↠ Lλ

Lλ is called a simple highest weight module.

Example sl2pCq

If λ ‰ 0, 1,... , Lλ “ ∆λ is inﬁnite dimensional. If λ “ 0, 1,... then Lλ is ﬁnite dimensional.

Basic problem

Describe the structure of ∆λ. Which simple modules occur with which multiplicity?

verma modules revisited

g is a complex semi-simple Lie algebra.

˚ “weight” λ P h ⇝ Verma module ∆λ.

26 Example sl2pCq

If λ ‰ 0, 1,... , Lλ “ ∆λ is inﬁnite dimensional. If λ “ 0, 1,... then Lλ is ﬁnite dimensional.

Basic problem

Describe the structure of ∆λ. Which simple modules occur with which multiplicity?

verma modules revisited

g is a complex semi-simple Lie algebra.

˚ “weight” λ P h ⇝ Verma module ∆λ.

∆λ has unique simple quotient ∆λ ↠ Lλ

Lλ is called a simple highest weight module.

26 Basic problem

Describe the structure of ∆λ. Which simple modules occur with which multiplicity?

verma modules revisited

g is a complex semi-simple Lie algebra.

˚ “weight” λ P h ⇝ Verma module ∆λ.

∆λ has unique simple quotient ∆λ ↠ Lλ

Lλ is called a simple highest weight module.

Example sl2pCq

If λ ‰ 0, 1,... , Lλ “ ∆λ is inﬁnite dimensional. If λ “ 0, 1,... then Lλ is ﬁnite dimensional.

26 verma modules revisited

g is a complex semi-simple Lie algebra.

˚ “weight” λ P h ⇝ Verma module ∆λ.

∆λ has unique simple quotient ∆λ ↠ Lλ

Lλ is called a simple highest weight module.

Example sl2pCq

If λ ‰ 0, 1,... , Lλ “ ∆λ is inﬁnite dimensional. If λ “ 0, 1,... then Lλ is ﬁnite dimensional.

Basic problem

Describe the structure of ∆λ. Which simple modules occur with which multiplicity?

26 (a) New paradigm in representation theory.

(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.

kazhdan-lusztig conjecture

∆λ : Verma module. Lλ : simple highest weight module.

Kazhdan-Lusztig conjecture (1979)

ÿ r∆λs “ Pλ,µp1qrLµs. µ

Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.

27 (b) Pλ,µ only depends on a pair of elements the Weyl group W of g.

kazhdan-lusztig conjecture

∆λ : Verma module. Lλ : simple highest weight module.

Kazhdan-Lusztig conjecture (1979)

ÿ r∆λs “ Pλ,µp1qrLµs. µ

Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.

(a) New paradigm in representation theory.

27 kazhdan-lusztig conjecture

∆λ : Verma module. Lλ : simple highest weight module.

Kazhdan-Lusztig conjecture (1979)

ÿ r∆λs “ Pλ,µp1qrLµs. µ

Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.

(a) New paradigm in representation theory.

(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.

27 kazhdan-lusztig conjecture

∆λ : Verma module. Lλ : simple highest weight module.

Kazhdan-Lusztig conjecture (1979)

ÿ r∆x¨0s “ Px¨0,y¨0p1qrLy¨0s. yPW

Here Pλ,µ P Zrvs is a Kazhdan-Lusztig polynomial.

(a) New paradigm in representation theory.

(b) Pλ,µ only depends on a pair of elements the Weyl group W of g.

28 29 1

1 30 v1

1 31 v1 v2

v1 1

1 32 v2 v1 v3 1

v2 v1

1 33 v2 v1

v3 v2 v3 v4 v1 1

v3 v2

v2 v1

1 34 v2 v1

v3 v2

v4 v3

v4 v5 v2 v1 v3 v4 v3 1

v2 v3 v2 v1

1 35 v2 v1

v3 v2

v3 v4 v3 v2

v4 v3 ` v5 v2 ` v4 v1

v5 v4 ` v6 v1 ` v3 v2 v3 v4 v3 ` v5 v2 ` v4 v1 1

v3 v4 v3 v2

v2 v3 v2 v1

1 36 v2 v3 v2 v1

v3 v4 v3 v2

v4 v5 v4 v3

v4 v5 v4 ` v6 v3 ` v5 v2 v1 v3 v6 v5 ` v7 v2 ` v4 v3 1

v4 v5 v4 ` v6 v3 ` v5 v2 v1

v4 v5 v4 v3

v3 v4 v3 v2

v2 v3 v2 v1

1 37 v2 v3 v2 v1

v3 v4 v3 v2

v3 v4 v3 ` v5 v2 ` v4 v3 v2

v4 v3 ` v5 2v4 ` v6 2v3 ` v5 v2 ` v4 v1

v5 v4 ` v6 2v5 ` v7 2v4 ` v6 v1 ` v3 v2 v3 v4 2v5 ` v7 2v6 ` v8 2v3 ` v5 2v2 ` v4 v1 1

v5 v4 ` v6 2v5 ` v7 2v4 ` v6 v1 ` v3 v2

v4 v3 ` v5 2v4 ` v6 2v3 ` v5 v2 ` v4 v1

v3 v4 v3 ` v5 v2 ` v4 v3 v2

v3 v4 v3 v2

38 v2 v3 1 v2 v1 Algebraic proof (’10s)

Elias-W. 2014, following Soergel 1990

Jantzen conjecture Beilinson-Bernstein W. 2016 following

(graded multiplicity “ Py,xpvq) Soergel 2008, Kübel 2012

Geometric proofs: D-modules, perverse sheaves, weights…

Algebraic proofs: “shadows of Hodge theory”, i.e. invariant forms (“geometric structures”) still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann

potted history

Geometric proof (’80s)

Kazhdan-Lusztig conjecture Brylinsky-Kashiwara

(multiplicity “ Py,xp1qq Beilinson-Bernstein

39 Algebraic proof (’10s)

Elias-W. 2014, following Soergel 1990

W. 2016 following Soergel 2008, Kübel 2012

Geometric proofs: D-modules, perverse sheaves, weights…

Algebraic proofs: “shadows of Hodge theory”, i.e. invariant forms (“geometric structures”) still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann

potted history

Geometric proof (’80s)

Kazhdan-Lusztig conjecture Brylinsky-Kashiwara

(multiplicity “ Py,xp1qq Beilinson-Bernstein

Jantzen conjecture Beilinson-Bernstein

(graded multiplicity “ Py,xpvq)

39 Geometric proofs: D-modules, perverse sheaves, weights…

Algebraic proofs: “shadows of Hodge theory”, i.e. invariant forms (“geometric structures”) still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann

potted history

Geometric proof (’80s) Algebraic proof (’10s)

Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,

(multiplicity “ Py,xp1qq Beilinson-Bernstein following Soergel 1990

Jantzen conjecture Beilinson-Bernstein W. 2016 following

(graded multiplicity “ Py,xpvq) Soergel 2008, Kübel 2012

39 Algebraic proofs: “shadows of Hodge theory”, i.e. invariant forms (“geometric structures”) still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann

potted history

Geometric proof (’80s) Algebraic proof (’10s)

Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,

(multiplicity “ Py,xp1qq Beilinson-Bernstein following Soergel 1990

Jantzen conjecture Beilinson-Bernstein W. 2016 following

(graded multiplicity “ Py,xpvq) Soergel 2008, Kübel 2012

Geometric proofs: D-modules, perverse sheaves, weights…

39 potted history

Geometric proof (’80s) Algebraic proof (’10s)

Kazhdan-Lusztig conjecture Brylinsky-Kashiwara Elias-W. 2014,

(multiplicity “ Py,xp1qq Beilinson-Bernstein following Soergel 1990

Jantzen conjecture Beilinson-Bernstein W. 2016 following

(graded multiplicity “ Py,xpvq) Soergel 2008, Kübel 2012

Geometric proofs: D-modules, perverse sheaves, weights…

Algebraic proofs: “shadows of Hodge theory”, i.e. invariant forms (“geometric structures”) still satisfying Poincaré duality, Hard Lefschetz, Hodge-Riemann 39 shadows of hodge theory beyond weyl groups

Weyl groups Ă Real reﬂection groups Ă Coxeter groups

41 Let W denote a real reﬂection group acting on hR. Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”).

the coinvariant ring

42 Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”).

the coinvariant ring

Let W denote a real reﬂection group acting on hR.

42 Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”).

the coinvariant ring

Let W denote a real reﬂection group acting on hR. Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

42 W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”).

the coinvariant ring

Let W denote a real reﬂection group acting on hR. Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

42 Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”).

the coinvariant ring

Let W denote a real reﬂection group acting on hR. Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

42 the coinvariant ring

Let W denote a real reﬂection group acting on hR. Example ˆ ˙ ˆ ˙ Sym ý R2 or Sym ý R3.

Let R denote the polynomial functions on hR. We view R as graded ˚ with hR in degree 2.

W Let R` denote the W-invariants of positive degree. Set W H :“ R{pR`q.

Remark If W is the Weyl group of a complex semi-simple Lie algebra g, then H is isomorphic to the cohomology of the ﬂag variety of g (the “Borel isomorphism”). 42 There exists a unique (up to scalar) bilinear form

x´, ´y : Hd´‚ ˆ Hd`‚ Ñ R

satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).

Remark x´, ´y is the analogue of the intersection form on cohomology.

the invariant form

W H :“ R{pR`q d :“ number of reﬂecting hyperplanes in hR “complex dimension of ﬂag variety”

43 Remark x´, ´y is the analogue of the intersection form on cohomology.

the invariant form

W H :“ R{pR`q d :“ number of reﬂecting hyperplanes in hR “complex dimension of ﬂag variety”

There exists a unique (up to scalar) bilinear form

x´, ´y : Hd´‚ ˆ Hd`‚ Ñ R

satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).

43 the invariant form

W H :“ R{pR`q d :“ number of reﬂecting hyperplanes in hR “complex dimension of ﬂag variety”

There exists a unique (up to scalar) bilinear form

x´, ´y : Hd´‚ ˆ Hd`‚ Ñ R

satisfying xγc, c1y “ xc, γc1y for all γ, c, c1 P H (the invariant form).

Remark x´, ´y is the analogue of the intersection form on cohomology.

43 (b) (Hodge-Riemann bilinear relations) The form pc, c1q “ xc, γic1y on Hd´i is p´1q?-deﬁnite on the kernel of γi`1.

H8 γ¨ Theorem (toy model) 6 For all γ P K and i ě 0: H (a) (Hard Lefschetz) Multiplication by γi induces an isomor- γ¨ phism Hd´i ÝÑ„ Hd`i „ „ H4 γ¨

H2 The theorem is identical to the hard Lefschetz theorem and Hodge-Riemann bilinear relations in complex algebraic geometry. γ¨ In the Weyl group case the theorem can be deduced from complex 0 algebraic geometry, but not in general. H

hodge theory for the coinvariant ring

˚ There exists an open cone K Ă hR (“Kähler cone”).

44 (b) (Hodge-Riemann bilinear relations) The form pc, c1q “ xc, γic1y on Hd´i is p´1q?-deﬁnite on the kernel of γi`1.

The theorem is identical to the hard Lefschetz theorem and Hodge-Riemann bilinear relations in complex algebraic geometry. In the Weyl group case the theorem can be deduced from complex algebraic geometry, but not in general.

hodge theory for the coinvariant ring

H2 γ¨

44 The theorem is identical to the hard Lefschetz theorem and Hodge-Riemann bilinear relations in complex algebraic geometry. In the Weyl group case the theorem can be deduced from complex algebraic geometry, but not in general.

hodge theory for the coinvariant ring

44 hodge theory for the coinvariant ring

H8 There exists an open cone K Ă h˚ (“Kähler cone”). R γ¨ Theorem (toy model) 6 For all γ P K and i ě 0: H (a) (Hard Lefschetz) Multiplication by γi induces an isomor- γ¨ phism Hdí ÝÑ„ Hdì „ „ H4 (b) (Hodge-Riemann bilinear relations) The form pc, c1q “ γ¨ xc, γic1y on Hdí is p´1q?-definite on the kernel of γi`1. H2 The theorem is identical to the hard Lefschetz theorem and Hodge-Riemann bilinear relations in complex algebraic geometry. γ¨ In the Weyl group case the theorem can be deduced from complex 0 algebraic geometry, but not in general. H

44 By appeal to the Decomposition Theorem (a deep theorem in algebraic geometry) he deduced the equality. In doing so he was

able to identify the Hw with the intersection cohomology of Schubert varieties. We provided an algebraic proof of (1) as a consequence of

Theorem (Elias-W.)

The hard Lefschetz and Hodge-Riemann relations hold for Hw.

connection to the kazhdan-lusztig conjecture

In 1990 Soergel deﬁned graded H-modules Hw for all w P W. Today they are known as “Soergel modules”. In the Weyl group case, he proved that the Kazhdan-Lusztig conjecture is equivalent to ÿ dimR Hw “ Py,wp1q. (1) yPW

45 We provided an algebraic proof of (1) as a consequence of

Theorem (Elias-W.)

The hard Lefschetz and Hodge-Riemann relations hold for Hw.

connection to the kazhdan-lusztig conjecture

By appeal to the Decomposition Theorem (a deep theorem in algebraic geometry) he deduced the equality. In doing so he was

able to identify the Hw with the intersection cohomology of Schubert varieties.

45 connection to the kazhdan-lusztig conjecture

By appeal to the Decomposition Theorem (a deep theorem in algebraic geometry) he deduced the equality. In doing so he was

able to identify the Hw with the intersection cohomology of Schubert varieties. We provided an algebraic proof of (1) as a consequence of

Theorem (Elias-W.)

The hard Lefschetz and Hodge-Riemann relations hold for Hw.

45 (a) The invariant form x´, ´y is unique up to scalar and satisﬁes the Hodge-Riemann relations (“uniqueness of geometric structure”). (b) The invariant form is our main tool in proving that Soergel modules decompose as they should (“semi-simplicity via introduction of geometric structure”).

Ideas of de Cataldo and Migliorini provide several useful clues. Diagrammatic algebra crucial to calculate and discover correct statements.

features of the proof

There is a resemblance to the semi-simple world:

46 (b) The invariant form is our main tool in proving that Soergel modules decompose as they should (“semi-simplicity via introduction of geometric structure”).

Ideas of de Cataldo and Migliorini provide several useful clues. Diagrammatic algebra crucial to calculate and discover correct statements.

features of the proof

There is a resemblance to the semi-simple world:

(a) The invariant form x´, ´y is unique up to scalar and satisﬁes the Hodge-Riemann relations (“uniqueness of geometric structure”).

46 Ideas of de Cataldo and Migliorini provide several useful clues. Diagrammatic algebra crucial to calculate and discover correct statements.

features of the proof

There is a resemblance to the semi-simple world:

(a) The invariant form x´, ´y is unique up to scalar and satisﬁes the Hodge-Riemann relations (“uniqueness of geometric structure”). (b) The invariant form is our main tool in proving that Soergel modules decompose as they should (“semi-simplicity via introduction of geometric structure”).

46 Diagrammatic algebra crucial to calculate and discover correct statements.

features of the proof

There is a resemblance to the semi-simple world:

Ideas of de Cataldo and Migliorini provide several useful clues.

46 features of the proof

There is a resemblance to the semi-simple world:

Ideas of de Cataldo and Migliorini provide several useful clues. Diagrammatic algebra crucial to calculate and discover correct statements.

46 Corollary (Elias-W. 2013)

The Kazhdan-Lusztig positivity conjecture holds.

A mystery for the 21st century?

Similar structures arise in the theory of non-rational polytopes (due to McMullen, Braden-Lunts, Karu, …) and in recent work of Apridisato-Huh-Katz on matroids. Why?

kazhdan-lusztig positivity

Kazhdan-Lusztig polynomials are deﬁned for any pair of elements in a Coxeter group. The Kazhdan-Luszig positivity conjecture (1979) is the statement that their coefﬁcients are always non-negative.

47 A mystery for the 21st century?

Similar structures arise in the theory of non-rational polytopes (due to McMullen, Braden-Lunts, Karu, …) and in recent work of Apridisato-Huh-Katz on matroids. Why?

kazhdan-lusztig positivity

Corollary (Elias-W. 2013)

The Kazhdan-Lusztig positivity conjecture holds.

47 kazhdan-lusztig positivity

Corollary (Elias-W. 2013)

The Kazhdan-Lusztig positivity conjecture holds.

A mystery for the 21st century?

Similar structures arise in the theory of non-rational polytopes (due to McMullen, Braden-Lunts, Karu, …) and in recent work of Apridisato-Huh-Katz on matroids. Why?

47 modular representations The approach of Soergel is also fruitful for studying modular representations. A major source of difﬁculty is that signature no longer makes sense, and Kazhdan-Lusztig like formulas do not always hold (such questions are tied to deciding when Lusztig’s conjecture holds). Invariant forms (now deﬁned over the integers) still play a decisive role in the theory.

modular representations

49 Invariant forms (now deﬁned over the integers) still play a decisive role in the theory.

modular representations

There are analogies between inﬁnite-dimensional representations of Lie algebras, and modular representations of algebraic groups. The analogue of the Kazhdan-Lusztig conjecture in this setting is the Lusztig conjecture (1980). The approach of Soergel is also fruitful for studying modular representations. A major source of difﬁculty is that signature no longer makes sense, and Kazhdan-Lusztig like formulas do not always hold (such questions are tied to deciding when Lusztig’s conjecture holds).

49 modular representations

49 (a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraic groups in characteristic p. (b) Weyl group ⇝ afﬁne Weyl group. (c) True for large p depending on root system (Kashiwara-Tanisaki, Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g. 100 true for p ą 10 for SL8. (d) False for primes growing exponentially in the rank (W. 2014,

following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.

lusztig conjecture

Lusztig conjecture (1980)

ÿ ˆ ˆ r∆As “ qA,Bp1qrLBs B

50 (b) Weyl group ⇝ afﬁne Weyl group. (c) True for large p depending on root system (Kashiwara-Tanisaki, Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g. 100 true for p ą 10 for SL8. (d) False for primes growing exponentially in the rank (W. 2014,

following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.

lusztig conjecture

Lusztig conjecture (1980)

ÿ ˆ ˆ r∆As “ qA,Bp1qrLBs B

(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraic groups in characteristic p.

50 (c) True for large p depending on root system (Kashiwara-Tanisaki, Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g. 100 true for p ą 10 for SL8. (d) False for primes growing exponentially in the rank (W. 2014,

following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.

lusztig conjecture

Lusztig conjecture (1980)

ÿ ˆ ˆ r∆As “ qA,Bp1qrLBs B

(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraic groups in characteristic p. (b) Weyl group ⇝ afﬁne Weyl group.

50 (d) False for primes growing exponentially in the rank (W. 2014,

following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.

lusztig conjecture

Lusztig conjecture (1980)

ÿ ˆ ˆ r∆As “ qA,Bp1qrLBs B

(a) Analogue of Kazhdan-Lusztig conjecture for reductive algebraic groups in characteristic p. (b) Weyl group ⇝ afﬁne Weyl group. (c) True for large p depending on root system (Kashiwara-Tanisaki, Kazhdan-Lusztig, Lusztig, Andersen-Jantzen-Soergel, Fiebig), e.g. 100 true for p ą 10 for SL8.

50 lusztig conjecture

Lusztig conjecture (1980)

ÿ ˆ ˆ r∆As “ qA,Bp1qrLBs B

following He-W. 2013), e.g. false for p “ 470 858 183 for SL100.

50 p Leads to p-Kazhdan-Lusztig polynomails qA,B.

Riche-W. (2018)

ÿ ˆ p ˆ r∆As “ qA,Bp1qrLBs B

Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.

p qA,B are computable via diagrammatic algebra + computer.

new character formula

Intersection cohomology sheaves ⇝ parity sheaves (Soergel, Juteau-Mautner-W.).

51 Riche-W. (2018)

ÿ ˆ p ˆ r∆As “ qA,Bp1qrLBs B

Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.

p qA,B are computable via diagrammatic algebra + computer.

new character formula

Intersection cohomology sheaves ⇝ parity sheaves (Soergel, Juteau-Mautner-W.).

p Leads to p-Kazhdan-Lusztig polynomails qA,B.

51 Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.

p qA,B are computable via diagrammatic algebra + computer.

new character formula

Intersection cohomology sheaves ⇝ parity sheaves (Soergel, Juteau-Mautner-W.).

p Leads to p-Kazhdan-Lusztig polynomails qA,B.

Riche-W. (2018)

ÿ ˆ p ˆ r∆As “ qA,Bp1qrLBs B

51 p qA,B are computable via diagrammatic algebra + computer.

new character formula

Intersection cohomology sheaves ⇝ parity sheaves (Soergel, Juteau-Mautner-W.).

p Leads to p-Kazhdan-Lusztig polynomails qA,B.

Riche-W. (2018)

ÿ ˆ p ˆ r∆As “ qA,Bp1qrLBs B

Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.

51 new character formula

Intersection cohomology sheaves ⇝ parity sheaves (Soergel, Juteau-Mautner-W.).

p Leads to p-Kazhdan-Lusztig polynomails qA,B.

Riche-W. (2018)

ÿ ˆ p ˆ r∆As “ qA,Bp1qrLBs B

Based on works of Achar-Makisumi-Riche-W. and Achar-Riche.

p qA,B are computable via diagrammatic algebra + computer.

51 V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

Basic problem

Determine multiplicities of simple modules in Fp bZ V.

The answer is only known for partitions with one or two rows! The following video illustrates the “billiards conjecture” (Lusztig-W. 2017), which predicts many new cases of this decomposition behaviour for partitions with three rows. The conjecture predicts that these numbers are given by a “discrete dynamical system”... Billiards and tilting characters: https://www.youtube.com/watch?v=Ru0Zys1Vvq4

billiards conjecture

52 Basic problem

Determine multiplicities of simple modules in Fp bZ V.

billiards conjecture

V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

52 The answer is only known for partitions with one or two rows! The following video illustrates the “billiards conjecture” (Lusztig-W. 2017), which predicts many new cases of this decomposition behaviour for partitions with three rows. The conjecture predicts that these numbers are given by a “discrete dynamical system”... Billiards and tilting characters: https://www.youtube.com/watch?v=Ru0Zys1Vvq4

billiards conjecture

V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

Basic problem

Determine multiplicities of simple modules in Fp bZ V.

52 The following video illustrates the “billiards conjecture” (Lusztig-W. 2017), which predicts many new cases of this decomposition behaviour for partitions with three rows. The conjecture predicts that these numbers are given by a “discrete dynamical system”... Billiards and tilting characters: https://www.youtube.com/watch?v=Ru0Zys1Vvq4

billiards conjecture

V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

Basic problem

Determine multiplicities of simple modules in Fp bZ V.

The answer is only known for partitions with one or two rows!

52 The conjecture predicts that these numbers are given by a “discrete dynamical system”... Billiards and tilting characters: https://www.youtube.com/watch?v=Ru0Zys1Vvq4

billiards conjecture

V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

Basic problem

Determine multiplicities of simple modules in Fp bZ V.

52 billiards conjecture

V a simple representation of Sn over Q.

⇝ reduce modulo p to get modular representation Fp bZ V.

Basic problem

Determine multiplicities of simple modules in Fp bZ V.