Categorifying q-Fock space
via graded cyclotomic q-Schur algebras
Ben Webster
U. of Oregon

November 20, 2010

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

1 / 44

Fock space

One of the most wonderful objects of representation theory is the Fock spaceof charge c. This is a vector space spanned by a basis sλ where λ ranges overall partitions of all sizes.
Thought of bosonically, this is symmetric polynomials, and sλ is the
Schur function.
Thought of fermionically, this is a semi-infinite wedge space, andsλ = xλ1 +c ∧ xλ2 −1+c ∧ xλ3 −2+c ∧ · · ·Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

2 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 1
0 1 2
2
1
0

n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 1
0 1 2
2 0
1

n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2 02
1

n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 1 2
0 1 2
2
1

n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

b n action
The slFix once and for all a positive integer n. Given a partition λ, we fill its boxes(i, j) with their content mod n which is i − j + c ∈ Z/nZ.
1 2 0 10 1 2
2
1n = 3, c = 1

b n action on Fock space, where we identify the Dynkin
There is a natural slbdiagram of sln with Z/nZ so that i and i + 1 are adjacent.
The action of Fi sums over adding a box of content i in all possible ways;
The action of Ei sums over removing a box of content i in all possible
ways.
These are obviously adjoint for the inner product where the sλ are
orthonormal.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

3 / 44

The Heisenberg actionb n . In fact, there
However, the Fock space is not irreducible as a module over sl
is an infinite-dimensional algebra of endomorphisms commuting with it.
Let pk denote the operator of multiplying by the degree k power sum function(in the bosonic realization) if k ≥ 0, and its adjoint if k < 0.
Theorem
The operators pnk define an action of the Heisenberg Lie algebra H on Fockb n and acts irreducibly on the space of highest
space which commutes with slweight vectors.
b n × H as glb n , so this says that Fock space is irreducible
One can think of slbover gln .

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

4 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations
There are several directions one can take this picture:
You can deform and obtain q-Fock space.
You can categorify and obtain q-Schur algebras.
Oddly enough, it was known for many years how to do both of theseseparately before we knew how to do both at once.
q-Fock space (’90)

Fock space (’32)

graded q-Schur algebras (’09)

q-Schur algebras (’89)

Since graded q-Schur algebras are so new, there are a lot of unresolved
questions about them:
relation to Rouquier-type categorification?
is there a natural graded presentation? a homogenous basis?
what about generalizations to higher level?
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

5 / 44

Generalizations

My aim in this talk to answer (parts of) these questions.
Main idea: Develop graded Rouquier-style categorification of q-Fock spacefrom first principles, and than check isomorphism with q-Schur algebra.
So, which principles would those be?

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

6 / 44

Categorification
It is an old observation that some numbers are really sets in disguise, andsome sets are categories in disguise. Of course, this added structure is achoice, but we know of oodles of instances where it “feels right.”
What has taken a little longer to develop is the linearized version of the same
story.
Sometimes, a number doesn’t seem to be the size of any particular set,
but is the dimension of a vector space.
Abelian groups can be gotten as the Grothendieck group of a category
with some notion of exact sequence.
Some very important and popular abelian groups, the semi-simple Liealgebras and their representations, managed to be the Grothendieck groups ofcategories for 100 years without anyone noticing.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

7 / 44

Categorification
It is an old observation that some numbers are really sets in disguise, andsome sets are categories in disguise. Of course, this added structure is achoice, but we know of oodles of instances where it “feels right.”
What has taken a little longer to develop is the linearized version of the same
story.
Sometimes, a number doesn’t seem to be the size of any particular set,
but is the dimension of a vector space.
Abelian groups can be gotten as the Grothendieck group of a category
with some notion of exact sequence.
Some very important and popular abelian groups, the semi-simple Liealgebras and their representations, managed to be the Grothendieck groups ofcategories for 100 years without anyone noticing.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

7 / 44

Categorification
It is an old observation that some numbers are really sets in disguise, andsome sets are categories in disguise. Of course, this added structure is achoice, but we know of oodles of instances where it “feels right.”
What has taken a little longer to develop is the linearized version of the same
story.
Sometimes, a number doesn’t seem to be the size of any particular set,
but is the dimension of a vector space.
Abelian groups can be gotten as the Grothendieck group of a category
with some notion of exact sequence.
Some very important and popular abelian groups, the semi-simple Liealgebras and their representations, managed to be the Grothendieck groups ofcategories for 100 years without anyone noticing.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

7 / 44

The Grothendieck group
The Grothendieck group (GG) of a category C with a distinguishedcollection of triples (A, B, C) ∈ Ob(C)3 is the abelian group generated bysymbols [A] for A ∈ Ob(C), modulo the relations[A] + [C] = [B].

The most popular choice is for C to be modules over some ring, withshort exact sequences 0 → A → B → C → 0.
Another option would be projective modules over a ring; now all short
exact sequences are split, so the triples are just (P1 , P1 ⊕ P2 , P2 ).
More exciting would be a derived category or homotopy category ofmodules. In this case, triples are objects with representatives that fit in anexact sequence.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

8 / 44

The Grothendieck group
The Grothendieck group (GG) of a category C with a distinguishedcollection of triples (A, B, C) ∈ Ob(C)3 is the abelian group generated bysymbols [A] for A ∈ Ob(C), modulo the relations[A] + [C] = [B].

The most popular choice is for C to be modules over some ring, withshort exact sequences 0 → A → B → C → 0.
Another option would be projective modules over a ring; now all short
exact sequences are split, so the triples are just (P1 , P1 ⊕ P2 , P2 ).
More exciting would be a derived category or homotopy category ofmodules. In this case, triples are objects with representatives that fit in anexact sequence.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

8 / 44

Universal enveloping algebras

Let g be your favorite symmetrizable Kac-Moody Lie algebra with Cartanb n ), root lattice Y andmatrix C (for today, you should probably choose sl
weight lattice X.
This Lie algebra has a presentation of the form
[Hj , Fi ] = −cij Fi[Hj , Ei ] = cij Ei[Ei , Fj ] = δji Hi

c +1

adEijjc +1

Ei = adFijj

Fi = 0

By definition, the universal enveloping algebra is the associative algebra
generated by these symbols subject to the relations above (where [−, −]means commutator).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

9 / 44

Universal enveloping algebras

Let g be your favorite symmetrizable Kac-Moody Lie algebra with Cartanb n ), root lattice Y andmatrix C (for today, you should probably choose sl
weight lattice X.
This Lie algebra has a presentation of the form
[Hj , Fi ] = −cij Fi[Hj , Ei ] = cij Ei[Ei , Fj ] = δji Hi

c +1

adEijjc +1

Ei = adFijj

Fi = 0

By definition, the universal enveloping algebra is the associative algebra
generated by these symbols subject to the relations above (where [−, −]means commutator).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

9 / 44

Universal enveloping algebras

Let g be your favorite symmetrizable Kac-Moody Lie algebra with Cartanb n ), root lattice Y andmatrix C (for today, you should probably choose sl
weight lattice X.
This Lie algebra has a presentation of the form
[Hj , Fi ] = −cij Fi[Hj , Ei ] = cij Ei[Ei , Fj ] = δji Hi

c +1

adEijjc +1

Ei = adFijj

Fi = 0

By definition, the universal enveloping algebra is the associative algebra
generated by these symbols subject to the relations above (where [−, −]means commutator).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

9 / 44

Universal enveloping algebras
We actually want a slightly bigger algebra U̇, with some extra idempotents 1λfor λ ∈ X. These satisfy the relations

1λ 1λ0 = δλλ0 1λNote that

Hi 1λ = 1λ Hi = λi 1λ .

1λ Ei = Ei 1λ−αi

1λ Fi = Fi 1λ+αi .

We can represent elements of this as pictures on a line

1λ Ei Fj Ej 1λ−αi =i

Ben Webster (U. of Oregon)λ − αi + αj

λ − αiλCategorifying q-Fock spacej

λ − αijNovember 20, 2010

10 / 44

Universal enveloping algebras
We actually want a slightly bigger algebra U̇, with some extra idempotents 1λfor λ ∈ X. These satisfy the relations

1λ 1λ0 = δλλ0 1λNote that

Hi 1λ = 1λ Hi = λi 1λ .

1λ Ei = Ei 1λ−αi

1λ Fi = Fi 1λ+αi .

We can represent elements of this as pictures on a line

1λ Ei Fj Ej 1λ−αi =i

Ben Webster (U. of Oregon)λ − αi + αj

λ − αiλCategorifying q-Fock spacej

λ − αijNovember 20, 2010

10 / 44Categorifying UEAs
The remarkable insight of Khovanov and Lauda was that one could makethese into the objects of a category U, with morphisms given by pictures in
the plane (Chuang and Rouquier had the same idea first, but never drew thepictures).
The morphisms of U are given by oriented 1-manifolds decorated with dots
and labeled with elements of [1, n − 1], whose boundaries are the given
objects (with orientations and labels), modulo certain relations.
λ − αjλλ

Ben Webster (U. of Oregon)iλ − αiλ − αj − αi

λ − αiλ − αi + αjjCategorifying q-Fock spacej

λ − αiNovember 20, 201011 / 44Categorifying UEAs
The remarkable insight of Khovanov and Lauda was that one could makethese into the objects of a category U, with morphisms given by pictures in
the plane (Chuang and Rouquier had the same idea first, but never drew thepictures).
The morphisms of U are given by oriented 1-manifolds decorated with dots
and labeled with elements of [1, n − 1], whose boundaries are the given
objects (with orientations and labels), modulo certain relations.
λ − αjλλ

Ben Webster (U. of Oregon)iλ − αiλ − αj − αi

λ − αiλ − αi + αjjCategorifying q-Fock spacej

λ − αiNovember 20, 201011 / 44

Relations in U (simply-laced case)
=

λ

b

X

a+b=α∨
i (λ)−1

Qij (u, v) = Qji (v, u) =(1
i 6= j ± 1
au + bv i = j + 1

λ
=

i

λ

=

+

λ

j−k

k

λ

=

(
1
0

j

b λ

a+b+c=α∨i (λ)−1

k

i

j

c

X

a

=

i

X

unless i = j+

ij = αi∨ (λ) − 1
j 6= αi∨ (λ) − 1

=

i

i

i

=

and

0

i

i

i

=

j

j

Ben Webster (U. of Oregon)

Qij (y1 , y2 )ijunless i = k 6= j

i

k

j

k

λ
=

i

i

j

=

λ

i

i

j

i

Categorifying q-Fock space

j

i

+

i

j

i

Qij (y3 , y2 ) − Qij (y1 , y2 )y3 − y1iNovember 20, 2010ji

12 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.

The relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.
Ei Fj = Fj Ei

λ

=

i

λ

j

i

j

The relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.

1λ Fi Ei = 1λ Ei Fi + 1λ λi
λ

=

λ

+

X

a+b+c=λi −1(λi > 0)cb λaThe relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.

1λ Ei Fi = 1λ Fi Ei − 1λ λi
λ

=

λ

+

X

a+b+c=−λi −1(λi < 0)cb λaThe relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.

(2)

Fi Fi+1 Fi = Fi+1 Fi
a·(2)+ Fi Fi+1

=

i

i+1

i

−

i

i+1

i

i

i+1

i

The relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44

Relations in U
These relations are all a bit overwhelming at first, but actually they match up
surprisingly well with the relations of the universal enveloping algebra. Of
course, this correspondence is a bit subtle.
The thing to look for is writing the identity element of any object as maps
factoring through another; this is how we find direct sum decompositions.

(2)

Fi Fi+1 Fi = Fi+1 Fi
a·(2)+ Fi Fi+1

=

i

i+1

i

−

i

i+1

i

i

i+1

i

The relations may look ad hoc, but actually every single one of them can beguessed by looking at the geometry of quiver varieties.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201013 / 44The category U

We let U be the idempotent completion of the category whose
objects are diagrams on a line shown above andmorphisms are the pictures in the plane, modulo the relations of theprevious slide.
Idempotent completion means adding a new object for each idempotent
which is the image of that idempotent as a projection.
Put another way, if we fix the weights at extreme left and right, then we get analgebra spanned by diagrams modulo these relations λ Uµ . We look at thecategories of projective modules over these.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201014 / 44The category U

We let U be the idempotent completion of the category whose
objects are diagrams on a line shown above andmorphisms are the pictures in the plane, modulo the relations of theprevious slide.
Idempotent completion means adding a new object for each idempotent
which is the image of that idempotent as a projection.
Put another way, if we fix the weights at extreme left and right, then we get analgebra spanned by diagrams modulo these relations λ Uµ . We look at thecategories of projective modules over these.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201014 / 44The monoidal structure
The category U is monoidal; it has a tensor product. Visually, it’s quite
simple. You just put diagrams next to each other if the label at the edges
match, and get 0 if they don’t.
λ1

A

λ2

⊗

µ1

B

µ2

=

λ1

A

λ2 = µ1Bµ2

Put another way, there is a 2-category where
objects are weights λ ∈ X, andmorphisms from λ to µ are the category of projective modules over λ Uµ .
The multiplication of morphisms is given by extension of scalars by the map
λ Uµ ⊗ µ Uν → λ Uν shown above.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201015 / 44The monoidal structure
The category U is monoidal; it has a tensor product. Visually, it’s quite
simple. You just put diagrams next to each other if the label at the edges
match, and get 0 if they don’t.
λ1

A

λ2

⊗

µ1

B

µ2

=

λ1

A

λ2 = µ1Bµ2

Put another way, there is a 2-category where
objects are weights λ ∈ X, andmorphisms from λ to µ are the category of projective modules over λ Uµ .
The multiplication of morphisms is given by extension of scalars by the map
λ Uµ ⊗ µ Uν → λ Uν shown above.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201015 / 44The Grothendieck group

Let Eλi =λ − αiλλ + αiλand Fiλ =iiTheorem (Khovanov-Lauda, W.)
The GG of U is U̇, via the isomorphism [Eλi ] 7→ 1λ Ei , [Fiλ ] 7→ 1λ Fi .
For example,
λ − αi + αj

λ − αihλijλ − αii7→ 1λ Ei Fj Ej 1λ−αijNote: I never imposed any of the relations of U̇! They all follow(non-obviously) from the relations in U.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201016 / 44The Grothendieck group

Let Eλi =λ − αiλλ + αiλand Fiλ =iiTheorem (Khovanov-Lauda, W.)
The GG of U is U̇, via the isomorphism [Eλi ] 7→ 1λ Ei , [Fiλ ] 7→ 1λ Fi .
For example,
λ − αi + αj

λ − αihλijλ − αii7→ 1λ Ei Fj Ej 1λ−αijNote: I never imposed any of the relations of U̇! They all follow(non-obviously) from the relations in U.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201016 / 44Grading
That probably all went by a little fast, but actually the relations I showed you
are homogeneous for a particular grading; the category U actually has agraded version Ũ.
= −hαi , αj i

deg
j

i

= hαi , αi i.

degi

Theorem (Khovanov-Lauda,W.)
The GG of Ũ is U̇q (g), the quantized universal enveloping corresponding to g.
As a general rule, it’s never harder to work with quantum groups in thispicture (sometimes, it even makes things easier); you just pay attention to thegrading.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201017 / 44Grading
That probably all went by a little fast, but actually the relations I showed you
are homogeneous for a particular grading; the category U actually has agraded version Ũ.
= −hαi , αj i

deg
j

i

= hαi , αi i.

degi

Theorem (Khovanov-Lauda,W.)
The GG of Ũ is U̇q (g), the quantized universal enveloping corresponding to g.
As a general rule, it’s never harder to work with quantum groups in thispicture (sometimes, it even makes things easier); you just pay attention to thegrading.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201017 / 44b n?
glThis is, of course, a beautiful story, and I’ll say a lot more about it before thistalk is over. But for our purposes, it’s not very satisfying, since it usesb n doesn’t.
extremely strongly that g has a Chevalley presentation, which gl
However, this is really a sign that our principles aren’t “first” enough yet.
Underlying all of this picture is geometry, which we haven’t delved into yet.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201018 / 44

Quiver varieties
Assume from now on that our Cartan matrix is symmetric, and let Γ be an
b n , of course, we can chooseorientation of the Dynkin graph of g. For sl
0

1

···

···

n

Choose a dimension vector d : V(Γ) → Z. LetM
Ed =Hom(Cdi , Cdj )i→jThis is the universal d-dimensionalof Γ since its points are
Lrepresentationdiexactly the representations of Γ on
C .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201019 / 44

Quiver varieties
Assume from now on that our Cartan matrix is symmetric, and let Γ be an
b n , of course, we can chooseorientation of the Dynkin graph of g. For sl
0

1

···

···

n

Choose a dimension vector d : V(Γ) → Z. LetM
Ed =Hom(Cdi , Cdj )i→jThis is the universal d-dimensionalof Γ since its points are
Lrepresentationdiexactly the representations of Γ on
C .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201019 / 44

Quiver flag varieties
Choose a sequence of dimension vectors D = (d1 , · · · , dn ) with d =and let FD denote the space of flags of subspaces

P

dj ,

· · · ⊂ Vij ⊂ Vij+1 ⊂ · · · ⊂ Cdi
jjsuch that dim Vij+1
Q /Vi = di . This is a parabolic homogeneous space for theaction of Gd = i∈Γ GLdi .

Definition
The quiver flag variety for the sequence QD is the subvariety{(f , V∗∗ ) ∈ Ed × FD |V∗j is a subrepresentation
and V∗j /V∗j−1 is a trivial representation for all j.}
This variety is Gd -invariant, and thus inherits a Gd -action.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

20 / 44

Quiver flag varieties
Choose a sequence of dimension vectors D = (d1 , · · · , dn ) with d =and let FD denote the space of flags of subspaces

P

dj ,

· · · ⊂ Vij ⊂ Vij+1 ⊂ · · · ⊂ Cdi
jjsuch that dim Vij+1
Q /Vi = di . This is a parabolic homogeneous space for theaction of Gd = i∈Γ GLdi .

Definition
The quiver flag variety for the sequence QD is the subvariety{(f , V∗∗ ) ∈ Ed × FD |V∗j is a subrepresentation
and V∗j /V∗j−1 is a trivial representation for all j.}
This variety is Gd -invariant, and thus inherits a Gd -action.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

20 / 44

Quiver flag varieties
There are natural projection maps
pD

QD

πD
FD

Ed

The map πD is an affine bundle; its fiber is the space of quiver
representations preserving the given flag.
The map pD is proper, and in some circumstances is a resolution of
singularities of a subvariety of Ed . The image this map necessarily lies inthe space of nilpotent quiver representations.
In finite or affine type, the nilpotent representations have finitely many
Gd -orbits, and the closure each of these has a resolution of singularities of theform QD , given by choosing the dimension vectors of the radical (or socle)
filtration.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

21 / 44

Quiver flag varieties
There are natural projection maps
pD

QD

πD
FD

Ed

The map πD is an affine bundle; its fiber is the space of quiver
representations preserving the given flag.
The map pD is proper, and in some circumstances is a resolution of
singularities of a subvariety of Ed . The image this map necessarily lies inthe space of nilpotent quiver representations.
In finite or affine type, the nilpotent representations have finitely many
Gd -orbits, and the closure each of these has a resolution of singularities of theform QD , given by choosing the dimension vectors of the radical (or socle)
filtration.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

21 / 44

Quiver flag varieties
There are natural projection maps
pD

QD

πD
FD

Ed

The map πD is an affine bundle; its fiber is the space of quiver
representations preserving the given flag.
The map pD is proper, and in some circumstances is a resolution of
singularities of a subvariety of Ed . The image this map necessarily lies inthe space of nilpotent quiver representations.
In finite or affine type, the nilpotent representations have finitely many
Gd -orbits, and the closure each of these has a resolution of singularities of theform QD , given by choosing the dimension vectors of the radical (or socle)
filtration.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

21 / 44

Quiver flag varieties
There are natural projection maps
pD

QD

πD
FD

Ed

The map πD is an affine bundle; its fiber is the space of quiver
representations preserving the given flag.
The map pD is proper, and in some circumstances is a resolution of
singularities of a subvariety of Ed . The image this map necessarily lies inthe space of nilpotent quiver representations.
In finite or affine type, the nilpotent representations have finitely many
Gd -orbits, and the closure each of these has a resolution of singularities of theform QD , given by choosing the dimension vectors of the radical (or socle)
filtration.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

21 / 44

Convolution and Borel-Moore homology
Faced with a situation like this, it’s quite popular to study the equivariant
Borel-Moore homology of the fiber products QD ×Ed QD0 .
Theorem
The vector space
AS ∼
=

M

H∗BM,Gd (QD ×Ed QD0 ; C)

D,D0 ∈S

for any set S of sequences is an algebra under convolution.
This algebra is isomorphic to
Ext•D(Ed /Gd )M(pD )∗ CQD

D∈S

where D(Ed /Gd ) denotes the equivariant derived category.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201022 / 44Borel-Moore homology and categorification

Given two sequences C = (c1 , . . . , cm ) and D = (d1 , . . . , dn ) , let C ∪ D be
their concatenation.
Definition
Let QC;D ⊂ QC∪D be the subvariety where the mth subspace (the last from C)
is the standard coordinate subspace.
Pull-push on the diagram
QC × QD ←− QC;D −→ QC∪D
defines a map AS ⊗ AS → AS for any set S closed under concatenation. We
call this map horizontal multiplication.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201023 / 44Borel-Moore homology and categorification

Given two sequences C = (c1 , . . . , cm ) and D = (d1 , . . . , dn ) , let C ∪ D be
their concatenation.
Definition
Let QC;D ⊂ QC∪D be the subvariety where the mth subspace (the last from C)
is the standard coordinate subspace.
Pull-push on the diagram
QC × QD ←− QC;D −→ QC∪D
defines a map AS ⊗ AS → AS for any set S closed under concatenation. We
call this map horizontal multiplication.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201023 / 44Borel-Moore homology and categorification
L
The object D∈S (pD )∗ CQD is semi-simple by the decomposition theorem,
and thus its simple summands are the intersection cohomology sheaves ofsome set of orbits of nilpotent orbits, and these sheaves categorify some pieceof the Hall algebra of nilpotent quiver representations.
Theorem (Lusztig)
If S = L is the set of sequences where every dj is a unit vector, AL
categorifies U̇(b), with product structure induced by horizontal
multiplication.
If S = M is all sequences, then the result is the same in finite type, but forb n , AM is a categorification of U̇(b) ⊗ Λ, an algebra whose twistedslb n ).
Drinfeld double is U̇(glSince Fock space is cyclic for the action of U̇(b) ⊗ Λ, we should expect ourcategorification of Fock space to be “cyclic” over the monoidal category ofmodules over AM .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201024 / 44Borel-Moore homology and categorification
L
The object D∈S (pD )∗ CQD is semi-simple by the decomposition theorem,
and thus its simple summands are the intersection cohomology sheaves ofsome set of orbits of nilpotent orbits, and these sheaves categorify some pieceof the Hall algebra of nilpotent quiver representations.
Theorem (Lusztig)
If S = L is the set of sequences where every dj is a unit vector, AL
categorifies U̇(b), with product structure induced by horizontal
multiplication.
If S = M is all sequences, then the result is the same in finite type, but forb n , AM is a categorification of U̇(b) ⊗ Λ, an algebra whose twistedslb n ).
Drinfeld double is U̇(glSince Fock space is cyclic for the action of U̇(b) ⊗ Λ, we should expect ourcategorification of Fock space to be “cyclic” over the monoidal category ofmodules over AM .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201024 / 44Borel-Moore homology and categorification
L
The object D∈S (pD )∗ CQD is semi-simple by the decomposition theorem,
and thus its simple summands are the intersection cohomology sheaves ofsome set of orbits of nilpotent orbits, and these sheaves categorify some pieceof the Hall algebra of nilpotent quiver representations.
Theorem (Lusztig)
If S = L is the set of sequences where every dj is a unit vector, AL
categorifies U̇(b), with product structure induced by horizontal
multiplication.
If S = M is all sequences, then the result is the same in finite type, but forb n , AM is a categorification of U̇(b) ⊗ Λ, an algebra whose twistedslb n ).
Drinfeld double is U̇(glSince Fock space is cyclic for the action of U̇(b) ⊗ Λ, we should expect ourcategorification of Fock space to be “cyclic” over the monoidal category ofmodules over AM .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201024 / 44Borel-Moore homology and categorification
L
The object D∈S (pD )∗ CQD is semi-simple by the decomposition theorem,
and thus its simple summands are the intersection cohomology sheaves ofsome set of orbits of nilpotent orbits, and these sheaves categorify some pieceof the Hall algebra of nilpotent quiver representations.
Theorem (Lusztig)
If S = L is the set of sequences where every dj is a unit vector, AL
categorifies U̇(b), with product structure induced by horizontal
multiplication.
If S = M is all sequences, then the result is the same in finite type, but forb n , AM is a categorification of U̇(b) ⊗ Λ, an algebra whose twistedslb n ).
Drinfeld double is U̇(glSince Fock space is cyclic for the action of U̇(b) ⊗ Λ, we should expect ourcategorification of Fock space to be “cyclic” over the monoidal category ofmodules over AM .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201024 / 44

Algebraic and geometric categorification

Now, you’ll note, I’ve now talked about categorifications arising from algebra(the pictorial categorification) and from geometry (Lusztig’s perverse
sheaves).
Question
How are these related? Does one really get two different pictures?

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201025 / 44

Algebraic and geometric categorification
If each dj is a unit vector, we can think of D as a sequence i = (i1 , . . . , in ) of
nodes. Associated to this sequence, we have an element Fi1 · · · Fin ∈ U(g)and its corresponding categorification Fi .
Theorem (Vasserot-Varagnolo/Rouquier)
We have an isomorphism H BM,Gd (Qi ×Ed Qi0 ) ∼= HomU − (Fi , Fi0 ) compatible
with horizontal and vertical multiplication. This isomorphism sends
a dot on the jth strand to the Euler class of the line bundle V j /V j−1 and⊂⊂

Ben Webster (U. of Oregon)⊂⊂a crossing on the kth and k + 1st strands to the fundamental class of
Qi,k , the space of flags with V k+1 /V k−1 trivial, the top of type i, and thebottom of type (k, k + 1) · i of the formVk
· · · ⊂ V k−2 ⊂ V k−1
V k+1 ⊂ V k+2 ⊂ · · ·0kVCategorifying q-Fock space

November 20, 2010

26 / 44bn
The case of glP
Thus AL (d) is actually the algebra Khovanov and Lauda call R( di αi ).
So, what about affine type A? Actually, it’s easier to come up with generatorsof AM , since we can use any flags we want. There are two essential types: thesplit and the join, which we denote pictorially by
c

c+d

d

c

c+d

d

These are given by pull-push on the diagram
Q(c,d) ←− Fc,d −→ Qc+d = pt.
Horizontal and vertical compositions of these are written exactly as the namesuggests.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201027 / 44bn
The case of glP
Thus AL (d) is actually the algebra Khovanov and Lauda call R( di αi ).
So, what about affine type A? Actually, it’s easier to come up with generatorsof AM , since we can use any flags we want. There are two essential types: thesplit and the join, which we denote pictorially by
c

c+d

d

c

c+d

d

These are given by pull-push on the diagram
Q(c,d) ←− Fc,d −→ Qc+d = pt.
Horizontal and vertical compositions of these are written exactly as the namesuggests.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201027 / 44bn
The case of gl
Of course, we have to also include multiplication by classes in
H∗BM,Gd (QD ) ∼= H∗BM,Gd (FD ) ∼= HP∗D (pt)where PD is the stabilizer of a flag of type D.
This ring is a polynomial ring freely generated by the Chern classes of
Vij /Vij−1 for all i, j.
Theorem (Stroppel-W.)
Under horizontal and vertical composition, the split, join and Chern classes
generate AM . In fact, we can give an explicit basis of AM using these
diagrams.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201028 / 44bn
The case of gl
Of course, we have to also include multiplication by classes in
H∗BM,Gd (QD ) ∼= H∗BM,Gd (FD ) ∼= HP∗D (pt)where PD is the stabilizer of a flag of type D.
This ring is a polynomial ring freely generated by the Chern classes of
Vij /Vij−1 for all i, j.
Theorem (Stroppel-W.)
Under horizontal and vertical composition, the split, join and Chern classes
generate AM . In fact, we can give an explicit basis of AM using these
diagrams.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201028 / 44

Relations
Now, you’ll note, I haven’t written down relations; that’s because I can’t.
In principle one can; we have a basis of the algebra, and an algorithm forwriting a product of basis vectors in terms of this basis, using localization inequivariant cohomology.
Thus, an arbitrarily patient person can work out any particular case, but no“closed form” is known.
QuestionIs there a simple set of relations for AM in terms of these generators?
Given the success (and simplicity) of Khovanov and Lauda’s relations, there’s
reason to be optimistic, but certainly it’s a ticklish problem.
Remarkably, for our purposes, this point won’t matter much.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201029 / 44

Relations
Now, you’ll note, I haven’t written down relations; that’s because I can’t.
In principle one can; we have a basis of the algebra, and an algorithm forwriting a product of basis vectors in terms of this basis, using localization inequivariant cohomology.
Thus, an arbitrarily patient person can work out any particular case, but no“closed form” is known.
QuestionIs there a simple set of relations for AM in terms of these generators?
Given the success (and simplicity) of Khovanov and Lauda’s relations, there’s
reason to be optimistic, but certainly it’s a ticklish problem.
Remarkably, for our purposes, this point won’t matter much.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201029 / 44

Representations
One of the reasons people like Kac-Moody algebras is that it has a nicerepresentation theory. Every integrable irrep is generated by a unique line
killed by all Ei , and the representation Vλ is determined by the weight λ ofthis line.
So, we can construct a representation Lλ of U by starting with a single object
V of weight λ with boring endomorphisms, and letting U act by horizontal
composition, subject to Ei ⊗ V = 0.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201030 / 44

Representations
One of the reasons people like Kac-Moody algebras is that it has a nicerepresentation theory. Every integrable irrep is generated by a unique line
killed by all Ei , and the representation Vλ is determined by the weight λ ofthis line.
So, we can construct a representation Lλ of U by starting with a single object
V of weight λ with boring endomorphisms, and letting U act by horizontal
composition, subject to Ei ⊗ V = 0.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201030 / 44

Representations
One of the reasons people like Kac-Moody algebras is that it has a nicerepresentation theory. Every integrable irrep is generated by a unique line
killed by all Ei , and the representation Vλ is determined by the weight λ ofthis line.
So, we can construct a representation Lλ of U by starting with a single object
V of weight λ with boring endomorphisms, and letting U act by horizontal
composition, subject to Ei ⊗ V = 0.

λ

Ben Webster (U. of Oregon)λ − αjλObjects:j

λ − αj − αijCategorifying q-Fock space

λ − αjiNovember 20, 201030 / 44

Representations
One of the reasons people like Kac-Moody algebras is that it has a nicerepresentation theory. Every integrable irrep is generated by a unique line
killed by all Ei , and the representation Vλ is determined by the weight λ ofthis line.
So, we can construct a representation Lλ of U by starting with a single object
V of weight λ with boring endomorphisms, and letting U act by horizontal
composition, subject to Ei ⊗ V = 0.
λ − αiλλ − αjλMorphisms:λλBen Webster (U. of Oregon)λ − αjj
Categorifying q-Fock space

λ − αj − αij

λ − αjiNovember 20, 201030 / 44

Relations
Theorem (Rouquier/Brundan-Kleshchev)
The GG of Lλ is the irreducible representation of U̇ with highest weight λ,and Lλ is essentially the unique such module category for U.
(You might think that this would give you the Verma module; it doesn’t!).
There’s just one new relation in this category, the “cyclotomic relation.”
−λj

−λj

0 = −

=
λ

j

αj∨ (λ)λj

The category Lλ is just projective modules over the algebra AL , the quotientof AλL by the two-sided ideal generated under vertical composition andhorizontal composition on the right by the relation above.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201031 / 44

Representations
b n , we can still try to
So, even if we don’t have a categorification of the full gldefine a categorification of Fock space as a representation over its lower half.
Definition
Let AλM the quotient of AM by the two-sided ideal generated by the relation
above by horizontal composition on the right and vertical composition.
The category Mλ of projective Aλm modules still has a monoidal action by U
given by adding black lines colored by simple roots (unit vectors).
Theorem (Stroppel-W.)
The graded GG of Mωi is isomorphic to q-deformed Fock space as ab n ⊗ Λ-module.
slL∞
The algebra Aωi ∼Sm (ζ) where ζ is a primitive nth root of unity.
=
M

Ben Webster (U. of Oregon)

m=0Categorifying q-Fock space

November 20, 201032 / 44

Representations
b n , we can still try to
So, even if we don’t have a categorification of the full gldefine a categorification of Fock space as a representation over its lower half.
Definition
Let AλM the quotient of AM by the two-sided ideal generated by the relation
above by horizontal composition on the right and vertical composition.
The category Mλ of projective Aλm modules still has a monoidal action by U
given by adding black lines colored by simple roots (unit vectors).
Theorem (Stroppel-W.)
The graded GG of Mωi is isomorphic to q-deformed Fock space as ab n ⊗ Λ-module.
slL∞
The algebra Aωi ∼Sm (ζ) where ζ is a primitive nth root of unity.
=
M

Ben Webster (U. of Oregon)

m=0Categorifying q-Fock space

November 20, 201032 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 2 32 3 4
34Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 2 32 3 4
34Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1

d1 = (0, 1, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
22d2 = (2, 0, 0)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
333d3 = (0, 2, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.

44d4 = (0, 1, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 2 32 3 4
34

d1 = (0, 1, 1)
d2 = (2, 0, 0)
d3 = (0, 2, 1)
d4 = (0, 1, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 1 12 2 2
3
4Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 1 12 2 2
3
4Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 1 1

f1 = (1, 2, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.

2 2 2

f2 = (1, 1, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.

3

f3 = (0, 0, 1)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.

4f4 = (0, 1, 0)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
At the core of the proof is the construction of a basis CS,T indexed by pairs
S, T of semi-standard tableaux of the same shape.
Let DT be the sequence of dimension vectors where the component of djT at
the node i is the number of boxes with label j in T with content i.
1 1 1 12 2 2
34

f1 = (1, 2, 1)
f2 = (1, 1, 1)
f3 = (0, 0, 1)
f4 = (0, 1, 0)

Let FT be the associated sequence for the ground-state tableau on the same
shape as T.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201033 / 44

A cellular basis
There is a natural element BT attached to each tableau; its top and bottom aremarked with DT and FT . The element itself is given by “connect each box toits row.”(0, 1, 1)(2, 0, 0)(0, 2, 1)(0, 1, 1)(1, 0, 0)BT =(1, 0, 0)(0, 1, 0)(1, 2, 1)(0, 1, 0)(1, 1, 1)(0, 0, 1)

1 1 2 3
2 3 4
T=
3
4

(0, 1, 0)

Of course, this involves choices; essentially the choice of a special reduced
word for the permutation which puts the reading word into order. Fix a choice
once and for all; this won’t make any difference for our purposes.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201034 / 44

A cellular basis
There is a natural element BT attached to each tableau; its top and bottom aremarked with DT and FT . The element itself is given by “connect each box toits row.”(0, 1, 1)(2, 0, 0)(0, 2, 1)(0, 1, 1)(1, 0, 0)BT =(1, 0, 0)(0, 1, 0)(1, 2, 1)(0, 1, 0)(1, 1, 1)(0, 0, 1)

1 1 2 3
2 3 4
T=
3
4

(0, 1, 0)

Of course, this involves choices; essentially the choice of a special reduced
word for the permutation which puts the reading word into order. Fix a choice
once and for all; this won’t make any difference for our purposes.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201034 / 44

A cellular basis
We let B∗T be the reflection of this diagram, and let CS,T = BS B∗T . Of course, if
this element is non-zero, S and T must have the same shape.
(1, 1, 1)(1, 2, 1)(0, 0, 1)(0, 1, 0)S=

1 1 1 2
2 2 2
3
4

T=

1 1 2 32 3 4
34(0, 1, 0)(1, 2, 1)(1, 1, 1)(0, 1, 0)(0, 0, 1)(0, 1, 0)(0, 1, 0)(1, 0, 0)(1, 0, 0)(0, 1, 1)(2, 0, 0)

Ben Webster (U. of Oregon)(0, 2, 1)(0, 1, 1)

Categorifying q-Fock space

November 20, 201035 / 44

A cellular basis
Theorem (Stroppel-W.)
The vectors CS,T form a basis of AωMi , which is cellular.
Under the isomorphism to the q-Schur algebra, the cellular ideals of this basis
coincide with those of the DJM cellular basis, and thus its cell modules aresent to the Weyl modules.
Proof.
Geometric considerations and a few calculations show that these vectors
span.
Reduce to the case where D is a unit vector (i.e. both tableaux are
standard).
This is almost the cellular basis of Hu and Mathas on the cyclotomic
Khovanov-Lauda algebra (=cyclotomic Hecke algebra).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201036 / 44

Graphical permutation modules
Let eν for ν = (ν1 , . . . , νn ) a composition be the idempotentPin AM whichpicks out components where the flag has total type ν, that is dji = νj .
For example e(1d ) AM e(1d ) = AL . Thus, eν AωMi e(1d ) is a right module overe(1d ) AωMi e(1d ) ∼= Hd (q) by Brundan and Kleshchev’s isomorphism.
Theorem (Stroppel-W.)
This right module is isomorphic to the signed permutation module over Hd (q)for ν. The left action of AωMi on AωMi e(1d ) induces the isomorphism to theq-Schur algebra.
The appearance of the signed rather than usual permutation module is a
“mistake” of Brundan and Kleshchev; it’s not particularly intrinsic. You also
have to be very careful to pick the right one of their isomorphisms.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201037 / 44

Graphical permutation modules
Let eν for ν = (ν1 , . . . , νn ) a composition be the idempotentPin AM whichpicks out components where the flag has total type ν, that is dji = νj .
For example e(1d ) AM e(1d ) = AL . Thus, eν AωMi e(1d ) is a right module overe(1d ) AωMi e(1d ) ∼= Hd (q) by Brundan and Kleshchev’s isomorphism.
Theorem (Stroppel-W.)
This right module is isomorphic to the signed permutation module over Hd (q)for ν. The left action of AωMi on AωMi e(1d ) induces the isomorphism to theq-Schur algebra.
The appearance of the signed rather than usual permutation module is a
“mistake” of Brundan and Kleshchev; it’s not particularly intrinsic. You also
have to be very careful to pick the right one of their isomorphisms.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201037 / 44

What’s the point?
What’s achieved by obtaining a categorification like this? For me, a big part ofthe motivation was categorified knot invariants, which I’ll talk about in Kyoto
next week. But this can also illuminate purely representation theoretic points.
Theorem
The functor HomAωi (−, AωMi ) where the action is twisted by the verticalM
reflection anti-automorphism categorifies the bar involution on q-Fock space.
This works for any module category over U with a nice enough equivalence toits opposite; we always get an involution which is anti-linear and commuteswith Fi and Ei .
The indecomposable projectives are always invariant under this functor (in thegrading so that their quotients are self-dual), and thus will be a canonical basis
in the sense of Lusztig whenever we have some upper-triangularity.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201038 / 44

What’s the point?
What’s achieved by obtaining a categorification like this? For me, a big part ofthe motivation was categorified knot invariants, which I’ll talk about in Kyoto
next week. But this can also illuminate purely representation theoretic points.
Theorem
The functor HomAωi (−, AωMi ) where the action is twisted by the verticalM
reflection anti-automorphism categorifies the bar involution on q-Fock space.
This works for any module category over U with a nice enough equivalence toits opposite; we always get an involution which is anti-linear and commuteswith Fi and Ei .
The indecomposable projectives are always invariant under this functor (in thegrading so that their quotients are self-dual), and thus will be a canonical basis
in the sense of Lusztig whenever we have some upper-triangularity.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201038 / 44

What’s the point?
What’s achieved by obtaining a categorification like this? For me, a big part ofthe motivation was categorified knot invariants, which I’ll talk about in Kyoto
next week. But this can also illuminate purely representation theoretic points.
Theorem
The functor HomAωi (−, AωMi ) where the action is twisted by the verticalM
reflection anti-automorphism categorifies the bar involution on q-Fock space.
This works for any module category over U with a nice enough equivalence toits opposite; we always get an involution which is anti-linear and commuteswith Fi and Ei .
The indecomposable projectives are always invariant under this functor (in thegrading so that their quotients are self-dual), and thus will be a canonical basis
in the sense of Lusztig whenever we have some upper-triangularity.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201038 / 44Canonical bases
This upper-triangularity isn’t free; in fact, we have that
Theorem (W.)
The indecomposables of Lλ give Lusztig’s canonical basis in Vλ if and only if
Lλ is graded equivalent to representations of a positively graded algebra.
The hypothesis fails in most non-symmetric types. However, in symmetrictype, it follows from geometry by work of Vasserot and Varagnolo.
Theorem (Stroppel-W.)
The indecomposable projectives of AωMi (graded so that their cosocle isself-dual) categorify the canonical basis of Leclerc-Thibon on q-Fock space.
In particular, this gives an (almost) purely combinatorial proof that the gradeddecomposition numbers of q-Schur algebras are the coefficients of a canonicalbasis (by BGG reciprocity).
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201039 / 44Canonical bases
This upper-triangularity isn’t free; in fact, we have that
Theorem (W.)
The indecomposables of Lλ give Lusztig’s canonical basis in Vλ if and only if
Lλ is graded equivalent to representations of a positively graded algebra.
The hypothesis fails in most non-symmetric types. However, in symmetrictype, it follows from geometry by work of Vasserot and Varagnolo.
Theorem (Stroppel-W.)
The indecomposable projectives of AωMi (graded so that their cosocle isself-dual) categorify the canonical basis of Leclerc-Thibon on q-Fock space.
In particular, this gives an (almost) purely combinatorial proof that the gradeddecomposition numbers of q-Schur algebras are the coefficients of a canonicalbasis (by BGG reciprocity).
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201039 / 44Higher levels
One very interesting thing to do is consider analogues of these results forhigher level q-Fock spaces. Of course, this would easier if people could agree
on what “higher level q-Fock spaces” meant.
One thing that you might mean is a highest weight representation ofb n ); that’s categorified by representations of Aλ for other weights λ.
Uq (glM
Another thing you could potentially mean is the tensor product (using theb n )) of level 1 Fock spaces (orusual Hopf algebra structure on Uq (slb n -irreps as above); this is bigger than the one discussed above. This isglλcategorified by a similar diagrammatic algebra AM which is isomorphic
to the cyclotomic q-Schur algebra.
This new diagrammatic algebra now has multiple red lines, which we forbid
to cross. It arises from the geometry of quiver varieties with shadow vertices.
Ei (vλ1 ⊗ Fj vλ2 ) ↔
λ2Ben Webster (U. of Oregon)

λ1 + λ2−αj

λ2 − αjλ2jCategorifying q-Fock space

λ1

λ1 + λ2−αj + αii
November 20, 2010

40 / 44Higher levels
One very interesting thing to do is consider analogues of these results forhigher level q-Fock spaces. Of course, this would easier if people could agree
on what “higher level q-Fock spaces” meant.
One thing that you might mean is a highest weight representation ofb n ); that’s categorified by representations of Aλ for other weights λ.
Uq (glM
Another thing you could potentially mean is the tensor product (using theb n )) of level 1 Fock spaces (orusual Hopf algebra structure on Uq (slb n -irreps as above); this is bigger than the one discussed above. This isglλcategorified by a similar diagrammatic algebra AM which is isomorphic
to the cyclotomic q-Schur algebra.
This new diagrammatic algebra now has multiple red lines, which we forbid
to cross. It arises from the geometry of quiver varieties with shadow vertices.
Ei (vλ1 ⊗ Fj vλ2 ) ↔
λ2Ben Webster (U. of Oregon)

λ1 + λ2−αj

λ2 − αjλ2jCategorifying q-Fock space

λ1

λ1 + λ2−αj + αii
November 20, 2010

40 / 44Higher levels
One very interesting thing to do is consider analogues of these results forhigher level q-Fock spaces. Of course, this would easier if people could agree
on what “higher level q-Fock spaces” meant.
One thing that you might mean is a highest weight representation ofb n ); that’s categorified by representations of Aλ for other weights λ.
Uq (glM
Another thing you could potentially mean is the tensor product (using theb n )) of level 1 Fock spaces (orusual Hopf algebra structure on Uq (slb n -irreps as above); this is bigger than the one discussed above. This isglλcategorified by a similar diagrammatic algebra AM which is isomorphic
to the cyclotomic q-Schur algebra.
This new diagrammatic algebra now has multiple red lines, which we forbid
to cross. It arises from the geometry of quiver varieties with shadow vertices.
Ei (vλ1 ⊗ Fj vλ2 ) ↔
λ2Ben Webster (U. of Oregon)

λ1 + λ2−αj

λ2 − αjλ2jCategorifying q-Fock space

λ1

λ1 + λ2−αj + αii
November 20, 2010

40 / 44Higher levels
One very interesting thing to do is consider analogues of these results forhigher level q-Fock spaces. Of course, this would easier if people could agree
on what “higher level q-Fock spaces” meant.
One thing that you might mean is a highest weight representation ofb n ); that’s categorified by representations of Aλ for other weights λ.
Uq (glM
Another thing you could potentially mean is the tensor product (using theb n )) of level 1 Fock spaces (orusual Hopf algebra structure on Uq (slb n -irreps as above); this is bigger than the one discussed above. This isglλcategorified by a similar diagrammatic algebra AM which is isomorphic
to the cyclotomic q-Schur algebra.
This new diagrammatic algebra now has multiple red lines, which we forbid
to cross. It arises from the geometry of quiver varieties with shadow vertices.
Ei (vλ1 ⊗ Fj vλ2 ) ↔
λ2Ben Webster (U. of Oregon)

λ1 + λ2−αj

λ2 − αjλ2jCategorifying q-Fock space

λ1

λ1 + λ2−αj + αii
November 20, 2010

40 / 44Tensor productsλThe new relations one needs to add to AM are=αi∨ (λ)

i

λ

=

a

b

a+b=λi −1

i

λ

X

+

i

i

i

i

i

i

= αi∨ (λ)
i

i

λ

=

λ

j

=
λ

i

λ

λ

j

i

i

=

=
i

i

the right-most redline is killed by Ei .

i

=

λ

Theorem (W.)λ

The GG of AM -mod is the tensor product of Fock spaces for the weights λi .
The classes of indecomposables give the canonical basis for the bar
involution discussed earlier.
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 2010

41 / 44Higher levels
Yet a third thing you could have in mind is the twisted higher-level spaces inthe sense of Uglov. These arise from choosing a list of integers (c1 , . . . , c` ),and q-deform the tensor product of level 1 Fock spaces with these charges.
The key difference between Uglov’s space and the tensor product is that they
are associated to different partial orders on the set of `-multipartitions.
multi-partitions
The tensor product quantization corresponds to the usual dominanceorder.
Uglov’s order corresponds to dominance order on anti-`-quotients (i.e.
λ ≺c λ0 if they are the `-quotients of partitions with the same relation indominance order).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201042 / 44Higher levels
Yet a third thing you could have in mind is the twisted higher-level spaces inthe sense of Uglov. These arise from choosing a list of integers (c1 , . . . , c` ),and q-deform the tensor product of level 1 Fock spaces with these charges.
The key difference between Uglov’s space and the tensor product is that they
are associated to different partial orders on the set of `-multipartitions.
multi-partitions
The tensor product quantization corresponds to the usual dominanceorder.
Uglov’s order corresponds to dominance order on anti-`-quotients (i.e.
λ ≺c λ0 if they are the `-quotients of partitions with the same relation indominance order).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201042 / 44Higher levels
Yet a third thing you could have in mind is the twisted higher-level spaces inthe sense of Uglov. These arise from choosing a list of integers (c1 , . . . , c` ),and q-deform the tensor product of level 1 Fock spaces with these charges.
The key difference between Uglov’s space and the tensor product is that they
are associated to different partial orders on the set of `-multipartitions.
multi-partitions
The tensor product quantization corresponds to the usual dominanceorder.
Uglov’s order corresponds to dominance order on anti-`-quotients (i.e.
λ ≺c λ0 if they are the `-quotients of partitions with the same relation indominance order).

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201042 / 44Higher levels
The categorifical manifestation of the first order is that the coarsest partial
order on the set of Weyl modules for which they form an exceptional
collection is the dominance order.
Thus, we can mutate our exceptional collection with respect to the change of
λorder from ≤ to ≺c , to obtain a new exceptional collection in Db (AM -mod)
which we call c-twisted Weyl modules.
In particular, this exceptional collection and its dual define a new t-structure
λon Db (AM -mod), whose heart Vc is a highest weight category with standards
given by the c-twisted Weyl modules.
Conjecture
Uglov’s higher level Fock spaces are the GG’s of Vc , with standard vectors
given by the classes of c-twisted Weyl modules and the canonical basis given
by projective modules in Vc .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201043 / 44Higher levels
The categorifical manifestation of the first order is that the coarsest partial
order on the set of Weyl modules for which they form an exceptional
collection is the dominance order.
Thus, we can mutate our exceptional collection with respect to the change of
λorder from ≤ to ≺c , to obtain a new exceptional collection in Db (AM -mod)
which we call c-twisted Weyl modules.
In particular, this exceptional collection and its dual define a new t-structure
λon Db (AM -mod), whose heart Vc is a highest weight category with standards
given by the c-twisted Weyl modules.
Conjecture
Uglov’s higher level Fock spaces are the GG’s of Vc , with standard vectors
given by the classes of c-twisted Weyl modules and the canonical basis given
by projective modules in Vc .
Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201043 / 44

Thanks for listening.

Ben Webster (U. of Oregon)

Categorifying q-Fock space

November 20, 201044 / 44