Categorifying higher su 3 knot polynomials David Clark [email protected] Randolph-Macon College Ashland, VA University of Virginia Topology Seminar March 29, 2011 David Clark Categorifying higher su 3 knot polynomials
Jul 16, 2020
Categorifying higher su3 knot polynomials
David [email protected]
Randolph-Macon CollegeAshland, VA
University of VirginiaTopology Seminar
March 29, 2011
David Clark Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
Using the skein relations,
7−→ q2 − q3
7−→ −q−3 + q−2
subject to Kuperberg’s su3 spider relations,
= q2 + 1 + q−2 = q + q−1
= +
David Clark Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
Using the skein relations,
7−→ q2 − q3
7−→ −q−3 + q−2
subject to Kuperberg’s su3 spider relations,
= q2 + 1 + q−2 = q + q−1
= +
David Clark Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
. . . we get an assignment
L 7−→ J su3(L),
a specialization of the HOMFLY polynomial.
From a representation theoretic standpoint, this polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.
David Clark Categorifying higher su3 knot polynomials
The quantum su3 link polynomial
. . . we get an assignment
L 7−→ J su3(L),
a specialization of the HOMFLY polynomial.
From a representation theoretic standpoint, this polynomialcomes from coloring the link with the fundamental vectorrepresentation V ∼= C3.
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1 L1Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1 L1Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
L 2
L1
L2
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
L 2
Σ
L1
L2
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
Original categorification
Khovanov categorified this polynomial
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
Theorem (Khovanov)
χ(Kh(L)) = Jsu3(L)
David Clark Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of thisinvariant, allowing us to linger in the realm of pictures a bitlonger.
Maps are now cobordisms between webs, called “foams.”
Categorified skein relations:
� //
(• // q2 // q3 // •
)
� //
(• // q−3 // q−2 // •
)
David Clark Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of thisinvariant, allowing us to linger in the realm of pictures a bitlonger.
Maps are now cobordisms between webs, called “foams.”
Categorified skein relations:
� //
(• // q2 // q3 // •
)
� //
(• // q−3 // q−2 // •
)
David Clark Categorifying higher su3 knot polynomials
“Algebra Light” categorification
Morrison and Nieh gave a “universal” categorification of thisinvariant, allowing us to linger in the realm of pictures a bitlonger.
Maps are now cobordisms between webs, called “foams.”
Categorified skein relations:
� //
(• // q2 // q3 // •
)
� //
(• // q−3 // q−2 // •
)
David Clark Categorifying higher su3 knot polynomials
Categorified spider relations (over Q)
= 0
= 3
+ + = 0
= 0
= 0
= 12 + 1
2
= 13 − 1
9 + 13 = −
David Clark Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.
David Clark Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosenalgebraic formulation.
“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.
David Clark Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.
“easy,” because it’s completely combinatorial.
David Clark Categorifying higher su3 knot polynomials
Useful properties
This view of Khovanov’s su3 theory is
“universal,” in that it’s independent of the chosenalgebraic formulation.“local,” in that it’s built with tangles in mind.“easy,” because it’s completely combinatorial.
David Clark Categorifying higher su3 knot polynomials
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
Theorem (C.)
The su3 Khovanov homology is properly functorial with respect tolink cobordisms, i.e.,
Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)
Functoriality allows us to explore the su3 link homology inmore subtle ways . . .
David Clark Categorifying higher su3 knot polynomials
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
Theorem (C.)
The su3 Khovanov homology is properly functorial with respect tolink cobordisms, i.e.,
Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)
Functoriality allows us to explore the su3 link homology inmore subtle ways . . .
David Clark Categorifying higher su3 knot polynomials
Useful properties
L1
L 2
Σ
L1
L2
Σ
Kh( )
Kh( )
Kh( )
Theorem (C.)
The su3 Khovanov homology is properly functorial with respect tolink cobordisms, i.e.,
Σ ' Σ′ ⇒ Kh(Σ) = Kh(Σ′)
Functoriality allows us to explore the su3 link homology inmore subtle ways . . .
David Clark Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies thepolynomial corresp to Vfund = C3.
Jsu3(K ) = Jsu3fund(K )
But there are polynomials obtained by coloring a link with anyirrep Vλof su3.
Jsu3λ (K )
Ben Webster has categorified these invariants in analgebro-geometric setting.
David Clark Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies thepolynomial corresp to Vfund = C3.
Jsu3(K ) = Jsu3fund(K )
But there are polynomials obtained by coloring a link with anyirrep Vλof su3.
Jsu3λ (K )
Ben Webster has categorified these invariants in analgebro-geometric setting.
David Clark Categorifying higher su3 knot polynomials
Bigger picture
The homology theory we’ve been discussing categorifies thepolynomial corresp to Vfund = C3.
Jsu3(K ) = Jsu3fund(K )
But there are polynomials obtained by coloring a link with anyirrep Vλof su3.
Jsu3λ (K )
Ben Webster has categorified these invariants in analgebro-geometric setting.
David Clark Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in thislocal, combinatorial setting.
Possible strategies:
Categorify the su3 Jones-Wenzl idempotents.Use representation theory, and work with the symmetricgroup.
David Clark Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in thislocal, combinatorial setting.
Possible strategies:Categorify the su3 Jones-Wenzl idempotents.
Use representation theory, and work with the symmetricgroup.
David Clark Categorifying higher su3 knot polynomials
Our goal
Our goal: to categorify these higher su3 polynomials in thislocal, combinatorial setting.
Possible strategies:Categorify the su3 Jones-Wenzl idempotents.Use representation theory, and work with the symmetricgroup.
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Fix a knot K, and consider its n-parallel cable:
K K(n)
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.
Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.
Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.
Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.
Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
Let Ri : K (n) → K (n) be the cobordism that swaps the ith and(i + 1)st cables via the right-hand rule.
Any composition of such cobordisms induces a map on theKhovanov homology of the n-cable:
Kh(Ri ) : Kh(K (n)) −→ Kh(K (n))
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
So let Sn act on Kh(Kn) via these maps!
Theorem (C.)
This is an honest group action, i.e., the map
Sn −→ End(Kh(K (n)))
σi 7−→ Kh(Ri )
is a homomorphism of groups.
David Clark Categorifying higher su3 knot polynomials
An action of the symmetric group
So let Sn act on Kh(Kn) via these maps!
Theorem (C.)
This is an honest group action, i.e., the map
Sn −→ End(Kh(K (n)))
σi 7−→ Kh(Ri )
is a homomorphism of groups.
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
Our action needs to satisfy the relations on transpositions in Sn:
1 σiσj = σjσi if j 6= i ± 1
2 σiσi+1σi = σi+1σiσi+1
3 σ2i = 1
For relations (1) and (2), we need to show that
Kh(RiRj) = Kh(RjRi ) if j 6= i ± 1
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
Our action needs to satisfy the relations on transpositions in Sn:
1 σiσj = σjσi if j 6= i ± 1
2 σiσi+1σi = σi+1σiσi+1
3 σ2i = 1
For relations (1) and (2), we need to show that
Kh(RiRj) = Kh(RjRi ) if j 6= i ± 1
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Sketch of proof.
Our action needs to satisfy the relations on transpositions in Sn:
1 σiσj = σjσi if j 6= i ± 1
2 σiσi+1σi = σi+1σiσi+1
3 σ2i = 1
For relations (1) and (2), we need to show that
Kh(RiRj) = Kh(RjRi ) if j 6= i ± 1
and
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1)
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (1):
K(4)
K(4)
R3 1R R31R
Functoriality⇒ Kh(RiRj) = Kh(RjRi ).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 2R 1R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Conveniently, these both follow directly from functoriality!
Relation (2):
K(4)
K(4)
1R 1R2R 1R 2R 2R
Functoriality⇒ Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1).
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
. . . but for relation (3), we need to show that
Kh(R2i ) = Id
K(4)
K(4)
1R Id2
So we need to look more carefully at the induced maps . . .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Mercifully, it will suffice to consider the 2-cable of K .
In particular, we need a movie of knot diagrams that describesthe cobordism R .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Mercifully, it will suffice to consider the 2-cable of K .
In particular, we need a movie of knot diagrams that describesthe cobordism R .
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).That’s a very nasty map to compute explicitly!
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).
That’s a very nasty map to compute explicitly!
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
T Tx Tx
Tx
ρ Tx
Tx
T
This is a pair of R2 moves on the ends, with 4c R3 movesin the middle (where c is the number of crossing in theoriginal knot K ).That’s a very nasty map to compute explicitly!
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Instead, consider the cobordism L:
R L
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
IdL R
1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials
Sketch of proof
However, with some work 1 one can show that
Kh(L) = Kh(R)
And notice that
IdL R
1using the Categorified Kauffman Trick, and other tricks.David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
Sketch of proof
Thus, we see that
Kh(R2) = Kh(R · R)
= Kh(L · R)
= Kh(Id)
= Id
So:
Kh(RiRj) = Kh(RjRi ) X
Kh(RiRi+1Ri ) = Kh(Ri+1RiRi+1) X
Kh(R2i ) = Id X
�
David Clark Categorifying higher su3 knot polynomials
So ...
Why do we care?
David Clark Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jsu3λ :
K ,Vλ Jsu3λ (K )
Basic idea:
1 For a knot K , we’ll find the (huge!) Khovanov complex ofone of its parallel cables.
2 Using our symmetric group action, we’ll project down to acomplex whose Euler characteristic is Jsu3λ (K ).
David Clark Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jsu3λ :
K ,Vλ Jsu3λ (K )
Basic idea:
1 For a knot K , we’ll find the (huge!) Khovanov complex ofone of its parallel cables.
2 Using our symmetric group action, we’ll project down to acomplex whose Euler characteristic is Jsu3λ (K ).
David Clark Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jsu3λ :
K ,Vλ Jsu3λ (K )
Basic idea:1 For a knot K , we’ll find the (huge!) Khovanov complex of
one of its parallel cables.
2 Using our symmetric group action, we’ll project down to acomplex whose Euler characteristic is Jsu3λ (K ).
David Clark Categorifying higher su3 knot polynomials
More categorification
Recall: our goal is to categorify the polynomials Jsu3λ :
K ,Vλ Jsu3λ (K )
Basic idea:1 For a knot K , we’ll find the (huge!) Khovanov complex of
one of its parallel cables.2 Using our symmetric group action, we’ll project down to a
complex whose Euler characteristic is Jsu3λ (K ).
David Clark Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3.
Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2
≥0.
Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.
There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.
The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as
sλ ∈ QSn.
David Clark Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3.
Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2
≥0.
Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.
There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.
The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as
sλ ∈ QSn.
David Clark Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3.
Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2
≥0.
Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.
There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.
The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as
sλ ∈ QSn.
David Clark Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3.
Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2
≥0.
Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.
There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.
The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as
sλ ∈ QSn.
David Clark Categorifying higher su3 knot polynomials
Some representation theory
Let V = C3 be the standard vector representation of su3.
Recall: there is a two-parameter family of irreps Vλ of su3,parameterized by λ = (λ1, λ2) ∈ Z2
≥0.
Fact: for n = λ1 + 2λ2, we know that Vλ is asubrepresentation of V⊗n.
There is an idempotent sλ ∈ End(V⊗n) that projects ontoVλ.
The map sλ, sometimes called the Schur functor, is reallyjust a linear combination of permutations of the tensorpowers of V⊗n, and can thus be viewed as
sλ ∈ QSn.
David Clark Categorifying higher su3 knot polynomials
Some representation theory
For example, to get the adjoint representation Vad, we canproject
sad : V ⊗ V ⊗ V −→ Vad
by letting
sad =1
3
(Id + τ(1 2) − τ(1 3) − τ(1 3 2)
)
David Clark Categorifying higher su3 knot polynomials
Some representation theory
For example, to get the adjoint representation Vad, we canproject
sad : V ⊗ V ⊗ V −→ Vad
by letting
sad =1
3
(Id + τ(1 2) − τ(1 3) − τ(1 3 2)
)
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
//
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Kh(K (n))
//
Vλ
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Kh(K (n))
sλ
����
//
Vλ “Khλ(K )”
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Kh(K (n))
sλ
����
//
Vλ “Khλ(K )”
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials
Proposed categorification for higher irreps
Viewing sλ ∈ CSn, we see it acts on Kh(K (n)).
V⊗n
sλ
����
Kh(K (n))
sλ
����
? //
Vλ “Khλ(K )”
Claim: χ(Khλ(K )) = Jsu3λ (K )
David Clark Categorifying higher su3 knot polynomials