The diagrammatic beauty of Rep(Uq

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The diagrammatic beauty of Rep(U

q

(sl

n

)): Part I

Daniel Tubbenhauer

The uncategorified story

March 2014

= =

Daniel Tubbenhauer March 2014 1 / 38

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1 The diagrammatic calculus sl2-webs

Thesl2-flow lines Algebra is rigid

2 The representation theory ofUq(sl2) The algebra

Connection to the diagrammatic calculus Invariant tensors

3 Connection to thesl_n-link-polynomials The Jones polynomial

Reshetikhin-Turaev: Jones is an intertwiner RT polynomials usingsl_n-webs

Daniel Tubbenhauer March 2014 2 / 38

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An old story: Rumer, Teller and Weyl (1932)

Daniel Tubbenhauer sl2-webs March 2014 3 / 38

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The sl

₂

-webs

Definition(Rumar, Teller, Weyl 1932)

Fix two numbersb,t∈Nwithb+t= 2ℓ. Asl2-webw withbbottom points and t top points is an embedding (non-intersecting!) of a finite number of lines and circles in a rectangle withb fixed points at the bottom andt at the top such that the two boundary points of the lines are some of the fixed points. Theset of all sl2-websw betweenbbottom points and t top points in denoted by ˜W2(b,t).

Example(b = 3 and t = 5)

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The sl

₂

-web space

Definition

Fix two numbersb,t∈Nwithb+t= 2ℓ. Thesl2-web spaceW2(b,t)is the free Q(q)-vector space¯ generatedby elements of ˜W2(b,t) modulo

Thecircle removal

= [2] = q+q⁻¹ Theisotopy relations

= =

Note thatW2(b,t) is afinite dimensional ¯Q(q)-vector space!

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The sl

₂

-web category

Definition(Kuperberg 1997)

Thesl₂-web category or web spiderSp(Uq(sl2))is the monoidal, ¯Q(q)-linear 1-category consisting of

Theobjectsare the natural numbersN={0,1,2, . . .}.

The1-cellsw: b→t are the elements ofW2(b,t).

The ¯Q(q)-linear composition isstacking.

The monoidal structure⊗is given byjuxtaposition, i.e.b⊗b^′ =b+b^′ and

⊗ =

As generatorssufficesthe identities, shifts, cups and caps

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The sl

₂

-web category - examples

Example

◦ =

and

◦ = = [2]·id(3,3)

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An extra information

Definition

Given asl2-webw ∈W2(b,t). Asl2-flow linef forw, denoted byw_f, is a choice of orientation for all lines and circles ofw.

If one ignores internal circles

and

then such a flow line iscompletelydetermine by its boundary. There it induces a state stringfor the bottom~Sb= (±, . . . ,±) and top~St = (±, . . . ,±) with a plus for outgoing flow lines and a minus for incoming.

Example

+ + −

− + − − +

Daniel Tubbenhauer Thesl2-flow lines March 2014 8 / 38

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The weight of a flow

Definition

Flows on the generators of thesl2-web categorySp(Uq(sl2)) are assigned a certainweightwtby the local rules

wt( ) = 0 wt( ) =−1 wt( ) = 0 wt( ) = +1

and always zero on identities and shifts. Theweightof anysl2-web with flow is the sum over the local weights.

Note that the weight isisotopy invariant, thus,well-definedforsl₂-webs without internal circles.

Example

wt= 0

wt=+1

wt=−1

= wt= 0

=

wt= 0

wt=0

Daniel Tubbenhauer Thesl2-flow lines March 2014 9 / 38

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Rigidity of sl

₂

-webs

A seemingly very small point turned out to bea crucial stepif we want to consider biggern: Topology is continuous and Algebra is rigid.

Definition, second try - rigid version

Thesl2-web category or web spiderSp(Uq(sl2))is the monoidal, ¯Q(q)-linear 1-category consisting of

Theobjectsare ordered compositions~k of 2ℓ∈Nwith only 0,1,2 as entries.

The1-cellsw:~k →~k^′arelabelled ladders(we use the convention and do not draw edges labelled 0 and use a dotted line for those labelled 2) generated by juxtaposition and vertical composition of (plus relations and restas before)

k1 k2 k3 k4

k1 k2

k1±k k2∓k k

k=0,1,2

2 0

1 1

0 2

=

2 0

0 2

What is theupshot? “Easy” to generalize to sl_n: Take labels 0,1, . . . ,n−1,nand

“directly” connected to the algebra (which I explain in a second!).

Daniel Tubbenhauer Algebra is rigid March 2014 10 / 38

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A rigid example

There is a small number of different ladders, namely theleft and right shifts

and (rigid)cups and caps

These suffice to generate allsl2-webs, e.g.

Daniel Tubbenhauer Algebra is rigid March 2014 11 / 38

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The quantum algebra U

_q

(sl

d

)

Definition

Ford∈N_>1thequantum special linear algebraUq(sld) is the associative, unital Q(q)-algebra generated by¯ K_i^±1 andEi andFi, fori= 1, . . . ,d−1, subject to some relations (that we do not need today).

Definition(Beilinson-Lusztig-MacPherson)

For each~k ∈Z^d−1 adjoin anidempotent1~k (think: projection to the~k-weight space!) toU_q(sld) and add some relations, e.g.

1~k1~k^′ =δ~k,~k^′1~k and K±i1~k =q^±^~^kⁱ1~k (noK^′s anymore!).

Theidempotented quantum special linear algebrais defined by U˙_q(sld) = M

~k,~k^′∈Z^d−¹

1~kU_q(sld)1~k^′.

Daniel Tubbenhauer The algebra March 2014 12 / 38

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The quantum algebra U

_q

(sl

d

) is a Hopf algebra

It is worth noting thatU_q(sld) is a Hopf algebrawithcoproduct∆ given by

∆(Ei) =E_i⊗K_i+ 1⊗E_i, ∆(Fi) =F_i⊗1 +K_i⁻¹⊗F_i and ∆(Ki) =K_i⊗K_i. Theantipode S and the counitεare given by

S(Ei) =−EiK_i⁻¹, S(Fi) =−KiFi, S(Ki) =K_i⁻¹, ε(Ei) =ε(Fi) = 0, ε(Ki) = 1.

The Hopf algebra structure allows toextendactions to tensor products of representations, to duals of representations and there is a trivial representation.

Example: U

_q

(sl

2

)-vector representation

Consider ¯Q²with basisx₋₁= (0,1),x+1= (1,0). These are called the weights−1 and +1 andK acts on them byq^∓1. The vector representation ofUq(sl2) is:

Think:K = 1 0

0 −1

(0,1)

E

##

(1,0)

F

cc

Think:

E= 0 1

0 0

F= 0 0

1 0

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The category Rep(U

q

(sl

2

))

Definition

Therepresentation categoryRep(Uq(sl2))is the monoidal, ¯Q(q)-linear 1-category consisting of:

Theobjectsare finite tensor products of theUq(sl2)-representations Λ^kQ¯². Denote them by~k = (k1, . . . ,km) withki ∈ {0,1,2}.

The1-cellsw:~k→~k^′ areU_q(sl2)-intertwiners.

Composition of 1-cells iscomposition of intertwinersand⊗is theordered tensor product.

It is worth noting that Λ⁰Q¯²= ¯Qis the trivialU_q(sl2)-representation, Λ²Q¯²∼= ¯Q its dual and Λ¹Q¯²= ¯Q² is the (self-dual)Uq(sl2)-vector representation.

Example

The 1-cells ofMor(~k, ~k^′)“are”(using the Hopf algebra structure!) the invariant tensorsInv_U_q₍sl₂)(~k^∗⊗~k^′) with~k^∗= (2−k1, . . . ,2−k_m).

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We have some control over the intertwiner

GivenV =N

iΛ^kⁱQ¯²denote thetensor basisofV (recall x₋₁= (0,1) and x+1= (1,0), setx_∅=x_{−1,+1}= 1) by{x_S |S = (S1, . . . ,S_m),S_i ⊂ {−1,+1}}.

Theorem(Kuperberg 1997, n > 3: Cautis-Kamnitzer-Morrison 2012)

Define two ¯Q-linear maps calledsplit and mergeby M_sâ,b: Λâ+bQ¯²→ΛâQ¯²⊗Λ^bQ¯², M_sâ,b(xS) = X

T⊂S

(−q)^ℓ(S,T)xT⊗xS−T

M_mâ,b: ΛâQ¯²⊗Λ^bQ¯²→Λâ+bQ¯², M_mâ,b(xS⊗x_T) =

((−q)^−ℓ(T,S)x_S∪T, S∩T =∅,

0, else.

for suitablea,b∈ {0,1,2}andℓ(S,T)∈ {−1,0,+1}. These are Uq(sl2)-intertwinerandgenerateRep(Uq(sl2)).

E.g.:M_m^1,1(x−1⊗x+1) = (−q)⁰,M_m^1,1(x+1⊗x₋₁) = (−q)⁻¹,M_m^1,1(x±1⊗x_±1) = 0.

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Intertwiner are pictures

Theorem(Kuperberg 1997, n > 3: Cautis-Kamnitzer-Morrison 2012)

The 1-categoriesRep(Uq(sl2)) andSp(Uq(sl2)) areequivalent. To be more precise, the equivalence Γ : Rep(Uq(sl2))→Sp(Uq(sl2)) is given by:

One objects: SendN

~kΛ^kⁱQ¯²to~k.

One 1-cells: We only need to consider the generators splitM_s^a,b and merge M_m^a,b. Send them to

M_s^a,b7→

a+b 0

a b

b and M_m^a,b7→

a b

0 a+b

a

Check that it iswell-defined!

I amlyinga little bit: One has to be a little more careful with objects and duals, but weignorethis for today.

Daniel Tubbenhauer Connection to the diagrammatic calculus March 2014 16 / 38

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Intertwiner are pictures: Some examples

Exempli gratia

What about the“left-plus-ladders”? They are acomposite!

k1 k2

k1+k k2−k k

k=0,1,2

=

k1 k2

k1+k k2−k k

k=0,1,2

e.g.

1 2

2 1

1 =

1 2

2 1

1

Generatethem by composition of merge and split!

M_m^k¹^,k

M_s^k,k²^−k = (M_m^k¹^,k⊗id(k2−k,k2−k))◦(id(k1,k1)⊗M_s^k,k²^−k) e.g.

= (M_m^1,1⊗id(1,1))◦(id(1,1)⊗M_s^1,1)

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How to prove it? Quantum skew Howe duality!

Theorem

There is anU˙q(sld)-actionon Sp(Uq(sl2))^d (objects of lengthd)!

1~k 7→

... ...

k1 ki−1 ki ki+1 ki+2 kd

E_i1~k, F_i1~k 7→

... ...

k1 ki−1 ki ki+1 ki+2 kd

k1 ki−1 ki±1 ki+1∓1 ki+2 kd

Thus,Sp(Uq(sl2))^d is aU˙_q(sld)-moduleandnot justaU_q(sl2)-module.

Even better: Since, weonlyneed “left-minus-ladders”, akaF’s, it can be realized as a ˙U_q(sld)-module of a certain highest weight: We can useU˙_q(sld)-highest weight theory to prove statements aboutU_q(sl2)-intertwiner!

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The invariant tensors suffice - a picture

The Hopf-structure says:Mor(~k, ~k^′)∼=Inv_U_q(sl₂)(~k^∗⊗~k^′). The picture says:

~k^′

~k

∼=

~k^′

~k^∗

Remaining question: How toidentifythe invariant tensors?

Daniel Tubbenhauer Invariant tensors March 2014 19 / 38

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This is not trivial...

First question: What do wemean by “identify” the invariant tensors?

Recall: Inv_U_q₍sl₂)(Λ^k¹Q¯²⊗ · · · ⊗Λ^k^dQ¯²)⊂Λ^k¹Q¯²⊗ · · · ⊗Λ^k^dQ¯², and Λ^k¹Q¯²⊗ · · · ⊗Λ^k^dQ¯²has aeasyto write down, buthorribleto work with basis: The elementary tensorsx~S!

Thus, “identify”v ∈InvUq(sl₂)(Λ^k¹Q¯²⊗ · · · ⊗Λ^k^dQ¯²) is writingv in terms ofx~S. Second question:Howto do it? Recall that the action on tensors is given by

∆(Ei) =E_i⊗K_i+ 1⊗E_i, ∆(Fi) =F_i⊗1 +K_i⁻¹⊗F_i and ∆(Ki) =K_i⊗K_i.

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Exempli gratia

Example: ¯ Q

²

⊗ Q ¯

²

Recall that ¯Q²has basisx+1= (1,0) andx−1= (0,1). Thus, ¯Q²⊗Q¯²has basis {x+1+1 =x+1⊗x+1,x+1−1=x+1⊗x−1,x−1+1=x−1⊗x+1,x−1−1=x−1⊗x−1}.

Test calculation:

F·x₊₁₋₁=F·x+1⊗x₋₁+K⁻¹·x+1⊗F·x₋₁

=x₋₁⊗x₋₁

F·x₋₁₊₁=F·x₋₁⊗x+1+K⁻¹·x₋₁⊗F·x+1

=q⁺¹x−1⊗x−1

Claim:x_+1,−1−q⁻¹x₋₁₊₁is invariant and spansInv_U_q₍sl₂)( ¯Q²⊗Q¯²).

How to do this ingeneral?

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The diagrammatic calculus helps

Theorem(Khovanov-Kuperberg 1997)

The decomposition of asl2-webw ∈Mor(∅, ~k) in terms of the elementary tensors x~S is encoded by the flow linesf onw in the following way:

Each flowf induces a state string ~Sf = (±, . . . ,±) at the boundary and has a weightwt(wf).

Then the coefficient forx_~_S

f is (−q)^wt(w^f⁾. Thus,w =P

f(−q)^wt(w^f⁾x_~_S

f.

(Onlyn= 2!) A basis Arc ofMor(∅, ~k) is given by all arc diagrams.

Example: ¯ Q

²

⊗ Q ¯

²

again

In this case there is exactly one arcuand it has the two flows

wt( ) = 0, ~S = (+1,−1) and wt( ) =−1, ~S= (−1,+1) Conclusion:u=x+1⊗x−1−q⁻¹x−1⊗x+1.

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The negative exponent property

Example

= +

+ +

=x+1⊗x+1⊗x₋₁⊗x₋₁−q⁻¹x+1⊗x₋₁⊗x+1⊗x₋₁

−q⁻¹x₋₁⊗x+1⊗x₋₁⊗x+1+q⁻²x₋₁⊗x₋₁⊗x+1⊗x+1

Observation: Oneleadingterm plus a rest with coefficients inq⁻¹Z[q⁻¹]! This is called thenegative exponent property. In fact, the arc basis is thedual canonical basisin the sense of Lusztig.

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n > 2? Use q -skew Howe!

Roughly:

Expressasln-web as a string ofF’s acting on a highest weight vectorv_n^ℓ. The action of theF’s isgiven bythe splits and merges. Read of the resulting vector inductively.

There is also apurely combinatorialway to do this!

3 3 3 0 0 0

F₃⁽²⁾

3 3 1 2 0 0

F₄⁽²⁾

3 3 1 0 2 0

F₅⁽²⁾

3 3 1 0 0 2

F2

3 2 2 0 0 2

F3

3 2 1 1 0 2

F4

3 2 1 0 1 2

F1

2 3 1 0 1 2

F2

2 2 2 0 1 2

F₃⁽²⁾

2 2 0 2 1 2

F2

2 1 1 2 1 2

F1

1 2 1 2 1 2

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Dual canonical sl

_n

-webs? Quantum skew Howe duality!

Counterexample

= + + Rest

=x₊₁₊₁² ⊗x₊₁₊₁¹ ⊗x₀₀² ⊗x₀₀¹ ⊗x₋₁₋₁² ⊗x₋₁₋₁¹

+x₊₁₋₁² ⊗x₊₁₋₁¹ ⊗x₊₁₋₁² ⊗x₊₁₋₁¹ ⊗x₊₁₋₁² ⊗x₊₁₋₁¹ + Rest Here Rest has coefficients inq⁻¹Z[q⁻¹].

Note that “most”n>2-webs do not have this property and this makes livevery complicated! But usingq-skew Howe duality one can obtain aniff-conditionfor a web to be dual canonical plus analgorithmto compute the dual canonical basis.

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The famous Jones polynomial

LetL_D be a diagram of an oriented link. Set [2] =q+q⁻¹ and n+= number of crossings n₋= number of crossings

Definition/Theorem(Jones 1984, Kauffman 1987)

Thebracket polynomialof the diagramL_D (without orientations) is a polynomial hL_Di ∈Z[q,q⁻¹] given by the following rules.

h∅i= 1 (normalization).

h i=h i −qh i (recursion step 1).

h ∐LDi= [2]· hLDi(recursion step 2).

[2]J(LD) = (−1)ⁿ⁻qⁿ⁺⁻²ⁿ⁻hLDi(Re-normalization).

The polynomialJ(·)∈Z[q,q⁻¹] is an invariantof oriented links.

Daniel Tubbenhauer The Jones polynomial March 2014 26 / 38

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Exempli gratia

* +

:

■■

▲▲

▲ rr rr r

✉✉

[2]² −2q·[2] +q²·[2]²

Thus,J(Hopf) =q⁵+q, i.e the Hopf link isnot trivial!

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“The Jones revolution”

Definition/Theorem(HOMFLY 1985, PT 1987)

Define a polynomialP_n(LD)∈Z[q,q⁻¹] uniquely determined by the property P_n() = 1 and the so-calledsl_nskein relations

q²ⁿ·Pn( )−q⁻²ⁿ·Pn( ) = (q+q⁻¹)·Pn( ). Thesln-HOMFLY-PT polynomialis alink invariantandP2(LD) =J(LD).

Shortlyafter Jones several authors independently found new knot polynomials.

One example is the HOMFLY-PT polynomial. Moreover, researches discovered connectionsto different parts of mathematics and physics. Before the “Jones revolution” there was alackof knot polynomials and after there wheretoo many.

The questions shifted to:

“Why do they exist? How can we order them?”

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A tangle is an intertwiner

Letgbeanyclassical Lie algebra. Denote byλi, µj theUq(g)-representation of highest weightVλi,Vµj. LetTD be a diagram of a,λi, µj-colored, oriented tangle.

λ1, λ2, λ3 λ4 λ5

µ1 µ2 µ3 µ4 µ5

V_λ₁⊗V_λ₂⊗V_λ₃⊗V_λ₄⊗V_λ₅ Vµ1⊗Vµ2⊗Vµ3⊗Vµ4⊗Vµ5

f(TD)

Daniel Tubbenhauer Reshetikhin-Turaev: Jones is an intertwiner March 2014 29 / 38

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Representation theory does the trick!

Definition(Reshetikhin-Turaev 1990)

Given the set-up from before we define a certainUq(g)-intertwiner f(TD) :V_λ₁⊗ · · · ⊗V_λ_k →V_λ_k+1⊗ · · · ⊗V_λ_l.

Theorem(Reshetikhin-Turaev 1990)

TheUq(g)-intertwinerf(TD) is aninvariantofTD.

Corollary(Reshetikhin-Turaev 1990)

In the case of colored, orientedlinksLD we have

f(LD) : ¯Q(q)→Q(q),¯ 17→PRT(LD)∈Z[q,q⁻¹],

that is each configuration as above gives apolynomial invariant of oriented links!

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This is powerful!

Example

We have the followinglistof examples!

Letg=sl2. If we restrictto theUq(sl2)-vector representation ¯Q², then the Reshetikhin-Turaev polynomialPRT(·) is the Jones orsl2-polynomial.

Letg=sl2. If we allowanykind of coloring withU_q(sl2)-representations, thenP_RT(·) is the so-calledcoloredJones polynomial.

Letg=sl_n. If werestrictto theUq(sln)-vector representation ¯Qⁿ, then the Reshetikhin-Turaev polynomialPRT(·) is thesl_n-polynomial.

But the Reshetikhin-Turaev polynomial is much moregeneralizethan all of them and“explains”them using one concept.

Moral: A lot of link polynomials arespecial instancesofsymmetries of the quantum groupsUq(g)!

Question: Can we do this more explicit forg=sln?

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“Straightening” again

Consider a diagram of an oriented tangle. Its components can becoloredwith colorsk ∈ {0, . . . ,n}. These colorscorrespond to the fundamental

U_q(sln)-representations Λ^kQ¯ⁿ.Straightening it into a Morse position.

2

1

1 2

4 5

7→

4 4 4 0 2 2 2 2 5 5 5 5

1 1 4 n 2 1 1 5 2 2 n 2

5 4 1 1 1 2 5 1 1 2 0 2

4 5 5 5 5 5 2 2 2 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

Daniel Tubbenhauer RT polynomials usingsln-webs March 2014 32 / 38

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Tangles to sl

_n

-webs

We can define as before the categorySp(Uq(sln)) consisting of 1-cells as

k1 k2 k3 k4

k1 k2

k1+k k k2−k

k=0,1,2,...,n

k1 k2

k1−k k2+k k

k=0,1,2,...,n

Letb≤a. Define anUq(sln)-intertwinerΛ^aQ¯ⁿ⊗Λ^bQ¯ⁿ→Λ^bQ¯ⁿ⊗Λ^aQ¯ⁿas follows.

b a

=

b

X

k=0

(−1)^k+(a+1)bq^−b+k

a b

a+k−b

k

a+k b−k

b a

,

“Morally” (up to some signs, shifts, re-orientations) the same fora<b and .

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Exempli gratia: Hopf link for sl

₂

2 1 1 1 1 1 1 1 1 0 0

0 1 2 2 1 1 1 0 0 1 0

2 2 1 0 1 1 1 1 1 2 2 1 2

0 0 0 1 1 1 1 1 1 1 2 2 2

7→

2 1 1 1 1 2 1 1 1 1 1 0 0

0 1 2 2 1 0 1 1 1 0 0 1 0

2 2 1 0 1 1 1 1 1 2 2 1 2

0 0 0 1 1 1 1 1 1 1 2 2 2

f10(Hopf) : Λ²Q¯²⊗Q¯ ⊗Λ²Q¯²⊗Q¯ →Q¯ ⊗Q¯ ⊗Λ²Q¯²⊗Λ²Q¯²is anintertwinerin Sp(Uq(sln)). In the end we get thesame polynomialas before (up to a shift).

Conclusion: Worksfine forn= 2. What aboutn>2?

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Quantum skew Howe duality helps

↑ U˙q(sld)−action

↓

U_q(sln)−web →

←²

1 1 1 1 2 1 1 1 1 1 0 0

0 1 2 2 1 0 1 1 1 0 0 1 0

2 2 1 0 1 1 1 1 1 2 2 1 2

0 0 0 1 1 1 1 1 1 1 2 2 2

F1

F2

F3

F2

E1

F1

1 1

F2

F3

F1

F2

Recall that we have anU˙_q(sld)-actiononSp(Uq(sln))^d. In the example above f10(Hopf) =F2F1F3F2F1E1F2F3F2F1F₂⁽²⁾v2200.

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The lower part ˙ U

⁻_q

(sl

d

) suffices!

↑

U˙⁻_q(sld)−action

↓

U_q(sln)−web →

2 ←

1 1 1 1 0 0 0 0 0 0 0 0

0 1 2 2 1 2 1 1 0 0 0 0 0

0 0 0 0 0 0 1 0 1 1 1 0 0

0 0 0 0 0 0 0 1 1 0 0 1 2

2 2 1 0 1 1 1 1 1 2 1 1 0

0 0 0 1 1 1 1 1 1 1 2 2 2

F1

F2 F3 F4

F5

F2 F3 F4

F1

F2

F3

F2

F4

F5

F3

F4

A crucial observation: We needonlyF’s!

f10(Hopf) =F₄⁽²⁾F4F3F5F4F2F3F2F1F4F3F2F5F4F3F2F1F₄⁽²⁾F₃⁽²⁾F₂⁽²⁾v220000.

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The sl

_n

-polynomials using sl

_d

-symmetries

Let ussummarizethe connection between (colored)sl_n-polynomials and the U˙_q(sld)-Uq(sln)-skew Howe duality.

Reshetikhin-Turaev: Thesl_n-polynomialsPn(·)areUq(sln)-intertwiner.

Uq(sln)-intertwinerarevectors in hom’s between ˙Uq(sld)-weight spaces.

OnlyF’s: The space of invariant U_q(sln)-tensors is aU˙_q(sld)-representation of somehighest weightvh and ˙U⁻_q(sld)suffices.

Conclusion: The (colored)sln-polynomialsPn(·) are instances of highest U˙q(sld)-weight representation theory!

IfL_D is a link diagram, thenP_n(LD) is obtained byjumping viaF’sfrom a highest ˙U_q(sld)-weightv_hto a lowest ˙U_q(sld)-weightv_l!

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There is stillmuchto do...

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Thanks for your attention!