From dualities to diagrams

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From dualities to diagrams

Or: the diagrammatic presentation machine Daniel Tubbenhauer

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Joint work with David Rose, Pedro Vaz and Paul Wedrich

May 2015

Daniel Tubbenhauer May 2015 1 / 29

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1 Exteriorgl_N-web categories

Graphical calculus via Temperley-Lieb diagrams Its cousins: theN-webs

Proof? Skew quantum Howe duality!

2 Symmetricgl₂-web categories More cousins: the green 2-webs

Proof? Symmetric quantum Howe duality!

3 Exterior-symmetricgl_N-web categories Even more cousins: the green-redN-webs Proof? Super quantum Howe duality!

Super-Super duality and even more cousins

Daniel Tubbenhauer May 2015 2 / 29

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The 2-web space

Definition(Rumer-Teller-Weyl 1932)

The 2-web spaceHom2-Web_g(b,t) is the freeC_q=C(q)-vector space generated by non-intersecting arc diagrams withb,t bottom/top boundary points modulo:

Thecircle removal:

1 =−q−q⁻¹=−[2]

Theisotopy relations:

1 1

=

1 1

=

1 1

Daniel Tubbenhauer Graphical calculus via Temperley-Lieb diagrams May 2015 3 / 29

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The 2-web category

Definition(Kuperberg 1995)

The 2-web category 2-Web_g is the (braided) monoidal,C_q-linear category with:

Objects are vectors~k = (1, . . . ,1) and morphisms areHom2-Web_g(~k,~l).

Composition◦:

1 1

◦

1 1

= ¹ ,

1 1

◦

1 1

=

1 1

Tensoring⊗:

1 1

⊗

1 1

=

1 1

1 1 1

1

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If you do not like quantum groups: q = 1 is fine for today

Recall thatgl₂is generated by E =

0 1 0 0

, F=

0 0 1 0

, H1=

1 0 0 0

, H2=

0 0 0 1

,

The elements ofU(gl₂) are polynomials inE,F,H1,H2,H=H1−H2modulo EF−FE =H, HE =EH+ 2E, HF =FH+ 2F.

The elements ofU_q(gl₂) are polynomials inE,F,L^±1_1,2,K =L1L⁻¹₂ modulo EF−FE =K −K⁻¹

q−q⁻¹ , KE =q²EK, KF =q⁻²FK.

Roughly:K =q^H and limq→1U_q(gl₂) =U(gl₂).

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Diagrams for intertwiners

Observe that there are (up to scalars) uniqueU_q(gl₂)-intertwiners cap:C²_q⊗C²_q →C_q and cup:C_q→C²_q⊗C²_q, projectingC²_q⊗C²_q ontoC_q respectively embeddingC_q intoC²_q⊗C²_q.

Letgl₂-Mod_e be the (braided) monoidal,C_q-linear category whose objects are tensor generated byC²_q. Define a functor Γ : 2-Web_g→gl₂-Mod_e:

On objects:~k = (1, . . . ,1) is send to (C²_q)^⊗k =C²_q⊗ · · · ⊗C²_q. On morphisms:

1 1

7→cap ,

1 1

7→cup

Theorem(Folklore)

Γ : 2-Web^⊕_g →gl₂-Mod_e is an equivalence of (braided) monoidal categories.

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The main step beyond gl

₂

: trivalent vertices

AnN-webis an oriented, labeled, trivalent graph locally made of

m^k+l_k,l =

k+l

k l

, s^k,l_k+l=

k+l

k l

k,l,k+l ∈N

(and no pivotal things today).

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer Its cousins: theN-webs May 2015 7 / 29

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Let us try the same for gl

_N

: the N -web space

Define the (braided) monoidal,C_q-linear categoryN-Web_g by using:

Definition(Cautis-Kamnitzer-Morrison 2012)

TheN-web space HomN-Web_g(~k,~l) is the freeC_q-vector space generated by N-webs with~k and~l at the bottom and top modulo isotopies and:

“glm ladder” relations like

l k

k−1 l+1 1 1

−

k l

k+1 l−1

1 1

= [k−l]

l k

Theexterior relations:

k = 0 , ifk>N.

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Diagrams for intertwiners - Part 2

Observe that there are (up to scalars) uniqueU_q(glN)-intertwiners

m^k+l_k,l :^V^k_qC^N_q ⊗^V^l_qC^N_q →^V^k+l_q C^N_q and s^k,l_k+l: ^V^k_q^+lC^N_q →^V^k_qC^N_q ⊗^V^l_qC^N_q given by projection and inclusion again.

Letgl_N-Mod_e be the (braided) monoidal,C_q-linear category whose objects are tensor generated by^V^k_qC^N_q. Define a functor Γ :N-Web_g→gl_N-Mod_e:

On objects:~k = (k1, . . . ,km) is send to^V^k_q¹C^N_q ⊗ · · · ⊗^V^k_q^mC^N_q. On morphisms:

k+l

k l

7→m^k+l_k,l ,

k+l

k l

7→s^k,l_k+l

Theorem(Cautis-Kamnitzer-Morrison 2012)

Γ :N-Web^⊕_g →gl_N-Mod_e is an equivalence of (braided) monoidal categories.

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“Howe” to prove this?

Howe: the commuting actions ofU_q(glm) andU_q(glN) on

VK

q(C^m_q ⊗C^N_q)∼= M

k1+···+km=K

(^V^k_q¹C^N_q ⊗ · · · ⊗^V^k_q^mC^N_q)

∼= M

l1+···+lN=K

(^V^l_q¹C^m_q ⊗ · · · ⊗^V^l_q^NC^m_q)

introduce anU_q(gl_m)-actionf on the first term with~k-weight space^V^~^k_qC^N_q. In particular, there is a functorial action

Φ^m_skew: ˙U_q(glm)→gl_N-Mod_e,

~k 7→^V^~^k_qC^N_q, X ∈1~lU_q(gl_m)1~k 7→f(X)∈Homgl_N-Mod_e(^V^~^k_qC^N_q,^V^~^l_qC^N_q).

Howe: Φ^m_skewis full. Or in words: all relations ingl_N-Mod_e follow from the ones in

˙

U_q(gl_m) and the ones in the kernel of Φ^m_skew.

Daniel Tubbenhauer Proof? Skew quantum Howe duality! May 2015 10 / 29

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Define the diagrams to make this work

Theorem(Cautis-Kamnitzer-Morrison 2012)

DefineN-Web_g such there is a commutative diagram U˙_q(glm) ^Φ

m

skew //

Υ^m

%%

❑❑

gl_N-Mod_e

N-Web_g

Γ

99

rr rr rr rr rr r

with

Υ^m(Fi1~k)7→

ki ki+1

ki−1 ki+1 +1

1 , Υ^m(Ei1~k)7→

ki+1 ki

ki+1−1 ki+1

1

Υ^minduces the “gl_m ladder” relations, ker(Υ^m) gives the exterior relations.

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Exempli gratia

The “glmladder” relation

l k

k−1 l+1 1 1

−

k l

k+1 l−1

1 1

= [k−l]

l k

is just

EF1_~_k−FE1_~_k = [k−l]1_~_k. The exterior relations are a diagrammatic version of

V>N

q C^N_q ∼= 0.

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The symmetric story is easier in some sense...

An 2-web is a labeled, trivalent graph locally made of

cap_k =

k k

, cup_k =

k k

, m^k+l_k,l =

k+l

k l

, s^k,l_k+l=

k+l

k l

Up to sign issues that I ignore today!

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer More cousins: the green 2-webs May 2015 13 / 29

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Never change a winning team

Define the (braided) monoidal,C_q-linear category 2-Web_rby using:

Definition

Given~k ∈Zⁿ

≥0and~l∈Zⁿ^′

≥0. The 2-web spaceHom2-Web_r(~k,~l) is the free C_q-vector space generated by 2-webs between~k and~l modulo isotopies and:

The “gl_nladder” relations again!

A circle evaluation and thedumbbell relation:

[2]

1 1

=

1 1

+

1 1

2

Butno(!)relation of the form

k = 0 , ifk>N.

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Diagrams for intertwiners - Part 3

Observe that there are (up to scalars) uniqueU_q(gl₂)-intertwiners

cap_k:Sym^k_qC²_q⊗Sym^k_qC²_q→C_q , m^k+l_k,l :Sym^k_qC²_q⊗Sym^l_qC²_q→Sym^k+l_q C²_q cupk:C_q→Sym^k_qC²_q⊗Sym^k_qC²_q , s^k,l_k+l:Sym^k+l_q C²_q→Sym^k_qC²_q⊗Sym^l_qC²_q (guess where they come from...)

Letgl₂-Mod_s be the (braided) monoidal, C_q-linear category whose objects are tensor generated bySym^k_qC^N_q. Define a functor Γ : 2-Web_r→gl₂-Mod_s:

On objects:~k = (k1, . . . ,kn) is send toSym^k_q¹C²_q⊗ · · · ⊗Sym^k_qⁿC²_q. On morphisms:

k k

7→cap_k ,

k k

7→cup_k ,

k+l

k l

7→m^k+l_k,l ,

k+l

k l

7→s^k_k^,l_+l

Theorem

Γ : 2-Web^⊕_r →gl₂-Mod_s is an equivalence of (braided) monoidal categories.

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“Howe” to prove this?

Howe: the commuting actions ofU_q(gl_n) andU_q(gl_N) on Sym^K_q(Cⁿ_q⊗C^N_q)∼= M

k₁+···+kn=K

(Sym^k_q¹C^N_q ⊗ · · · ⊗Sym^k_qⁿC^N_q)

∼= M

l1+···+lN=K

(Sym^l_q¹Cⁿ_q⊗ · · · ⊗Sym^l_q^NCⁿ_q)

introduce anU_q(gln)-actionf on the first term with~k-weight spaceSym^~^k_qC^N_q. In particular, there is a functorial action

Φⁿ_sym: ˙U_q(gl_n)→gl₂-Mod_s,

~k 7→Sym^~^k_qC²_q, X ∈1~lU_q(gln)1~k 7→f(X)∈Homgl₂-Mod_s(Sym^~^kqC²_q,Sym^~^l_qC²_q).

Howe: Φⁿ_sym is full. Or in words: all relations ingl₂-Mod_s follow from the ones in U˙_q(gln) and the ones in the kernel of Φⁿsym.

Daniel Tubbenhauer Proof? Symmetric quantum Howe duality! May 2015 16 / 29

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Copy-paste

Theorem

Define 2-Web_rsuch that there is a commutative diagram U˙_q(gl_n) ^Φ

n

sym //

Υⁿ

%%

❏❏

❏

gl₂-Mod_s

2-Web_r

Γ

99

ss ss ss ss ss

with

Υⁿ(Fi1~k)7→

k l

k−1 l+1

1 , Υⁿ(Ei1~k)7→

l k

l−1 k+1

1

Υⁿinduces the “gln ladder” relations, ker(Υⁿ) gives the circle/dumbbell relation.

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Exempli gratia

The dumbbell relation

[2]

1 1

=

1 1

+

1 1

2

is a diagrammatic version of

C²_q⊗C²_q∼=C_q⊕Sym²_qC²_q. No relations of the form

k = 0 , ifk>N,

because

Sym^>N_q C^N_q 6∼= 0.

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Could there be a pattern?

An green-redN-web is a colored, labeled, trivalent graph locally made of

m^k+l_k,l =

k+l

k l

, m^k+l_k,l =

k+l

k l

, m^k+l_k_,1 =

k+ 1

k 1

, m^k+l_k,1 =

k+ 1

k 1

And of course splits and some mirrors as well!

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer Even more cousins: the green-redN-webs May 2015 19 / 29

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The green-red N -web category

Define the (braided) monoidal,C_q-linear categoryN-Web_grby using:

Definition

Given~k ∈Z^m+n

≥0 ,~l ∈Z^m^′⁺ⁿ^′

≥0 . The green-redN-web space HomN-Web_gr(~k,~l) is the freeC_q-vector space generated byN-webs between~k and~l modulo isotopies and:

The “glm+glnladder” relations.

The dumbbell relation:

[2]

1 1

=

1 1

2 +

1 1

2

Theexterior relations:

k = 0 , ifk>N.

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Diagrams for intertwiners - Part 4

Observe that there are (up to scalars) uniqueU_q(slN)-intertwiners

m^k+1_k,1 :^V^k_qC^N_q ⊗C^N_q →^V^k_q⁺¹C^N_q and m^k+1_k,1 :Sym^k_qC^N_q ⊗C^N_q →Sym^k+1_q C^N_q plus others as before.

Letgl_N-Mod_es be the (braided) monoidal,C_q-linear category whose objects are tensor generated by^V^k_qC^N_q,Sym^k_qC^N_q. Define a functor Γ :N-Web_gr→gl_N-Mod_es:

On objects:~k = (k1, . . . ,km+n) is send to^V^k_q¹C^N_q ⊗ · · · ⊗Sym^k_q^m+nC^N_q. On morphisms:

k+1

k 1

7→m^k+1_k,1 ,

k+1

k 1

7→m^k+1_k,1 , · · ·

Theorem

Γ :N-Web^⊕_gr→gl_N-Mod_esis an equivalence of (braided) monoidal categories.

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Super gl( m | n )

Definition

Thequantum general linear superalgebraU_q(gl(m|n)) is generated byL^±1_i and Fi,Ei subject the some relations, most notably, thesuper relations:

F_m² = 0 =E_m² , LmL⁻¹_m+1−L⁻¹_m Lm+1

q−q⁻¹ =FmEm+EmFm, [2]FmFm+1Fm−1Fm=FmFm+1FmFm−1+Fm−1FmFm+1Fm

+Fm+1FmFm−1Fm+FmFm−1FmFm+1 (plus an E version).

There is a Howe pair (U_q(gl(m|n)),U_q(glN)) with~k = (k1, . . . ,km+n)-weight space under theU_q(gl(m|n))-action on^V^K_q(C^m|n_q ⊗C^N_q) given by

Vk1

qC^N_q ⊗ · · ·^V^k_q^mC^N_q ⊗Sym^k_q^m+1C^N_q ⊗ · · · ⊗Sym^k_q^m+nC^N_q.

Daniel Tubbenhauer Proof? Super quantum Howe duality! May 2015 22 / 29

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Define the diagrams to make this work

Theorem

DefineN-Web_gr such there is a commutative diagram U˙_q(gl(m|n)) ^Φ

m|n

su //

Υ◆^m|n_su◆◆◆◆◆◆◆&&

◆◆

◆

gl_N-Mod_es

N-Web_gr

Γ

88

qq qq qq qq qq q

with

Υ^m|n_su (Fm1~k)7→

km km+1

km−1 km+1 +1

1 , Υ^m|n_su (Em1~k)7→

km+1 km

km+1−1 km+1

1

Υ^m|nsu induces the “gl(m|n) ladder” relations, ker(Υ^m|nsu ) gives the exterior relations.

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Exempli gratia

The dumbbell relation is the super commutator relation:

[2]1(1,1)=FmEm1(1,1)+EmFm1(1,1)

[2]

1 1

=

1 1

2 0

1 1

+

1 1

2 0

1 1

C^N_q ⊗C^N_q ∼=^V²qC^N_q ⊕Sym²_qC^N_q. All other super relations are consequences!

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Another meal for our machine

Howe: the commuting actions ofU_q(gl(m|n)) and U_q(gl(N|M)) on

VK

q(C^m|n_q ⊗C^N|M_q )∼= M

k1+···+kn=K

(^V^~^k_q⁰C^N|M_q ⊗Sym^~^k_q¹C^N|M_q )

∼= M

l1+···+lN=K

(^V^~^lq⁰C^m|n_q ⊗Sym^~^lq¹C^m|n_q )

introduce anU_q(gl(m|n))-actionf with~k-weight space^V^~^k_q⁰C^N|M_q ⊗Sym^~^k_q¹C^N|M_q . In particular, there is a functorial action

Φ^m|n_susu: ˙U_q(gl(m|n))→gl(N|M)-Mod_es,

~k 7→^V^~^k_q⁰C^N|M_q ⊗Sym^~^k_q¹C^N|M_q , etc..

Howe: Φ^m|nsusuis full. Or in words: all relations ingl(N|M)-Mod_es follow from the ones in ˙U_q(gl(m|n)) and the ones in the kernel of Φ^m|nsusu.

Daniel Tubbenhauer Super-Super duality and even more cousins May 2015 25 / 29

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The definition of the diagrams is already determined

Theorem

DefineN|M-Web_gr such there is a commutative diagram

˙

U_q(gl(m|n)) ^Φ

m|n

susu //

Υ❖❖^m|n_susu❖❖❖❖❖❖❖''

❖❖

gl(N|M)-Mod_es

N|M-Web_gr

Γ

66

♥♥

♥

with

Υ^m|n_susu(Fm1~k)7→

km km+1

km−1 km+1 +1

1 , Υ^m|n_susu(Em1~k)7→

km+1 km

km+1−1 km+1

1

Υ^m|nsusuinduces “gl(m|n) ladder” relations, ker(Υ^m|nsusu) gives a “not-a-hook” relation.

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The machine spits this out

The (braided) monoidal,C_q-linear categoryN|M-Web_gr by using:

Definition

Given~k ∈Z^m+n

≥0 and~l∈Z^m^′⁺ⁿ^′

≥0 . TheN|M-web space HomN|M-Web_gr(~k,~l) is the freeC_q-vector space generated byN|M-webs between~k,~l modulo isotopies and:

The “gl_m+gl_nladder” relations.

The dumbbell relation:

[2]

1 1

=

1 1

2 +

1 1

2

Thenot-a-hook relations (given by killing an idempotent corresponding to a box-shaped Young diagram).

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Some concluding remarks

TakingN,M→ ∞, one obtains a diagrammatic presentation∞-Web_grof some form of the Hecke algebroid. Roughly: the machine spits it out, if you feed it with Schur-Weyl duality.

∞-Web_gr is completely symmetric in green-red which allows us to prove a symmetry of HOMFLY-PT polynomials

P^a,q(L(~λ)) = (−1)^co P^a,q⁻¹(L(~λ^T)).

diagrammatically.

Homework: feed the machine with you favorite duality (e.g. Howe dualities in other types) and see what it spits out.

Everything is amenable to categorification!

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There is stillmuchto do...

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Thanks for your attention!