Webs and q-Howe dualities in types BCD

Webs and q -Howe dualities in types BCD

Or: A story about “howe” I failed Daniel Tubbenhauer

5 8

3 4

3 9

3 6

3 1

1 1 1 1

7 -web

A-web new part

1 1

2 3 2

2 2

7 9

2 2

6 5

6 4

1 2

-web

A-web new part

Joint work with Antonio Sartori

1 The typeAstory

Classical Schur-Weyl duality Howe’s dualities in typeA

2 The typeBCD story

Classical Schur-Weyl-Brauer duality Howe’s dualities in typesBCD

3 The quantum story Various quantizations Concluding remarks

A pioneer of representation theory

Schur’s remarkable relationship between gl_nand the symmetric groupSk:

Schur∼1901. LetV=V^gl =Cⁿ. There are commuting actions U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

generating each other’s centralizer. TheU(gl_n)-C[S_k]-bimodule decomposes as M

λ∈P

L(gl_n, λ)⊗L(S_k, λ^T).

Theλ’s are partitions (Young diagrams) of k with at mostnrows.

First statement

Second statement Third statement

The precise form does not matter for today. It is only important that one can make it explicit.

A pioneer of representation theory

Schur’s remarkable relationship between gl_nand the symmetric groupSk:

Schur∼1901. LetV=V^gl =Cⁿ. There are commuting actions U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

generating each other’s centralizer. TheU(gl_n)-C[S_k]-bimodule decomposes as M

λ∈P

L(gl_n, λ)⊗L(S_k, λ^T).

Theλ’s are partitions (Young diagrams) of k with at mostnrows.

First statement

Second statement Third statement

The precise form does not matter for today. It is only important that one can make it explicit.

A pioneer of representation theory

Schur’s remarkable relationship between gl_nand the symmetric groupSk:

Schur∼1901. LetV=V^gl =Cⁿ. There are commuting actions U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

generating each other’s centralizer. TheU(gl_n)-C[S_k]-bimodule decomposes as M

λ∈P

L(gl_n, λ)⊗L(S_k, λ^T).

Theλ’s are partitions (Young diagrams) of k with at mostnrows.

First statement

Second statement

Third statement

The precise form does not matter for today. It is only important that one can make it explicit.

A pioneer of representation theory

Schur’s remarkable relationship between gl_nand the symmetric groupSk:

Schur∼1901. LetV=V^gl =Cⁿ. There are commuting actions U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

generating each other’s centralizer. TheU(gl_n)-C[S_k]-bimodule decomposes as M

λ∈P

L(gl_n, λ)⊗L(S_k, λ^T).

Theλ’s are partitions (Young diagrams) of k with at mostnrows.

First statement

Second statement Third statement

The precise form does not matter for today. It is only important that one can make it explicit.

A pioneer of representation theory

Schur’s remarkable relationship between gl_nand the symmetric groupSk:

Schur∼1901. LetV=V^gl =Cⁿ. There are commuting actions U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

generating each other’s centralizer. TheU(gl_n)-C[S_k]-bimodule decomposes as M

λ∈P

L(gl_n, λ)⊗L(S_k, λ^T).

Theλ’s are partitions (Young diagrams) of k with at mostnrows.

First statement

Second statement Third statement

The precise form does not matter for today. It is only important that one can make it explicit.

The diagrammatic presentation machine

U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

fix use

Schur’sfirst statementgives a functor

S Φ Rep(gl_n)

Categorical version of the symmetric group

The diagrammatic presentation machine

U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

fix use

Schur’ssecond statementgives a full functor

S Φ Rep(gl_n) full

Categorical version of the symmetric group

The diagrammatic presentation machine

U(gl_n)

V⊗ · · · ⊗V

| {z }

ktimes

C[S_k]

fix use

Schur’sthird statement gives a full functor

S Rep(gl_n)

S/“ ker(Φ)” Rep(gln) Φ

full

Φ fully faithful

whose “kernel ker(Φ)” can be calculated.

Hence, up to taking duals and Karoubi closures,Schur gave us a diagrammatic

presentation of the representation categoryRep(gl_n) ofgl_n.

Categorical version of the symmetric group

“Thick” Schur-Weyl duality

One of Howe’s remarkable relationships betweengl_nandgl_k: Howe∼1975. LetV=Cⁿ. There are commuting actions

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

generating each other’s centralizer, andVa1V⊗ · · · ⊗VakV is the (a1, . . . ,ak)th weight space as regardsU(gl_k). TheU(gl_n)-U(gl_k)-bimodule decomposes as

M

λ∈P

L(gl_n, λ)⊗L(gl_k, λ^T).

Theλ’s are partitions with at mostk columns and nrows.

1¹/₂th statement

“Thick” Schur-Weyl duality

One of Howe’s remarkable relationships betweengl_nandgl_k: Howe∼1975. LetV=Cⁿ. There are commuting actions

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

generating each other’s centralizer, andVa1V⊗ · · · ⊗VakV is the (a1, . . . ,ak)th weight space as regardsU(gl_k). TheU(gl_n)-U(gl_k)-bimodule decomposes as

M

λ∈P

L(gl_n, λ)⊗L(gl_k, λ^T).

Theλ’s are partitions with at mostk columns and nrows.

1¹/2th statement

Again: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

fix use

Howe’sfirst statementgives a functor

U(gl˙ _k) Φ^ext_A Rep(gln)

Dot version generated by weight space idempotents 1λ,

and Eiand Fi

Again: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

fix use

Howe’ssecond statementgives a full functor

U(gl˙ _k) Φ^ext_A Rep(gln) full

Dot version generated by weight space idempotents 1λ,

and Eiand Fi

Again: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

fix use

Howe’sthird statementgives a full functor

U(gl˙ _k) Rep(gln)

U(gl˙ _k)/“ ker(Φ^ext_A )” Rep(gl_n) Φ^ext_A

full

Φ^ext_A fully faithful whose “kernel ker(Φ^ext_A )” we can calculate.

Dot version generated by weight space idempotents 1λ,

and Eiand Fi

Again: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(gl_k)

fix use

Howe’s1¹/2th statement ^defines a diagrammatic categoryWeb such that

U(gl˙ _k) Rep(gl_n)

S Web

Φ^ext_A full

Γ^ext_A full β_A

fully faithful

commutes. In particular,Web is a ^thick version of the symmetric group.

Dot version generated by weight space idempotents 1λ,

and Eiand Fi

The presentation functor

Observe that there are (up to scalars) uniqueU(gl_n)-intertwiners

a+b

a,b :VaV⊗VbVVa+bV, ^a,b_a+b: Va+bV,→VaV⊗VbV given by projection and inclusion.

The presentation functor is

Γ^ext_A :Web →Rep(gl_n), a7→Va

V,

a a+b

b

7→ ^a+ba,b ,

a

a+b b

7→ ^a,ba+b

The (co)associativity relations say thatV•V is a (co)algebra with (co)multiplication ^a+b_a,b ( ^a,b_a+b).

We can play the game the other way around as well by defining Howe’s action via:

a a+1

b b−1

E =

a a+1

1 b−1

b−1

◦

a

a 1 b−1

VaV⊗VbV→VaV⊗V⊗Vb−1V→Va+1b V⊗Vb−1V etc.

The presentation functor

Observe that there are (up to scalars) uniqueU(gl_n)-intertwiners

a+b

a,b :VaV⊗VbVVa+bV, ^a,b_a+b: Va+bV,→VaV⊗VbV given by projection and inclusion.

The presentation functor is

Γ^ext_A :Web →Rep(gl_n), a7→Va

V,

a a+b

b

7→ ^a+ba,b ,

a

a+b b

7→ ^a,ba+b

The (co)associativity relations say thatV•V is a (co)algebra with (co)multiplication ^a+b_a,b ( ^a,b_a+b).

We can play the game the other way around as well by defining Howe’s action via:

a a+1

b b−1

E =

a a+1

1 b−1

b−1

◦

a

a 1 b−1

VaV⊗VbV→VaV⊗V⊗Vb−1V→Va+1b V⊗Vb−1V etc.

The presentation functor

Observe that there are (up to scalars) uniqueU(gl_n)-intertwiners

a+b

a,b :VaV⊗VbVVa+bV, ^a,b_a+b: Va+bV,→VaV⊗VbV given by projection and inclusion.

The presentation functor is

Γ^ext_A :Web →Rep(gl_n), a7→Va

V,

a a+b

b

7→ ^a+ba,b ,

a

a+b b

7→ ^a,ba+b

The (co)associativity relations say thatV•V is a (co)algebra with (co)multiplication ^a+b_a,b ( ^a,b_a+b).

We can play the game the other way around as well by defining Howe’s action via:

a a+1

b b−1

E =

a a+1

1 b−1

b−1

◦

a

a 1 b−1

VaV⊗VbV→VaV⊗V⊗Vb−1V→Va+1b V⊗Vb−1V etc.

Another pioneer of representation theory

Brauer’s remarkable relationship betweengn=son,sp_nand the Brauer algebraBr^k_n:

Brauer∼1937. LetV=Cⁿ. There are commuting actions U(g_n)

V⊗ · · · ⊗V

| {z }

ktimes

Br^k_n

generating each other’s centralizer. TheU(g_n)-Br^k_n-bimodule decomposes as M

λ∈P

L(g_n, λ)⊗L(Br^k_n, λ^T).

Theλ’s are partitions ofk,k−2,k−4, . . . whose precise form depend ongn.

Be careful: One needs to work witho_nin typeD. Today, I silently stay withso_n, and thus, in typeB.

Another pioneer of representation theory

Brauer’s remarkable relationship betweengn=son,sp_nand the Brauer algebraBr^k_n:

Brauer∼1937. LetV=Cⁿ. There are commuting actions U(g_n)

V⊗ · · · ⊗V

| {z }

ktimes

Br^k_n

generating each other’s centralizer. TheU(g_n)-Br^k_n-bimodule decomposes as M

λ∈P

L(g_n, λ)⊗L(Br^k_n, λ^T).

Theλ’s are partitions ofk,k−2,k−4, . . . whose precise form depend ongn.

Be careful: One needs to work witho_nin typeD.

Today, I silently stay withso_n, and thus, in typeB.

The diagrammatic presentation machine – it still works fine

U(g_n)

V⊗ · · · ⊗V

| {z }

ktimes

Br^k_n

fix use

As usual, Brauer’s insights give a full functor

Br_n Rep(g_n)

Br_n/“ ker(Φ)” Rep(g_n)

Φ full

Φ fully faithful

whose “kernel ker(Φ)” can be calculated.

Hence, up to Spin’s and Karoubi closures,Brauer gave us a diagrammatic

Categorical version of the Brauer algebra

“Thick” Schur-Weyl-Brauer duality

Another one of Howe’s remarkable relationships:

Howe∼1975. LetV=Cⁿ. There are commuting actions U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

generating each other’s centralizer, andVa1V⊗ · · · ⊗VakV is the (a1, . . . ,ak)th weight space ofU(so_2k). TheU(so_n)-U(so_2k)-bimodule decomposes as

M

λ∈P

L(so_n, λ)⊗L(so_2k,Pk

j=1(λ^T_j −ⁿ/₂)ε_j).

Theλ’s again satisfy certain explicit conditions andai =ai+ⁿ/2.

Note that the action ofU(so_2k) is not as clear as it was forU(gl_k)!

“Thick” Schur-Weyl-Brauer duality

Another one of Howe’s remarkable relationships:

Howe∼1975. LetV=Cⁿ. There are commuting actions U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

generating each other’s centralizer, andVa1V⊗ · · · ⊗VakV is the (a1, . . . ,ak)th weight space ofU(so_2k). TheU(so_n)-U(so_2k)-bimodule decomposes as

M

λ∈P

L(so_n, λ)⊗L(so_2k,Pk

j=1(λ^T_j −ⁿ/₂)ε_j).

Theλ’s again satisfy certain explicit conditions andai =ai+ⁿ/2.

Note that the action ofU(so_2k) is not as clear as it was forU(gl_k)!

Still alive: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V U(gl_k) U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

⊂ ⊃=

fix use

Howe’s1¹/2th statement ^defines a diagrammatic categoryWeb such that

U(so˙ _2k) Rep(son)

Br_n Web Φ^ext_BD

full

Γ^ext_BD full fully faithful

β

commutes. In particular,Web is a ^thick version of the Brauer algebra.

Restricting the action on one side

Increases the centralizer on the other Hence, we get

“old diagram generators” and

“new diagram generators”

Still alive: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V U(gl_k) U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

⊂ ⊃=

fix use

Howe’s1¹/2th statement ^defines a diagrammatic categoryWeb such that

U(so˙ _2k) Rep(son)

Br_n Web Φ^ext_BD

full

Γ^ext_BD full fully faithful

β

commutes. In particular,Web is a ^thick version of the Brauer algebra.

Restricting the action on one side

Increases the centralizer on the other Hence, we get

“old diagram generators” and

“new diagram generators”

Still alive: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V U(gl_k) U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

⊂ ⊃=

fix use

Howe’s1¹/2th statement ^defines a diagrammatic categoryWeb such that

U(so˙ _2k) Rep(son)

Br_n Web Φ^ext_BD

full

Γ^ext_BD full fully faithful

β

commutes. In particular,Web is a ^thick version of the Brauer algebra.

Restricting the action on one side

Increases the centralizer on the other

Hence, we get

“old diagram generators” and

“new diagram generators”

Still alive: The diagrammatic presentation machine

U(gl_n) V•V⊗ · · · ⊗V•V U(gl_k) U(so_n) V•V⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

⊂ ⊃=

fix use

Howe’s1¹/2th statement ^defines a diagrammatic categoryWeb such that

U(so˙ _2k) Rep(son)

Br_n Web Φ^ext_BD

full

Γ^ext_BD full fully faithful

β

commutes. In particular,Web is a ^thick version of the Brauer algebra.

Restricting the action on one side

Increases the centralizer on the other Hence, we get

“old diagram generators”

and

“new diagram generators”

Some delicate quantizations

U(gl_n) V_•

V ⊗ · · · ⊗V_•

V U(gl_k) U(so_n) V•V ⊗ · · · ⊗V•V

| {z }

ktimes

U(so_2k)

⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U(gl_k) U(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U(so_2k)

⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U(so_2k)

=

Quantum skew Howe duality:

Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

6⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

6 ???

6⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

6 ???

6⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n].

The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB web relations∼1995.

We wanted to generalize Kuperberg’s

results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

6 ???

6⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n].

The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB web relations∼1995.

We wanted to generalize Kuperberg’s

results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do.

Using a coideal subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U⁰_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do.

Using a coideal subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Some delicate quantizations

U_q(gl_n) V_•

qV_q⊗ · · · ⊗V_•

qV_q U_q(gl_k) U⁰_q(so_n) V•

qV_q⊗ · · · ⊗V• qV_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

Using aq-monoidal diagrammatic category Web_q,qn we can ^define a full Howe functor Φ^ext_BD such that we get a commuting diagram

U˙_q(so_2k) Rep⁰_q(so_n)

Brq,qⁿ Web_q,qn

Φ^ext_BD

Γ^ext_BD fully faithful

β

define

Quantum skew Howe duality: Lehrer–Zhang–Zhang∼2009.

(But its quite easy and not their main point.)

Does not quantize! Quantizes easily

No action at all. Action unclear.

The quantum dimension ofV^gl_q is [n]. The quantum dimension ofV^so_q is [n−1]+1.

Hence,V^soq does not come fromV^glq ! This “flaw” propagates all the way through:Va

qV^soq have “weird” quantum dimensions.

The quantum dimension ofVso5

↑ q

Above: Kuperberg’sB2web relations∼1995. We wanted to

generalize Kuperberg’s results. We failed because quantization

is hard outside of typeA.

But let me explain what we can do. Using a coideal

subalgebra does the trick.

The action is constructed using

the unquantized diagrammatics.

Further directions

U_q(gl_n) V_•

qV^gl_q ⊗ · · · ⊗V_•

qV^gl_q U_q(gl_k) U⁰_q(so_n) V•

qV^gl_q ⊗ · · · ⊗V• qV^gl_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)

I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.

I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.

I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)

∗=ⁿ⁻¹/2for typeB,

∗=ⁿ/2for typeD.

This should give the quantum group story, but it is much trickier since e.g.

V^so_q ∼=V^gl_q ⊕(V_q^gl)^∗⊕C asU_q(gl_∗)-modules in typeB.

Thus, the above is not the usualU(gl_∗)-U(gl_k) duality.

a a b b

=q^∗

a a b b

q-interchange law

∗=some power depending ona,b c

b a+b a a+b

+c

Singular cobordisms (“foams”,

`

a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.

Theq-monoidal property has to be smartly encoded.

2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli

whose pictures I shamelessly stole.)

Further directions

U_q(gl_∗) V_•

qV_q^so⊗ · · · ⊗V_•

qV^so_q U_q(gl_k) U_q(so_n) V•

qV_q^so⊗ · · · ⊗V• qV^so_q

| {z }

ktimes

?U⁰_q(so_2k)?

⊂

⊃ =

I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)

I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.

I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.

I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)

∗=ⁿ⁻¹/₂for typeB,

∗=ⁿ/2for typeD.

This should give the quantum group story, but it is much trickier since e.g.

V^so_q ∼=V^gl_q ⊕(V_q^gl)^∗⊕C asU_q(gl_∗)-modules in typeB.

Thus, the above is not the usualU(gl_∗)-U(gl_k) duality.

a a b b

=q^∗

a a b b

q-interchange law

∗=some power depending ona,b c

b a+b a a+b

+c

Singular cobordisms (“foams”,

`

a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.

Theq-monoidal property has to be smartly encoded.

2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli

whose pictures I shamelessly stole.)

Further directions

U_q(gl_n) V_•

qV^gl_q ⊗ · · · ⊗V_•

qV^gl_q U_q(gl_k) U⁰_q(so_n) V•

qV^gl_q ⊗ · · · ⊗V• qV^gl_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)

I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.

I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.

I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)

∗=ⁿ⁻¹/2for typeB,

∗=ⁿ/2for typeD.

This should give the quantum group story, but it is much trickier since e.g.

V^so_q ∼=V^gl_q ⊕(V_q^gl)^∗⊕C asU_q(gl_∗)-modules in typeB.

Thus, the above is not the usualU(gl_∗)-U(gl_k) duality.

a a b b

=q^∗

a a b b

q-interchange law

∗=some power depending ona,b

c b a+b a a+b

+c

Singular cobordisms (“foams”,

`

a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.

Theq-monoidal property has to be smartly encoded.

2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli

whose pictures I shamelessly stole.)

Further directions

U_q(gl_n) V_•

qV^gl_q ⊗ · · · ⊗V_•

qV^gl_q U_q(gl_k) U⁰_q(so_n) V•

qV^gl_q ⊗ · · · ⊗V• qV^gl_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)

I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.

I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.

I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)

∗=ⁿ⁻¹/2for typeB,

∗=ⁿ/2for typeD.

This should give the quantum group story, but it is much trickier since e.g.

V^so_q ∼=V^gl_q ⊕(V_q^gl)^∗⊕C asU_q(gl_∗)-modules in typeB.

Thus, the above is not the usualU(gl_∗)-U(gl_k) duality.

a a b b

=q^∗

a a b b

q-interchange law

∗=some power depending ona,b

c b a+b a a+b

+c

Singular cobordisms (“foams”,

`

a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.

Theq-monoidal property has to be smartly encoded.

2-q-monoidal foams. (Maybe connected to Beliakova–Putyra–Wehrli

whose pictures I shamelessly stole.)

Further directions

U_q(gl_n) V_•

qV^gl_q ⊗ · · · ⊗V_•

qV^gl_q U_q(gl_k) U⁰_q(so_n) V•

qV^gl_q ⊗ · · · ⊗V• qV^gl_q

| {z }

ktimes

U_q(so_2k)

⊂ ⊃=

I Use a similar approach to get the quantum group to work. (Needs probably some mixed Howe duality `a la Queffelec–Sartori.)

I q-monoidal categories are probably very useful to study representation categories of coideal subalgebras. An abstract formulation `a la Brundan–Ellis (“super monoidal”) should be useful.

I Coideal subalgebras are amenable to categorification, cf. Ehrig–Stroppel or Bao–Shan–Wang–Webster. Similarly, their representation categories should be amenable to categorification.

I Formulate everything in a “2-q-monoidal language”. (Again, `a la Brundan–Ellis.)

∗=ⁿ⁻¹/2for typeB,

∗=ⁿ/2for typeD.

This should give the quantum group story, but it is much trickier since e.g.

V^so_q ∼=V^gl_q ⊕(V_q^gl)^∗⊕C asU_q(gl_∗)-modules in typeB.

Thus, the above is not the usualU(gl_∗)-U(gl_k) duality.

a a b b

=q^∗

a a b b

q-interchange law

∗=some power depending ona,b c

b a+b a a+b

+c

Singular cobordisms (“foams”,

`

a la Khovanov–Rozansky and Mackaay–Stoˇsi´c–Vaz) categorify webs.

Theq-monoidal property has to be smartly encoded.

2-q-monoidal foams.

(Maybe connected to Beliakova–Putyra–Wehrli whose pictures I shamelessly stole.)