from schur-weyl duality to quantum symmetric pairs · schur-weyl duality. . . . . . . quantization....

. . . . . .Schur-Weyl duality

. . . . . . .Quantization

. . . . . . . . .q-Schur duality

. . . . . . .BLM construction

. . . . . . .Quantum symmetric pairs

From Schur-Weyl duality to quantum symmetric pairs

Chun-Ju Lai

Max Planck Institute for Mathematics in Bonn

[email protected]

Dec 8, 2016






Outline

..1 Schur-Weyl duality

..2 Quantization

..3 q-Schur duality

..4 BLM construction

..5 Quantum symmetric pairs






Background

• GLn := GLn(C) = general linear group of Cn

• Schur’s 1901 dissertation on the polynomial representations of GLn:− Each polynomial representation V is semi-simple

(i.e. V =⊕

irreducible modules)− Each polynomial representation is homogeneous for some degree d ∈ N− There is a correspondence{

polynomial representationof GLn of degree d

}1:1←→

{representation of

symmetric group Sd

}{irreducibles} ↔ {irreducibles}↔

{partitions λ = (λ1, . . . , λn) ⊢ d}






Background

• Weyl’s 1926 research on representation of semi-simple Lie group− based on repn theory of Lie algebra, integration over compact form− has no counterpart to Sd in Schur’s method

• In 1927, Schur re-derived his 1901 dissertation by showing the doublecentralizer property for (GLn,Sd), which was publicized by Weyl in his1939 book “The Classic Groups”.

• The methods used in the classical Schur-Weyl are still important inrepresentation theory today






Dual actions

• gln := gln(C): general linear Lie algebraV := Cn natural representation of gln⇒ action on V ⊗d by the Leibniz rule, e.g.,

g · (v1 ⊗ v2) = (g · v1)⊗ v2 + v1 ⊗ (g · v2)

so that its exponential etg acts group-like –

etg · (v1 ⊗ v2) = etg · v1 ⊗ etg · v2

• Sd has a (right) action on V ⊗d by permuting tensor factors, e.g.,

(v1 ⊗ v2) · (12) = v2 ⊗ v1

glnψ↷ V ⊗d

φ↶ Sd

general linear Lie algebra tensor space symmetric group






Example: commutivity

Here’s an example showing(g(v1 ⊗ v2)

)(12) = g

((v1 ⊗ v2)(12)

)V ⊗ V

(12) - V ⊗ V

v1 ⊗ v2 - v2 ⊗ v1

⟳

gv1 ⊗ v2 + v1 ⊗ gv2?

- gv2 ⊗ v1 + v2 ⊗ gv1?

V ⊗ V

g

?

(12)- V ⊗ V

g

?






Double Centralizer Property

• We have ψ : gln → End(V ⊗d), φ : Sd → End(V ⊗d) and

glnψ↷ V ⊗d

φ↶ Sd

general linear Lie algebra tensor space symmetric group

Schur-Weyl duality (1927)

..1 The actions of gln and Sd on the tensor space V ⊗d commute

..2 The algebras generated by the actions of gln and Sd in End(V ⊗d) arecentralizing algebras of each other, i.e.,

Endφ(A)(V⊗d) = ψ(B), Endψ(B)(V

⊗d) = φ(A),

where A = C[Sd] = group algebra, B = U(gln) = univ env algebra






Double centralizer property

• The double centralizer property leads to

Corollary

There is a decomposition

V ⊗d =⊕λ⊢d

Vλ ⊗ Lλ,

where {Vλ} = Irrep(Sd);{Lλ} are distinct irreducibles of gln or 0.






Quantization

U(gln)univ env algebra

↷ (Cn)⊗d ↶ C[Sd]group algebra of sym group

Uq(gln)quantum group

quantization

?

............↷ (Q(v)n)⊗d ↶ H(Sd)

Hecke algebra

quantization

?

............

Here the quantized algebras are over Q(v) with v = q1/2: indeterminate






Hecke algebra

• Recall that the symmetric group Sd is generated by simple reflections

s1, s2, . . . , sd−1

subject to the braid relations and s2i = 1.

• Hecke algebra H(Sd) is a Q(v)-algebra generated by

T1, T2, . . . , Td−1

subject to braid relations and the Hecke relations

(Ti + 1)(Ti − q) = 0. (1)

• Specializing q → 1, (1) recovers s2i = 1






Hecke algebra

• H(Sd) has a linear basis (called standard basis)

{Tw | w ∈ Sd}

• H(Sd) has a involution ¯ : H(Sd)→ H(Sd) sending

v 7→ v−1, Tw 7→ T−1w−1

• H(Sd) has a bar-invariant basis {Cw}w∈Sdcalled the Kazhdan-Lusztig(KL)

basis

• The famous Kazhdan-Lusztig theory (’79) offered a solution the the difficultproblem of determining irreducible character problem for category O ofsemisimple Lie algebras using the KL basis






Quantum groups

• Recall that the universal enveloping algebra U(gln) is an associative algebragenerated by

{ei, fi, dj , d−1j }, subject to

Chevalley/Serre relations (e.g. e21e2 + e2e21 = 2e1e2e1)

• Around 1985, Drinfeld and Jimbo introduced the quantum groupU = Uq(gln) as a Q(v)-algebra generated by

{Ei, Fi, Dj , D−1j }, subject to

q-Chevellay/Serre relations (e.g. E21E2 + E2E

21 = (v + v−1)E1E2E1)






Quantum groups

• U provides solutions to the Yang-Baxter equation

• U is a Hopf algebra. Particularly it has a comultiplication ∆ : U→ U⊗U

• There is a triangular decomposition U = U−U0U+

• There is a modified (i.e. idempotented) quantum group U:

Uquantum group

replacing 1 with idempotents−−−−−−−−−−−−−−−−−⇀↽−−−−−−−−−−−−−−−−−take certain infinite sum

Umodified quantum group

{U-modules} 1:1↔ {weight modules of U}

• U is viewed as a preadditive category in categorification






Canonical basis

• U has a sheaf theoretic interpretation

⇒ bar involution ¯ : U→ U from the Verdier duality

• U− admits canonical basis, i.e., U− has a standard basis {ai}i∈I that is“unitriangular” and “integral”:

ai ∈ ai +∑j<i

Z[v, v−1]aj ,

and a (unique) canonical basis {bi}i∈I that is “bar-invariant” and“positive”:

bi = bi ∈ ai +∑j<i

v−1N[v−1]aj

• This is analogous to (dual) Kazhdan-Lusztig basis for Hecke algebra






Canonical basis

• Canonical basis = Kashiwara’s global crystal basis, i.e., for any weight λ,

{bivλ} is a basis of irreducible integrable U-module L(λ) := U−vλ

• Canonical basis {bi} has positivity, i.e.

bibj ∈∑k

N[v, v−1]bk

• U does not admit canonical basis, while U does

• Existence of canonical basis has connection/application in− Categorification− Algebraic combinatorics− Category O− Cluster algebra− Geometric representation theory






Outline


..2 Quantization

..3 q-Schur duality








q-Schur duality of type A

Our goal here is to describe the quantized Schur-Weyl duality below inexamples:


ψ↷ V⊗d

φ↶ H(Sd)

Hecke algebra

We start with describing first a tensor space V⊗d admitting natural actionsfrom both sides






Natural representation V of Uq(gln)

• V :=n∑i=1

Q(v)vi: natural representation of Uq(gln), e.g. n = 3

v1F1⇄E1

v2F2⇄E2

v3

D3 ⟳×1

⟳×1

⟳×q

D−12 ⟳×1

⟳×q−1

⟳×1

• Uq(gln) acts on V⊗d via comultiplication, e.g.

∆(F2) = D−12 D3 ⊗ F2 + F2 ⊗ 1

Hence

F2(v3 ⊗ v2) = D−12 D3v3 ⊗ F2v2 +��F2v3 ⊗ v2 = qv3 ⊗ v3F2(v2 ⊗ v3) = D−12 D3v2 ⊗��F2v3 + F2v2 ⊗ v3 = v3 ⊗ v3






Hecke algebra action

• For H(S2), we have the following equalities in one-line notation(1 = [12], s1 = [21])

T[12]T1 = T[21]T[21]T1 = qT[12] + (q − 1)T[21]

• This leads to a natural Hecke algebra action on V⊗2 respecting themultiplication rule, e.g.

(va ⊗ vb)T1 =

vb ⊗ va if b > a

qvb ⊗ va + (q − 1)va ⊗ vb if b < a

qva ⊗ va if b = a






Example: commutivity

Here’s an example showing(F2(v3 ⊗ v2)

)T1 = F2

((v3 ⊗ v2)T1

)V⊗ V

T1 - V⊗ V

v3 ⊗ v2 - (q − 1)v3 ⊗ v2 + qv2 ⊗ v3

⟳

qv3 ⊗ v3?

- q2v3 ⊗ v3 = (q2 − q)v3 ⊗ v3 + qv3 ⊗ v3?

V⊗ V

F2

?

T1- V⊗ V

F2

?






q-Schur duality


ψ↷ V⊗d

φ↶ H(Sd)

Hecke algebra

q-Schur duality of type A (Jimbo’86)

The algebras Uq(gln) and H(Sd) satisfy the double centralizer property:

Endφ(A)(V⊗d) = ψ(B), Endψ(B)(V⊗d) = φ(A),

where A = H(Sd), B = Uq(gln)






q-Schur algebra

• In other words, there is a surjection, for each d, from Uq(gln) to theq-Schur algebra

Sn,d = EndH(Sd)(V⊗d),

and hence we have

U(gln)quantum group↠

Sn,dq-Schur algebra

↷ V⊗dtensor space

↶ H(Sd)Hecke algebra






q-Schur duality for other types

• It is known that Uq(gln) is a type A quantum group. There are (q-)Schurduality for other types, e.g.,

Onorthogonal group↠

Son,d

orthogonal Schur algebra

↷ V ⊗d ↶ BdBrauer algebra

Uq(son)quantum group↠

Son,d

orthogonal q-Schur algebra

↷ V⊗d ↶ BMWdBirman-Murakami-Wenzl algebra

• In these pictures the “types” are associated to the classical/quantum groups






q-Schur duality for other types

• If we associate the “types” to the Hecke algebra instead, we obtain anotherfamily of (q-)Schur duality:

SXn,d := EndHX

d(V⊗d)

q-Schur algebra of type X

↷ V⊗d ↶ HXd

type X Hecke algebra

• Question: Is there a bottom-top construction that recovers the quantumgroup from the Schur algebras?






BLM construction

• The answer is Yes, the construction for type A is provided in 1990 byBeilinson-Lusztig-MacPherson (BLM) geometrically using partial flags and aso-called stabilization procedure. We can paraphrase it algebraically asbelow:

KAn

stabilization algebra of type A

⇒

SAn,d := EndHA

d(V⊗d)

q-Schur algebra of type A

↷ V⊗d ↶ HAd := H(Sd)

type A Hecke algebra

• Stabilization algebra KAn ≈ inverse limit lim←−

d∈NSAn,d






BLM construction

KAn

stabilization algebra of type A⇒

SAn,d := EndHA

d(V⊗d)

q-Schur algebra of type A

↷ V⊗d ↶ HAd := H(Sd)

type A Hecke algebra

• Canonical bases for SAn,d(d ∈ N) lift “compatibly” to canonical basis of KA

n

• KAn ≃ U(gln)

⇒ A concrete realization of canonical basis of U(gln)






BLM construction

• A BLM construction (for type X) produces from a family of q-Schur algebra{SX

n,d | d ∈ N} an Q(v)-algebra KXn that enjoys similar properties

KXn

stabilization algebra of type X

⇒

SXn,d := EndHX

d(V⊗d)

q-Schur algebra of type X

↷ V⊗d ↶ HXd

type X Hecke algebra






New “quantum groups”

• In type B/C, there are two types of BLM constructions due toBao-Kujawa-Li-Wang (’14) associated to involutions (type ı, ȷ) of theDynkin diagram of type A

..

...1

...2

..· · · ...

n2 − 1

...

n2

. . . . . .

...n

...

n− 1

..· · · ...n2 + 2

...n2 + 1

............. ..

. ...1

..· · · ...

n−12

. . . . ...

n+12

. ...n

..· · · ...n+32

.........

type ȷ : n even type ı : n odd

B KBCn is NOT the modified Drinfeld quantum group of type B/C; it is a

modified quantization of certain product gl• × sl•






Applications of new “quantum groups”

• Depending on parity of n, KBCn is isomorphic to involutive quantum groups

Uın or Uȷ

n introduced by Bao-Wang (’13). The canonical basis theory forUın,U

ȷn

− gives a new formulation of KL theory for Lie algebra of type B/C− establishes for the first time KL theory for Lie superalgebra osp(2m+ 1|2n)

(i.e. super type B/C)

⇒ reformulation of KL theory for Lie algebra of type D and for Li superalgebraosp(2m|2n) (i.e. super type D) by Bao (’16)

• We expect similar applications for other types (e.g. affine classical)






BLM-type constructions

• Here’s a summary on what are known

BLM construction Geometric Algebraic(method) (dimension counting on flags) (combinatorics)Affine A Ginzburg-Vasserot (’93) Du-Fu (’14)

Lusztig (’99)Finite B/C BKLW (’14)Finite D Fan-Li (’14)Affine C Fan-Lai-Li-Luo-Wang1 (’16) FLLLW2 (’16)






Coideal subalgebras

• It is worth mentioning that Uın,U

ȷn are coideal subalgebras of U(gln), i.e.,

the comultiplication ∆ : U(gln)→ U(gln)⊗U(gln) satisfies that

∆(Uın) ⊂ Uı

n ⊗U(gln)

∆(Uȷn) ⊂ Uȷ

n ⊗U(gln)

B A coideal subalgebra is different from a Hopf subalgebra B of U s.t.∆(B) ⊂ B⊗B






Outline


..2 Quantization

..3 q-Schur duality








Symmetric pairs

• A symmetric pair (g, gθ) consists of a Lie algebra g and its fixed-pointsubalgebra gθ with respect to an involution

θ : g→ g

• It plays an important role in the study of real reductive groups

• Classification of symmetric pairs of finite type= Classification of real simple Lie algebras↔ Satake diagrams (of finite type)






Quantum symmetric pairs

• The quantum symmetric pair(QSP) is introduced by Letzter (’02) for finitetype; and by Kolb (’14) for symmetrizable Kac-Moody as a quantization ofthe symmetric pair (g, gθ)

• The pair (U,B) is a QSP if U = U(g) and B = B(θ) is certain coidealsubalgebra of U, i.e., the comultiplication ∆ : U→ U⊗U satisfies that

∆(B) ⊂ B⊗U






Examples

The following are examples of QSP:

• (Uq(gln), twisted Yangian) from Yang-Baxter equation

• (Uq(g),B) where g is of classical type, and B is constructed based onsolutions of the reflection equation

RKRK = KRKR

• (Uq(sl2), q-Onsager algebra) from Ising model

• Quantum algebras from BLM construction. In other words, quantumalgebras arising from Schur-Weyl duality






QSP from BLM construction

Recall that

{SXn,d | d ∈ N} =========

stabilization⇒ KX

n =========take inf sum

⇒ KXn

The resulting quantum algebras are:

type quantum algebra KXn

Finite A Uq(gln)Finite B/C KBC

n ≃ Uın,U

ȷn

Affine A Uq(gln)

Affine C KCn (four variants)

Proposition

The pairs (U(gln),Uın), (U(gln),U

ȷn) and all four variants of (U(gln),K

Cn) are

quantum symmetric pairs.






Applications and future work

• A canonical basis theory for QSP of Satake diagrams finite type is developedrecently by Bao-Wang (’16). It may lead to similar application as in(U(gln),K

Xn) with further study

? canonical basis theory for affine QSP

? BLM construction from {SXn,d} when X is of affine B/D, finite and affine

exceptional type

? BLM-type construction for other Schur-type dualities (e.g., for Braueralgebras, partition algebras)

? BLM-type construction for Howe duality






Thank you for your attention

from schur-weyl duality to quantum symmetric pairs · schur-weyl duality. . . . . . . quantization....

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