q-Deformed Representation Theory At The Limit · The study of the representation theory of...

q-Deformed Representation Theory AtThe Limit

Jonas Wahl

based on work in progress with Alexey Bufetov

Hausdorff Center for Mathematics, Bonn

August 9, 2019

Outline

1 Characters of U(∞)

2 The Gelfand-Tsetlin graph and its boundary

3 A quantum group point of view

4 Tensor product decomposition

5 Some questions

Outline





5 Some questions

The study of the representation theory of inductive limit groupssuch as U(∞) =

⋃∞N=1 U(N) or S(∞) =

⋃N∈N S(N) has

applications and connections to

I symmetric functions;

I combinatorics of partitions;

I random matrices;

I planar tilings;

I stochastic processes.

DefinitionA character on U(∞) is a continuous, positive definite mapχ : U(∞)→ C that is constant on conjugacy classes andnormalized, (i.e. χ(e) = 1).


⋃∞N=1 U(N) or S(∞) =

⋃N∈N S(N) has




I random matrices;

I planar tilings;




⋃∞N=1 U(N) or S(∞) =

⋃N∈N S(N) has




I random matrices;

I planar tilings;



The set of characters on U(∞) is convex and every charactercan be uniquely disintegrated into extreme points of this set, i.e.extreme characters.

There are several approaches to the classification of extremecharacters:

I Voiculescu, 1976: list of extreme characters and conjecturethat this list is complete.

I Boyer 1983: Classification follows from a theorem of Edrei(1953) on the classification of totally positive Toeplitzmatrices;

I Vershik-Kerov, 1982: Extreme characters are limits ofnormalized characters on U(N) as N →∞;
















Classification of extreme characters

I Okounkov-Olshanski, 1998: Full details of theVershik-Kerov proof + generalization;

I Vershik-Kerov, Olshanski and others: Identification ofcharacters on U(∞) with central measures on theboundary of the Gelfand-Tsetlin graph GT.

Classification of extreme characters

I Okounkov-Olshanski, 1998: Full details of theVershik-Kerov proof + generalization;

I Vershik-Kerov, Olshanski and others: Identification ofcharacters on U(∞) with central measures on theboundary of the Gelfand-Tsetlin graph GT.

Theorem (Voiculescu, Edrei, Boyer, Vershik-Kerov)Extreme characters of U(∞) are parametrized by sextupels

(α+, α−, β+, β−, δ+, δ−) ∈ R∞ × R∞ × R∞ × R∞ × R× R

such that

α± = (α±1 ≥ α±2 ≥ · · · ≥ 0), β± = (β±1 ≥ β±2 ≥ · · · ≥ 0)

and

∞∑i=1

(α±i + β±i ) ≤ δ±, β+1 + β−1 ≤ 1.

Of course, there is also an explicit formula for the characterassociated to such a sextuple.

Are there q-analogues of these results?

Problem: Although there is a proposed definition for Uq(∞) asa σ-C∗-quantum group due to Mahanta and Mathai (2011), it isnot clear how to study its representation theory intrinsically.

However: Gorin, 2011: The approach of Kerov-Vershik and theGelfand-Tsetlin graph can be q-deformed in a natural way thatadmits classification results.

Are there q-analogues of these results?

Problem: Although there is a proposed definition for Uq(∞) asa σ-C∗-quantum group due to Mahanta and Mathai (2011), it isnot clear how to study its representation theory intrinsically.

However: Gorin, 2011: The approach of Kerov-Vershik and theGelfand-Tsetlin graph can be q-deformed in a natural way thatadmits classification results.

Outline





5 Some questions

Recall: The (equivalence classes of) irreducible representationsof the unitary group U(N) are indexed by decreasing N-tupels ofintegers (signatures)

λ = (λ1, . . . , λN) ∈ ZN , λ1 ≥ λ2 ≥ · · · ≥ λN .

Ifλ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · · ≥ µN−1 ≥ λN

for

µ = (µ1, . . . , µN−1) ∈ SignN−1, λ = (λ1, . . . , λN) ∈ SignN ,

we write µ ≺ λ and say µ interlaces λ.

Recall: The (equivalence classes of) irreducible representationsof the unitary group U(N) are indexed by decreasing N-tupels ofintegers (signatures)

λ = (λ1, . . . , λN) ∈ ZN , λ1 ≥ λ2 ≥ · · · ≥ λN .

Ifλ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · · ≥ µN−1 ≥ λN

for

µ = (µ1, . . . , µN−1) ∈ SignN−1, λ = (λ1, . . . , λN) ∈ SignN ,

we write µ ≺ λ and say µ interlaces λ.

The Gelfand-Tsetlin graph

DefinitionThe Gelfand-Tsetlin graph GT is the rooted graded graph withvertex set V =

⋃∞N=0 SignN and an edge between µ ∈ SignN−1

and λ ∈ SignN if and only if µ ≺ λ.

The boundary of GT is the Borel space (Ω,F) of infinite paths

∗ = φ(0) ≺ φ(1) ≺ φ(2) ≺ φ(3) ≺ . . .

on GT endowed with the product σ-algebra F coming fromΩ ⊂

∏∞N=1 SignN .

The Gelfand-Tsetlin graph

DefinitionThe Gelfand-Tsetlin graph GT is the rooted graded graph withvertex set V =

⋃∞N=0 SignN and an edge between µ ∈ SignN−1

and λ ∈ SignN if and only if µ ≺ λ.

The boundary of GT is the Borel space (Ω,F) of infinite paths

∗ = φ(0) ≺ φ(1) ≺ φ(2) ≺ φ(3) ≺ . . .

on GT endowed with the product σ-algebra F coming fromΩ ⊂

∏∞N=1 SignN .

Let 0 < q < 1. For an edge µ ≺ λ from level N − 1 to level N,set

w(µ ≺ λ) = qN|µ|−(N−1)|λ|

2 ,

where |λ| =∑N

i=1 λi .

For a finite pathφ = ∗ ≺ φ(1) ≺ φ(2) ≺ · · · ≺ φ(N), define the weight

w(φ) =N−1∏i=1

w(φ(i) ≺ φ(i + 1)) = q∑N

i=1 |φ(i)|−N−1

2 |φ(N)|.

Note: For λ ∈ SignN , ∑∗≺φ(1)≺φ(2)≺···≺φ(N)|φ(N)=λ

w(φ) = dimq λ.


w(µ ≺ λ) = qN|µ|−(N−1)|λ|

2 ,

where |λ| =∑N

i=1 λi . For a finite pathφ = ∗ ≺ φ(1) ≺ φ(2) ≺ · · · ≺ φ(N), define the weight

w(φ) =N−1∏i=1

w(φ(i) ≺ φ(i + 1)) = q∑N

i=1 |φ(i)|−N−1

2 |φ(N)|.


w(φ) = dimq λ.


w(µ ≺ λ) = qN|µ|−(N−1)|λ|

2 ,

where |λ| =∑N

i=1 λi . For a finite pathφ = ∗ ≺ φ(1) ≺ φ(2) ≺ · · · ≺ φ(N), define the weight

w(φ) =N−1∏i=1

w(φ(i) ≺ φ(i + 1)) = q∑N

i=1 |φ(i)|−N−1

2 |φ(N)|.


w(φ) = dimq λ.

q-central measures

DefinitionA probability measure P on (Ω,F) is q-central if for every finitepath φ as above, we have

P(Zφ) = P((ωi)i≥0|ωN = φ(N)) w(φ)

dimq(φ(N)),

where Zφ denotes the finite cylinder corresponding to the finitepath φ = ∗ ≺ φ(1) ≺ φ(2) ≺ · · · ≺ φ(N), i.e.

Zφ = (ωi)i≥0 ∈ Ω | ωk = φ(k) for k = 0, . . . ,N.

Connection to charactersHow do these measures relate to characters?

If q = 1, then the restriction of the character χ : U(∞)→ C toU(N) can be interpreted as a Schur generating function for aprobability measure Pχ

N on SignN :

χ|U(N)=

∑λ∈SignN

PχN (λ)

χU(N)λ

dimλ,

where χU(N)λ is the character of the representation πλ of U(N).

Then, there is central measure Pχ on (Ω,F), satisfying

Pχ(ωN = λ) = PχN for all N.



N on SignN :

χ|U(N)=

∑λ∈SignN

PχN (λ)

χU(N)λ

dimλ,






N on SignN :

χ|U(N)=

∑λ∈SignN

PχN (λ)

χU(N)λ

dimλ,




The map χ 7→ Pχ is a bijection between characters and centralmeasures on the boundary of GT that identifies extremecharacters on U(∞) with extreme points of the convex set ofcentral measures.

Hence, the following question arises:

Can one classify extreme q-central measures on the boundaryof GT?

Theorem (Gorin, 2011)The extreme q-central measures on (Ω,F) are parametrized bythe set

N = (ν1 ≤ ν2 ≤ ν3 ≤ . . . ) ⊂ Z∞

The N-th q-Schur generating function

QνN(x1, . . . , xN) =

∑λ∈SignN

Pν(ωN = λ)χ

U(N)λ (q−

N−12 x1, . . . , q−

N−12 xN)

dimq λ

is given as a limit

limk→∞

χU(k)λ(k)(q−

k−12 x1, . . . , q−

k−12 xN , qN− k−1

2 , . . . , qk−1

2 )

dimq λ(k),

where λ(k)k−i+1 → νi as k →∞ for i = 1, . . . ,N.

Theorem (Gorin, 2011)The extreme q-central measures on (Ω,F) are parametrized bythe set

N = (ν1 ≤ ν2 ≤ ν3 ≤ . . . ) ⊂ Z∞

The N-th q-Schur generating function

QνN(x1, . . . , xN) =

∑λ∈SignN

Pν(ωN = λ)χ

U(N)λ (q−

N−12 x1, . . . , q−

N−12 xN)

dimq λ

is given as a limit

limk→∞

χU(k)λ(k)(q−

k−12 x1, . . . , q−

k−12 xN , qN− k−1

2 , . . . , qk−1

2 )

dimq λ(k),

where λ(k)k−i+1 → νi as k →∞ for i = 1, . . . ,N.

Outline





5 Some questions

Fix 0 < q < 1. Some facts:

I There is a q-deformation of U(N), the compact quantumgroup Uq(N) = (C(Uq(N)),∆);

I The irreducible representations of Uq(N) are indexed bythe set SignN as well;

I There is a natural inclusion Uq(N) ⊂ Uq(N + 1) andrestrictions of representations of Uq(N + 1) to Uq(N)decompose as in the group case.

Also for a general compact quantum group G, let us fix thenotation

C∗r (G) :=c0∏

λ∈IrrG

B(Hλ), W ∗(G) :=`∞∏

λ∈IrrG

B(Hλ).






C∗r (G) :=c0∏

λ∈IrrG

B(Hλ), W ∗(G) :=`∞∏

λ∈IrrG

B(Hλ).






C∗r (G) :=c0∏

λ∈IrrG

B(Hλ), W ∗(G) :=`∞∏

λ∈IrrG

B(Hλ).






C∗r (G) :=c0∏

λ∈IrrG

B(Hλ), W ∗(G) :=`∞∏

λ∈IrrG

B(Hλ).

The Stratila-Voiculescu algebra

If. . .GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .

is an inductive system of cqg’s implemented byφN : C(GN)→ C(GN−1), we can dualize this to injective unital∗-homomorphisms ΦN : W ∗(GN−1)→ W ∗(GN).

Denote the C∗-inductive limit of the inductive system(W ∗(GN),ΦN)N≥0 by M.

DefinitionThe Stratila-Voiculescu algebra of the inductive system(GN , φN)N≥0 is the sub-C∗-algebra A of M generated by thecopies of C∗r (GN), N ≥ 0 inside M.


If. . .GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .





If. . .GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .




I For all N ≥ 0, we set GN = Uq(N), we fix representatives(Uλ,Hλ) for λ ∈ SignN ;

I we denote the Stratila-Voiculescu algebra of. . .Uq(N − 1) ⊂ Uq(N) ⊂ . . . by A(Uq);

I by Fλ, λ ∈ SignN , we denote the unique invertible positiveoperator in Mor(Uλ,Ucc

λ ) such that Tr(Fλ) = Tr(F−1λ );

I Let pλ, λ ∈ SignN be the projection C∗r (GN)→ B(Hλ);I Finally, let χq

λ be the state on C∗r (GN) defined by

χqλ( · ) =

Tr(Fλpλ( · ))

dimq(λ).







χqλ( · ) =

Tr(Fλpλ( · ))

dimq(λ).







χqλ( · ) =

Tr(Fλpλ( · ))

dimq(λ).





I Let pλ, λ ∈ SignN be the projection C∗r (GN)→ B(Hλ);

I Finally, let χqλ be the state on C∗r (GN) defined by

χqλ( · ) =

Tr(Fλpλ( · ))

dimq(λ).







χqλ( · ) =

Tr(Fλpλ( · ))

dimq(λ).

Recall that (Ω,F) was the space of infinite paths on the graphGT.

Theorem (Sato, 2018)Let P be a q-central measure on (Ω,F). Then, there exists astate χ on A(Uq) such that for all N

χ|C∗r (GN )=

∑λ∈SignN

P(ωN = λ)χqλ.

If χ is a state on A(Uq) such that for all N

χ|C∗r (GN )=

∑λ∈SignN

cNλχ

qλ for some cN

λ ≥ 0,∑

λ∈SignN

cNλ = 1,

then cNλ = P(ωN = λ) for some uniquely determined q-central

P.



χ|C∗r (GN )=

∑λ∈SignN

P(ωN = λ)χqλ.


χ|C∗r (GN )=

∑λ∈SignN

cNλχ

qλ for some cN

λ ≥ 0,∑

λ∈SignN

cNλ = 1,


P.



χ|C∗r (GN )=

∑λ∈SignN

P(ωN = λ)χqλ.


χ|C∗r (GN )=

∑λ∈SignN

cNλχ

qλ for some cN

λ ≥ 0,∑

λ∈SignN

cNλ = 1,


P.

Moreover, Sato characterized the states appearing in thetheorem intrinsically (i.e. without referring to q-centralmeasures) as the KMS-states with respect to a direct limitscaling group action R yτ A(Uq).

Outline





5 Some questions

There is also a notion of convolution of q-central measures thatemulates the tensor product between representations.

TheoremLet P,Q be q-central measures on (Ω,F). There exists aq-central measure P ∗ Q which is uniquely determined by theidentities

(P ∗ Q)N(ξ) =∑η,ν∈SignN

PN(η)QN(ν)mult(ξ, η ⊗ ν) dimq ξ

dimq η dimq ν

for all N ≥ 0, where PN(ξ) = P((ωi)i≥0 ∈ Ω | ωN = ξ) andwhere mult(ξ, η ⊗ ν) denotes the multiplicity of therepresentation Uξ in Uη ⊗ Uν .

Recall that extreme q-central measures are indexed by the set

N = (ν1 ≤ ν2 ≤ ν3 ≤ . . . ) ⊂ Z∞.

If m, n ∈ N have constant tails, one can give explicit formulasfor Pm ∗ Pn.

ExampleLet m = (−m, 0, 0, . . . ) and n = (−n, 0, 0, . . . ), where m ≥ n.Then Pm ∗ Pn decomposes as

Pm ∗ Pn =n∑

i=0

ci P(−(m+n−i),−i,0,0,... )

where ci = limk→∞χ(m+n−i,i,0,...,0)(1,q,...,q

k−1)

χ(m,0,...,0)(1,q,...,qk−1)χ(n,0,...,0)(1,q,...,,qk−1).

Outline





5 Some questions

I Is there a proper dual object to A(Uq)? Is there a goodgeneral theory of topological (non-lc) quantum groups or atleast direct limit quantum groups?

I Is it possible to classify central measures on otherbranching graphs?

I Is there a proper dual object to A(Uq)? Is there a goodgeneral theory of topological (non-lc) quantum groups or atleast direct limit quantum groups?

I Is it possible to classify central measures on otherbranching graphs?

Thank you!

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