q -Deformed Representation Theory At The Limit Jonas Wahl based on - - PowerPoint PPT Presentation

q-Deformed Representation Theory At The 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

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

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

N=1 U(N) or S(∞) = N∈N S(N) has

applications and connections to

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

N=1 U(N) or S(∞) = N∈N S(N) has

applications and connections to

◮ symmetric functions; ◮ combinatorics of partitions; ◮ random matrices; ◮ planar tilings; ◮ stochastic processes.

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

N=1 U(N) or S(∞) = N∈N S(N) has

applications and connections to

◮ symmetric functions; ◮ combinatorics of partitions; ◮ random matrices; ◮ planar tilings; ◮ stochastic processes.

Definition

A character on U(∞) is a continuous, positive definite map

χ : U(∞) → C that is constant on conjugacy classes and

normalized, (i.e. χ(e) = 1).

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

The set of characters on U(∞) is convex and every character can be uniquely disintegrated into extreme points of this set, i.e. extreme characters. There are several approaches to the classification of extreme characters:

◮ Voiculescu, 1976: list of extreme characters and conjecture

that this list is complete.

The set of characters on U(∞) is convex and every character can be uniquely disintegrated into extreme points of this set, i.e. extreme characters. There are several approaches to the classification of extreme characters:

◮ Voiculescu, 1976: list of extreme characters and conjecture

that this list is complete.

◮ Boyer 1983: Classification follows from a theorem of Edrei

(1953) on the classification of totally positive Toeplitz matrices;

The set of characters on U(∞) is convex and every character can be uniquely disintegrated into extreme points of this set, i.e. extreme characters. There are several approaches to the classification of extreme characters:

◮ Voiculescu, 1976: list of extreme characters and conjecture

that this list is complete.

◮ Boyer 1983: Classification follows from a theorem of Edrei

(1953) on the classification of totally positive Toeplitz matrices;

◮ Vershik-Kerov, 1982: Extreme characters are limits of

normalized characters on U(N) as N → ∞;

Classification of extreme characters

◮ Okounkov-Olshanski, 1998: Full details of the

Vershik-Kerov proof + generalization;

Classification of extreme characters

◮ Okounkov-Olshanski, 1998: Full details of the

Vershik-Kerov proof + generalization;

◮ Vershik-Kerov, Olshanski and others: Identification of

characters on U(∞) with central measures on the boundary 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 character associated to such a sextuple.

Are there q-analogues of these results?

Problem: Although there is a proposed definition for Uq(∞) as a σ-C∗-quantum group due to Mahanta and Mathai (2011), it is not clear how to study its representation theory intrinsically.

Are there q-analogues of these results?

Problem: Although there is a proposed definition for Uq(∞) as a σ-C∗-quantum group due to Mahanta and Mathai (2011), it is not clear how to study its representation theory intrinsically. However: Gorin, 2011: The approach of Kerov-Vershik and the Gelfand-Tsetlin graph can be q-deformed in a natural way that admits classification results.

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

Recall: The (equivalence classes of) irreducible representations

• f the unitary group U(N) are indexed by decreasing N-tupels of

integers (signatures)

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

Recall: The (equivalence classes of) irreducible representations

• f the unitary group U(N) are indexed by decreasing N-tupels of

integers (signatures)

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

If

λ1 ≥ 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

Definition

The Gelfand-Tsetlin graph GT is the rooted graded graph with vertex set V = ∞

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

and λ ∈ SignN if and only if µ ≺ λ.

The Gelfand-Tsetlin graph

Definition

The Gelfand-Tsetlin graph GT is the rooted graded graph with vertex 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) ≺ . . .

• n GT endowed with the product σ-algebra F coming from

Ω ⊂ ∞

N=1 SignN.

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

N|µ|−(N−1)|λ| 2

,

where |λ| = N

i=1 λi.

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

N|µ|−(N−1)|λ| 2

,

where |λ| = N

i=1 λi. For a finite path

φ = ∗ ≺ 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)|.

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

N|µ|−(N−1)|λ| 2

,

where |λ| = N

i=1 λi. For a finite path

φ = ∗ ≺ 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.

q-central measures

Definition

A probability measure P on (Ω, F) is q-central if for every finite path 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 finite path φ = ∗ ≺ φ(1) ≺ φ(2) ≺ · · · ≺ φ(N), i.e. Zφ = {(ωi)i≥0 ∈ Ω | ωk = φ(k) for k = 0, . . . , N}.

Connection to characters

How do these measures relate to characters?

Connection to characters

How do these measures relate to characters? If q = 1, then the restriction of the character χ : U(∞) → C to U(N) can be interpreted as a Schur generating function for a probability measure Pχ

N on SignN:

χ|U(N) =

• λ∈SignN

Pχ

N(λ)χ U(N)

λ

dim λ,

where χ

U(N)

λ

is the character of the representation πλ of U(N).

Connection to characters

How do these measures relate to characters? If q = 1, then the restriction of the character χ : U(∞) → C to U(N) can be interpreted as a Schur generating function for a probability measure Pχ

N on SignN:

χ|U(N) =

• λ∈SignN

Pχ

N(λ)χ U(N)

λ

dim λ,

where χ

U(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) = Pχ

N

for all N.

The map χ → Pχ is a bijection between characters and central measures on the boundary of GT that identifies extreme characters on U(∞) with extreme points of the convex set of central measures. Hence, the following question arises: Can one classify extreme q-central measures on the boundary

• f GT?

Theorem (Gorin, 2011)

The extreme q-central measures on (Ω, F) are parametrized by the set

N = {(ν1 ≤ ν2 ≤ ν3 ≤ . . . )} ⊂ Z∞

Theorem (Gorin, 2011)

The extreme q-central measures on (Ω, F) are parametrized by the set

N = {(ν1 ≤ ν2 ≤ ν3 ≤ . . . )} ⊂ Z∞

The N-th q-Schur generating function Qν

N(x1, . . . , xN) =

• λ∈SignN

Pν(ωN = λ)χ

U(N)

λ

(q− N−1

2 x1, . . . , q− N−1 2 xN)

dimq λ

is given as a limit

lim

k→∞

χ

U(k)

λ(k)(q− k−1

2 x1, . . . , q− k−1 2 xN, qN− k−1 2 , . . . , q k−1 2 )

dimq λ(k) ,

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

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

Fix 0 < q < 1. Some facts:

◮ There is a q-deformation of U(N), the compact quantum

group Uq(N) = (C(Uq(N)), ∆);

Fix 0 < q < 1. Some facts:

◮ There is a q-deformation of U(N), the compact quantum

group Uq(N) = (C(Uq(N)), ∆);

◮ The irreducible representations of Uq(N) are indexed by

the set SignN as well;

Fix 0 < q < 1. Some facts:

◮ There is a q-deformation of U(N), the compact quantum

group Uq(N) = (C(Uq(N)), ∆);

◮ The irreducible representations of Uq(N) are indexed by

the set SignN as well;

◮ There is a natural inclusion Uq(N) ⊂ Uq(N + 1) and

restrictions of representations of Uq(N + 1) to Uq(N) decompose as in the group case.

Fix 0 < q < 1. Some facts:

◮ There is a q-deformation of U(N), the compact quantum

group Uq(N) = (C(Uq(N)), ∆);

◮ The irreducible representations of Uq(N) are indexed by

the set SignN as well;

◮ There is a natural inclusion Uq(N) ⊂ Uq(N + 1) and

restrictions 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 the notation C∗

r (G) := c0

• λ∈Irr G

B(Hλ), W ∗(G) :=

ℓ∞

• λ∈Irr G

B(Hλ).

The Stratila-Voiculescu algebra

If

. . . GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .

is an inductive system of cqg’s implemented by

φN : C(GN) → C(GN−1), we can dualize this to injective unital ∗-homomorphisms ΦN : W ∗(GN−1) → W ∗(GN).

The Stratila-Voiculescu algebra

If

. . . GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .

is an inductive system of cqg’s implemented by

φN : 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.

The Stratila-Voiculescu algebra

If

. . . GN−1 ⊂ GN ⊂ Gn+1 ⊂ . . .

is an inductive system of cqg’s implemented by

φN : 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.

Definition

The Stratila-Voiculescu algebra of the inductive system

(GN, φN)N≥0 is the sub-C∗-algebra A of M generated by the

copies of C∗

r (GN), N ≥ 0 inside M.

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

◮ For all N ≥ 0, we set GN = Uq(N), we fix representatives (Uλ, Hλ) for λ ∈ SignN; ◮ we denote the Stratila-Voiculescu algebra of . . . Uq(N − 1) ⊂ Uq(N) ⊂ . . . by A(Uq);

◮ For all N ≥ 0, we set GN = Uq(N), we fix representatives (Uλ, Hλ) for λ ∈ SignN; ◮ we denote the Stratila-Voiculescu algebra of . . . Uq(N − 1) ⊂ Uq(N) ⊂ . . . by A(Uq); ◮ by Fλ, λ ∈ SignN, we denote the unique invertible positive

• perator in Mor(Uλ, Ucc

λ ) such that Tr(Fλ) = Tr(F −1 λ );

◮ For all N ≥ 0, we set GN = Uq(N), we fix representatives (Uλ, Hλ) for λ ∈ SignN; ◮ we denote the Stratila-Voiculescu algebra of . . . Uq(N − 1) ⊂ Uq(N) ⊂ . . . by A(Uq); ◮ by Fλ, λ ∈ SignN, we denote the unique invertible positive

• perator in Mor(Uλ, Ucc

λ ) such that Tr(Fλ) = Tr(F −1 λ );

◮ Let pλ, λ ∈ SignN be the projection C∗

r (GN) → B(Hλ);

◮ For all N ≥ 0, we set GN = Uq(N), we fix representatives (Uλ, Hλ) for λ ∈ SignN; ◮ we denote the Stratila-Voiculescu algebra of . . . Uq(N − 1) ⊂ Uq(N) ⊂ . . . by A(Uq); ◮ by Fλ, λ ∈ SignN, we denote the unique invertible positive

• perator in Mor(Uλ, Ucc

λ ) such that Tr(Fλ) = Tr(F −1 λ );

◮ Let pλ, λ ∈ SignN be the projection C∗

r (GN) → B(Hλ);

◮ Finally, let χq

λ be the state on C∗

r (GN) defined by

χq

λ( · ) = Tr(Fλpλ( · ))

dimq(λ) .

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

GT.

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

GT.

Theorem (Sato, 2018)

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

χ|C∗

r (GN ) =

• λ∈SignN

P(ωN = λ)χq

λ.

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

GT.

Theorem (Sato, 2018)

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

χ|C∗

r (GN ) =

• λ∈SignN

P(ωN = λ)χq

λ.

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

χ|C∗

r (GN ) =

• λ∈SignN

cN

λ χq λ

for some cN

λ ≥ 0,

• λ∈SignN

cN

λ = 1,

then cN

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

P.

Moreover, Sato characterized the states appearing in the theorem intrinsically (i.e. without referring to q-central measures) as the KMS-states with respect to a direct limit scaling group action R τ A(Uq).

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

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

Theorem

Let P, Q be q-central measures on (Ω, F). There exists a q-central measure P ∗ Q which is uniquely determined by the identities

(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}) and where mult(ξ, η ⊗ ν) denotes the multiplicity of the representation 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 formulas for Pm ∗ Pn.

Example

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

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

◮ Is there a proper dual object to A(Uq)? Is there a good

general theory of topological (non-lc) quantum groups or at least direct limit quantum groups?

◮ Is there a proper dual object to A(Uq)? Is there a good

general theory of topological (non-lc) quantum groups or at least direct limit quantum groups?

◮ Is it possible to classify central measures on other

branching graphs?

Thank you!