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 Characters of U ( ∞ ) 1 The Gelfand-Tsetlin graph and its boundary 2 A quantum group point of view 3 Tensor product decomposition 4 Some questions 5

Outline Characters of U ( ∞ ) 1 The Gelfand-Tsetlin graph and its boundary 2 A quantum group point of view 3 Tensor product decomposition 4 Some questions 5

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 ± β ± = ( β ± 1 ≥ α ± 1 ≥ β ± 2 ≥ · · · ≥ 0 ) , 2 ≥ · · · ≥ 0 ) and ∞ � ( α ± i + β ± i ) ≤ δ ± , β + 1 + β − 1 ≤ 1 . i = 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 U q ( ∞ ) 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 U q ( ∞ ) 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 Characters of U ( ∞ ) 1 The Gelfand-Tsetlin graph and its boundary 2 A quantum group point of view 3 Tensor product decomposition 4 Some questions 5

Recall : The (equivalence classes of) irreducible representations of the unitary group U ( N ) are indexed by decreasing N -tupels of integers ( signatures ) λ = ( λ 1 , . . . , λ N ) ∈ Z N , λ 1 ≥ λ 2 ≥ · · · ≥ λ N .

Recall : The (equivalence classes of) irreducible representations of the unitary group U ( N ) are indexed by decreasing N -tupels of integers ( signatures ) λ = ( λ 1 , . . . , λ N ) ∈ Z N , λ 1 ≥ λ 2 ≥ · · · ≥ λ N . If λ 1 ≥ µ 1 ≥ λ 2 ≥ µ 2 ≥ · · · ≥ µ N − 1 ≥ λ N for µ = ( µ 1 , . . . , µ N − 1 ) ∈ Sign N − 1 , λ = ( λ 1 , . . . , λ N ) ∈ Sign N , 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 Sign N and an edge between µ ∈ Sign N − 1 and λ ∈ Sign N 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 Sign N and an edge between µ ∈ Sign N − 1 and λ ∈ Sign N 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 from Ω ⊂ � ∞ N = 1 Sign N .

Let 0 < q < 1. For an edge µ ≺ λ from level N − 1 to level N , set N | µ |− ( N − 1 ) | λ | w ( µ ≺ λ ) = q , 2 where | λ | = � N i = 1 λ i .

Let 0 < q < 1. For an edge µ ≺ λ from level N − 1 to level N , set N | µ |− ( N − 1 ) | λ | w ( µ ≺ λ ) = q , 2 where | λ | = � N i = 1 λ i . For a finite path φ = ∗ ≺ φ ( 1 ) ≺ φ ( 2 ) ≺ · · · ≺ φ ( N ) , define the weight N − 1 � N i = 1 | φ ( i ) |− N − 1 � 2 | φ ( N ) | . w ( φ ( i ) ≺ φ ( i + 1 )) = q w ( φ ) = i = 1

Let 0 < q < 1. For an edge µ ≺ λ from level N − 1 to level N , set N | µ |− ( N − 1 ) | λ | w ( µ ≺ λ ) = q , 2 where | λ | = � N i = 1 λ i . For a finite path φ = ∗ ≺ φ ( 1 ) ≺ φ ( 2 ) ≺ · · · ≺ φ ( N ) , define the weight N − 1 � N i = 1 | φ ( i ) |− N − 1 � 2 | φ ( N ) | . w ( φ ( i ) ≺ φ ( i + 1 )) = q w ( φ ) = i = 1 Note : For λ ∈ Sign N , � w ( φ ) = dim q λ. ∗≺ φ ( 1 ) ≺ φ ( 2 ) ≺···≺ φ ( N ) | φ ( N )= 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 w ( φ ) P ( Z φ ) = P ( { ( ω i ) i ≥ 0 | ω N = φ ( N ) } ) dim q ( φ ( N )) , where Z φ denotes the finite cylinder corresponding to the finite path φ = ∗ ≺ φ ( 1 ) ≺ φ ( 2 ) ≺ · · · ≺ φ ( N ) , i.e. Z φ = { ( ω i ) i ≥ 0 ∈ Ω | ω k = 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 P χ N on Sign N : U ( N ) N ( λ ) χ � P χ λ χ | U ( N ) = dim λ, λ ∈ Sign N U ( N ) where χ is the character of the representation π λ of U ( N ) . λ

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 P χ N on Sign N : U ( N ) N ( λ ) χ � P χ λ χ | U ( N ) = dim λ, λ ∈ Sign N U ( N ) where χ is the character of the representation π λ of U ( N ) . λ Then, there is central measure P χ on (Ω , F ) , satisfying P χ ( ω N = λ ) = P χ for all N . N

The map χ �→ P χ 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 of GT ?

Theorem (Gorin, 2011) The extreme q-central measures on (Ω , F ) are parametrized by the set N = { ( ν 1 ≤ ν 2 ≤ ν 3 ≤ . . . ) } ⊂ Z ∞

Theorem (Gorin, 2011) The extreme q-central measures on (Ω , F ) are parametrized by the set N = { ( ν 1 ≤ ν 2 ≤ ν 3 ≤ . . . ) } ⊂ Z ∞ The N-th q-Schur generating function U ( N ) ( q − N − 1 2 x 1 , . . . , q − N − 1 2 x N ) P ν ( ω N = λ ) χ � Q ν λ N ( x 1 , . . . , x N ) = dim q λ λ ∈ Sign N is given as a limit U ( k ) λ ( k ) ( q − k − 1 2 x 1 , . . . , q − k − 1 2 x N , q N − k − 1 k − 1 2 , . . . , q 2 ) χ lim , dim q λ ( k ) k →∞ where λ ( k ) k − i + 1 → ν i as k → ∞ for i = 1 , . . . , N.