Diaconis Shahshahani Upper Bound Lemma for Finite Quantum Groups

Transcription

1 Diaconis Shahshahani Upper Bound Lemma for Finite Quantum Groups J.P. McCarthy Cork Institute of Technology 30 August 2018 Irish Mathematical Society Meeting, UCD

2 Finite Classical Groups aka Finite Groups A finite group is an object G FinSet together with morphisms m, e, and 1. Associativity, identity, and inverse are given by G G G I G m G G m I G m G G G m m G G G G G e I G = = I G e { } G G G { } m m G G G G G S I G G G G G G m e ε G I G S G G

3 The C-Functor The C-Functor, C : FinSet FinVec C, is a covariant functor mapping a set X to a vector space CX (the finite-dimensional vector space with basis {δ x : x X }), and a morphism f : X Y, x f (x) to a morphism Cf : CX CY, δ x δ f (x).

4 The C-Functor The C-Functor, C : FinSet FinVec C, is a covariant functor mapping a set X to a vector space CX (the finite-dimensional vector space with basis {δ x : x X }), and a morphism f : X Y, x f (x) to a morphism Cf : CX CY, δ x δ f (x). Applying the C-Functor to a group G yields the group algebra, CG. As the vector space is finite dimensional, C(G G) = CG CG and so : CG CG CG, δ s δ t δ st.

5 The C-Functor The C-Functor, C : FinSet FinVec C, is a covariant functor mapping a set X to a vector space CX (the finite-dimensional vector space with basis {δ x : x X }), and a morphism f : X Y, x f (x) to a morphism Cf : CX CY, δ x δ f (x). Applying the C-Functor to a group G yields the group algebra, CG. As the vector space is finite dimensional, C(G G) = CG CG and so : CG CG CG, δ s δ t δ st. As the group axioms are commutative diagrams, the group axioms are translated into C -group axioms. For example, associativity: CG CG CG I CG CG CG I CG CG CG CG

6 The Dual Endofunctor The Dual Endofunctor, D : FinVec C FinVec C, is a contravariant functor mapping a vector space U to its dual U (recall everything is in finite dimensions), and a morphism (linear map) T : U V, to its transpose (ϕ V ): D(T ) : V U, ϕ ϕ T.

7 The Dual Endofunctor The Dual Endofunctor, D : FinVec C FinVec C, is a contravariant functor mapping a vector space U to its dual U (recall everything is in finite dimensions), and a morphism (linear map) T : U V, to its transpose (ϕ V ): D(T ) : V U, ϕ ϕ T. Applying the Dual Endofunctor to a group algebra CG yields the algebra of functions on G, F (G), with basis {δ g : g G}. This carries a commutative C -algebra structure, but inherits from the group axioms via the functor composition Q := D C an encoding of the group axioms.

8 The Dual Endofunctor The Dual Endofunctor, D : FinVec C FinVec C, is a contravariant functor mapping a vector space U to its dual U (recall everything is in finite dimensions), and a morphism (linear map) T : U V, to its transpose (ϕ V ): D(T ) : V U, ϕ ϕ T. Applying the Dual Endofunctor to a group algebra CG yields the algebra of functions on G, F (G), with basis {δ g : g G}. This carries a commutative C -algebra structure, but inherits from the group axioms via the functor composition Q := D C an encoding of the group axioms. This encoding has maps, the comultiplication, := Qm; the counit, ε := Qe; and the antipode, S := Q( 1 ), that satisfy three commutative diagrams that encode associativity, identity, and inverses.

9 The Encoded Group Axioms (Hopf (1940s); Kac (1960s)) The comultiplication, for example: : F (G) F (G) F (G) is a linear map (δ g ) = δ gt 1 δ t. t G

10 The Encoded Group Axioms (Hopf (1940s); Kac (1960s)) The comultiplication, for example: : F (G) F (G) F (G) is a linear map (δ g ) = δ gt 1 δ t. t G The group axiom of associativity is, for example, encoded by coassociativity (note the reversal of arrows): F (G) F (G) F (G) I F (G) F (G) F (G) I F (G) F (G) F (G) F (G) The encoded group axioms are called Hopf-algebra axioms.

11 The Encoded Group Axioms (Hopf (1940s); Kac (1960s)) The comultiplication, for example: : F (G) F (G) F (G) is a linear map (δ g ) = δ gt 1 δ t. t G The group axiom of associativity is, for example, encoded by coassociativity (note the reversal of arrows): F (G) F (G) F (G) I F (G) F (G) F (G) I F (G) F (G) F (G) F (G) The encoded group axioms are called Hopf-algebra axioms. The interaction between this structure, and the C -algebra structure gives the algebra of functions on a group, F (G), the structure of what is called a C -Hopf algebra.

12 Quantum Groups (Drinfeld, Jimbo, Woronowicz (1980s)) There are, however, finite dimensional spaces together with morphisms that also satisfy these axioms but are not the algebra of functions on any group because the multiplication is no longer commutative multi-matrix algebras.

13 Quantum Groups (Drinfeld, Jimbo, Woronowicz (1980s)) There are, however, finite dimensional spaces together with morphisms that also satisfy these axioms but are not the algebra of functions on any group because the multiplication is no longer commutative multi-matrix algebras. These are the algebras of functions on (finite) quantum groups: F (G) Q G Q(group axioms) but not ab=ba F (G) G Q These quantum spaces do not actually exist and are referred to as virtual objects

14 Quantum Groups (Drinfeld, Jimbo, Woronowicz (1980s)) There are, however, finite dimensional spaces together with morphisms that also satisfy these axioms but are not the algebra of functions on any group because the multiplication is no longer commutative multi-matrix algebras. These are the algebras of functions on (finite) quantum groups: F (G) Q G Q(group axioms) but not ab=ba F (G) G Q These quantum spaces do not actually exist and are referred to as virtual objects yet many questions that can be posed and resolved in the classical setting may also be posed and hopefully resolved in the quantum case.

15 Classical Random Walks (Markov (1906); Borel (1940)) Given a finite group, G, and G-valued random variables ζ i iid ν Mp (G), the sequence of random variables {ξ i } k i=1 given by ξ i := ζ i ζ 2 ζ 1, is called a (right-invariant) random walk on G driven by ν.

16 Classical Random Walks (Markov (1906); Borel (1940)) Given a finite group, G, and G-valued random variables ζ i iid ν Mp (G), the sequence of random variables {ξ i } k i=1 given by ξ i := ζ i ζ 2 ζ 1, is called a (right-invariant) random walk on G driven by ν. The distribution of ξ k all of interest in this work is given by where (noting E ν (δ g ) = ν(g)) (ν ν)(g) := t G ν} {{ ν ν} =: ν k, k copies ν(gt 1 )ν(t) = (E ν E ν ) (δ }{{} g ). =:E ν E ν

17 Classical Random Walks (Markov (1906); Borel (1940)) Given a finite group, G, and G-valued random variables ζ i iid ν Mp (G), the sequence of random variables {ξ i } k i=1 given by ξ i := ζ i ζ 2 ζ 1, is called a (right-invariant) random walk on G driven by ν. The distribution of ξ k all of interest in this work is given by where (noting E ν (δ g ) = ν(g)) (ν ν)(g) := t G ν} {{ ν ν} =: ν k, k copies ν(gt 1 )ν(t) = (E ν E ν ) (δ }{{} g ). =:E ν E ν Denote by π M p (G) the uniform distribution; which is invariant in the sense that for any ν M p (G), ν π = π = π ν.

18 Quantum Random Walks (Franz & Gohm (2005)) A probability ν M p (G) gives rise to a state (norm one, positive linear functional) on F (G): f t G f (t) ν(t) =: E ν(f ). Therefore the quantum probabilistic identifications are made: ν M p (G) a state E ν on F (G); (distribution of) random walk on G {E ν k } := {E k ν }

19 Quantum Random Walks (Franz & Gohm (2005)) A probability ν M p (G) gives rise to a state (norm one, positive linear functional) on F (G): f t G f (t) ν(t) =: E ν(f ). Therefore the quantum probabilistic identifications are made: ν M p (G) a state E ν on F (G); (distribution of) random walk on G {E ν k } := {E k ν } Note that for f F (G): E π (f ) = f (t)π(t) = 1 f (t) = f, G t G t G the average over all points t G of f.

20 Quantum Random Walks (Franz & Gohm (2005)) A probability ν M p (G) gives rise to a state (norm one, positive linear functional) on F (G): f t G f (t) ν(t) =: E ν(f ). Therefore the quantum probabilistic identifications are made: ν M p (G) a state E ν on F (G); (distribution of) random walk on G {E ν k } := {E k ν } Note that for f F (G): E π (f ) = f (t)π(t) = 1 f (t) = f, G t G t G the average over all points t G of f. Note that a (finite) quantum group also has a uniform distribution. Given by the Haar state, E π, it is also invariant in the sense that E π E ν = E π = E ν E π for all ν M p (G).

21 Classical Distance to Random Classical Random Walks of interest are primarily those in which the ν k converge to uniform, to random. Thus, the question is posed: when is ν k close to random?

22 Classical Distance to Random Classical Random Walks of interest are primarily those in which the ν k converge to uniform, to random. Thus, the question is posed: when is ν k close to random? The distance to random is given by ν k π = sup ν k (S) π(s) = sup E ν k (1 S ) E π (1 S ). S G S G

23 Classical Distance to Random Classical Random Walks of interest are primarily those in which the ν k converge to uniform, to random. Thus, the question is posed: when is ν k close to random? The distance to random is given by ν k π = sup ν k (S) π(s) = sup E ν k (1 S ) E π (1 S ). S G S G Probabilities ν M p (G) have densities, that is there is an element f ν F (G) such that E ν (f ) = E π (f ν f ) for all f F (G) (indeed f ν (g) = G ν(g)).

24 The presence of this one-norm allows a Cauchy Schwarz inequality to be used: this becomes crucial. Classical Distance to Random Classical Random Walks of interest are primarily those in which the ν k converge to uniform, to random. Thus, the question is posed: when is ν k close to random? The distance to random is given by ν k π = sup ν k (S) π(s) = sup E ν k (1 S ) E π (1 S ). S G S G Probabilities ν M p (G) have densities, that is there is an element f ν F (G) such that E ν (f ) = E π (f ν f ) for all f F (G) (indeed f ν (g) = G ν(g)). Where Theorem f 1 = 1 G f (t) = E π ( f ), t G ν k π = 1 2 f ν k f π 1

25 Quantum Distance to Random There is a one-to-one correspondence between subsets S G and projections p F (G) S 1 S and so for ν M p (G) ν k π = sup E ν k (p) E π (p). (1) p F (G), a projection

26 Quantum Distance to Random There is a one-to-one correspondence between subsets S G and projections p F (G) S 1 S and so for ν M p (G) ν k π = sup E ν k (p) E π (p). (1) p F (G), a projection Probabilities ν M p (G) also have densities, that is there is an element a ν F (G) such that E ν (a) = E π (a ν a) for all a F (G).

27 Quantum Distance to Random There is a one-to-one correspondence between subsets S G and projections p F (G) S 1 S and so for ν M p (G) ν k π = sup E ν k (p) E π (p). (1) p F (G), a projection Probabilities ν M p (G) also have densities, that is there is an element a ν F (G) such that E ν (a) = E π (a ν a) for all a F (G). Theorem In the quantum case, the total variation distance is also equal to half the one norm: ν k π = 1 2 a ν k a π 1 := 1 2 E π( a ν k a π ). (Freslon (2018))

28 Quantum Distance to Random There is a one-to-one correspondence between subsets S G and projections p F (G) S 1 S and so for ν M p (G) ν k π = sup E ν k (p) E π (p). (1) p F (G), a projection Probabilities ν M p (G) also have densities, that is there is an element a ν F (G) such that E ν (a) = E π (a ν a) for all a F (G). Theorem In the quantum case, the total variation distance is also equal to half the one norm: ν k π = 1 2 a ν k a π 1 := 1 2 E π( a ν k a π ). (Freslon (2018)) This one norm also has an associated Cauchy Schwarz inequality, and this allows (1) to be used in the quantum case.

29 Classical Diaconis Shahshahani Theory Every group representation ρ : G GL(V ) splits into a direct sum of irreducible representations where ρ α : G GL(V α ) with dim(v α ) =: d α.

30 Classical Diaconis Shahshahani Theory Every group representation ρ : G GL(V ) splits into a direct sum of irreducible representations where ρ α : G GL(V α ) with dim(v α ) =: d α. Definition: (used by Diaconis) The Fourier Transform of ν M p (G) is a linear map: ν L(V α ); α Irr(G) where the Fourier Transform of ν at the representation ρ α is ν Vα =: ν(α) = t G ν(t)ρ α (t).

31 Classical Diaconis Shahshahani Theory Every group representation ρ : G GL(V ) splits into a direct sum of irreducible representations where ρ α : G GL(V α ) with dim(v α ) =: d α. Definition: (used by Diaconis) The Fourier Transform of ν M p (G) is a linear map: ν L(V α ); α Irr(G) where the Fourier Transform of ν at the representation ρ α is ν Vα =: ν(α) = t G ν(t)ρ α (t). Upper Bound Lemma: Where τ is the trivial representation, ν k π 2 1 [ d α Tr ( ν (α) ) k ν (α) k]. 4 α Irr(G)\{τ} (Diaconis & Shahshahani (1981))

32 Applications Simple Random Walk on Circle, Z n step left/right with equal probability; close to random in about k = n 2 steps. Random Walk on the Hypercube, Z n 2 stick or move to one 1 of the nearest neighbours with equal probability, n+1 ; close to random in k = 1 4n ln n steps. Random Transposition Shuffle of S n swap two cards chosen at random; close to random in k = 1 2n ln n steps (k 102 for n = 52). Diaconis (1988)

33 Quantum Diaconis Shahshahani Theory? (Wills, (2010)) Consider again the Upper Bound Lemma: ν k π α Irr(G)\{τ} [ d α Tr ( ν (α) ) k ν (α) k]. Note there is no reference to points in the space G. While it appears that ν(α) is defined with respect to points, it is actually a sum over all points, a rôle played by the Haar state E π : 1 f (t) = E π (f ). G }{{} t G }{{} quantum: no reference to points classical: references points t G

34 Quantum Diaconis Shahshahani Theory? (Wills, (2010)) Consider again the Upper Bound Lemma: ν k π α Irr(G)\{τ} [ d α Tr ( ν (α) ) k ν (α) k]. Note there is no reference to points in the space G. While it appears that ν(α) is defined with respect to points, it is actually a sum over all points, a rôle played by the Haar state E π : 1 f (t) = E π (f ). G }{{} t G }{{} quantum: no reference to points classical: references points t G...and (finite) quantum groups have a representation theory remarkably similar to that of classical groups; e.g. there is a Peter Weyl Theorem (Woronowicz, (1987)) (their algebras of functions have corepresentations).

35 Diaconis Van Daele Theory (McCarthy, 2017) Using results of Van Daele ( ), it can be shown that the properties of the classical Fourier Transform ν(ρ α ), that are used to prove the classical Upper Bound Lemma, are also shared by a quantum Fourier Transform ν(κ α ) (this κ α is a corepresentation).

36 Diaconis Van Daele Theory (McCarthy, 2017) Using results of Van Daele ( ), it can be shown that the properties of the classical Fourier Transform ν(ρ α ), that are used to prove the classical Upper Bound Lemma, are also shared by a quantum Fourier Transform ν(κ α ) (this κ α is a corepresentation). For example, the sum over irreducible representations comes from the classical ν(δ e ) = 1 d α Tr [ ν(α)] ; G α Irr(G) which has a generalisation to finite quantum groups: ĥ(ν) = d α Tr [ ν(α)], α Irr(G) where ĥ is an unnormalised but invariant state on the dual of F (G).

37 Upper Bound Lemma (McCarthy, 2017) Leaning heavily on the finiteness assumption, the Upper Bound Lemma for Finite Quantum Groups follows in a similar manner to that of the classical result of Diaconis and Shahshahani. In the notation that is used, the classical Upper Bound Lemma: ν k π α Irr(G)\{τ} and the quantum Upper Bound Lemma: ν k π α Irr(G)\{τ} are essentially the same thing. [ d α Tr ( ν (α) ) k ν (α) k], [ d α Tr ( ν (α) ) k ν (α) k].

38 References 1. Markov (1906), Extension of the law of large numbers to dependent events 2. Borel (1940), Théorie Mathématique du Bridge à la Portée de Tous 3. Diaconis & Shahshahani (1981), Generating a Random Permutation with Random Transpositions 4. Diaconis (1988), Group Representations in Probability and Statistics 5. Franz & Gohm (2005), Random Walks on Finite Quantum Groups 6. Freslon (2018), Cut-off phenomenon for random walks on free orthogonal quantum groups 7. McCarthy (2017), Random Walks on Finite Quantum Groups: Diaconis Shahshahani Theory for Quantum Groups 8. Van Daele (2006), The Fourier Transform in Quantum Group Theory