Random Walks on Finite Quantum Groups

Transcription

1 Random Walks on Finite Quantum Groups Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups J.P. McCarthy Cork Institute of Technology 17 May 2017 Seoul National University

2 The Classical Problem Given a finite group, G, and elements ζ i G chosen according to a fixed probability distribution (ν M p (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 is given by where ν} {{ ν ν} =: ν k, k copies µ ν(s) = t G µ(st 1 )ν(t). Denote by π M p (G) the uniform or random distribution. The distance to random is measured by the total variation distance: ν k π TV = sup ν k (S) π(s) = 1 S G 2 ν k π l 1.

3 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.

4 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 ); ν 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).

5 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 ); ν 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 TV 1 [ d α Tr ( ν (α) ) k ν (α) k]. 4 α Irr(G)\{τ} (Diaconis & Shahshahani (1981))

6 Examples Simple Random Walk on Circle, Z n step left/right with equal probability; close to random in O(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 O ( 1 4 n ln n) steps. Random Transposition Shuffle of S n swap two cards chosen at random; close to random in O ( 1 2 n ln n) steps. Diaconis (1988)

7 Quantum Diaconis-Shahshahani Theory? 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 points, a rôle played by the Haar state h: 1 f (t) = h(f ). G }{{} t G }{{} quantum: no reference to points classical: references points t G

8 Quantum Diaconis-Shahshahani Theory? 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 points, a rôle played by the Haar state h: 1 f (t) = h(f ). G }{{} t G }{{} quantum: no reference to points classical: references points t G...and quantum groups have (co)representations. (This work uses the κ : V V F (G), e j i e i ρ ij formulation).

9 Random Walks on Finite Quantum Groups Definition: The algebra of functions on a (finite) quantum group G, is a finite-dimensional C -Hopf algebra A =: F (G). Early work on quantum stochastic processes by various authors led to random walks on duals of compact groups (particularly Biane) and other examples, but Franz & Gohm (2005) define with clarity a random walk on a finite quantum group. Definition: The sequence of random variables {j i } k i=0 j k = (k) : F (G) F (G), with distributions given by k+1 copies Ψ k j k = (ν k ε) (k) = ν k, (ν M p (G)), is called a right-invariant random walk on G driven by ν.

10 Distance to Random Definition: The random distribution on a finite quantum group, G, is given by the Haar state: π := := h : F (G) C. G Therefore the distance to random might be defined with respect to some norm,, on the quantum group ring CG: ν k π.

11 Distance to Random Definition: The random distribution on a finite quantum group, G, is given by the Haar state: π := := h : F (G) C. G Therefore the distance to random might be defined with respect to some norm,, on the quantum group ring CG: ν k π. The very important question: What are necessary and sufficient conditions on ν M p (G) to ensure that ν k π? is open. In the classical case ν must not be concentrated on a subgroup (irreducibility) nor a coset of a normal subgroup (aperiodicity).

12 Distance to Random For ν k π to be called a quantum total variation distance, such a norm must have three properties: Agreement in the classical case: ν k π = ν k π TV, A Cauchy Schwarz-type inequality (UBL is for a 2 ): ν k π c ν k π 2, For lower bounds, a presentation as a supremum: ν k π = sup F (s, ν k π). s S

13 Distance to Random Consider the one-norm on F (G): a F (G) 1 = G a.

14 Distance to Random Consider the one-norm on F (G): a F (G) 1 = G a. The bijective map F : F (G) CG, defined by F(a)b = ba, allows the norm of elements of CG to be calculated back in F (G), and indeed the norm ν k π := 1 2 F 1 (ν k π) F (G) 1, has the three properties required in order for ν k π to be defined as a quantum total variation distance. G

15 Distance to Random Agreement in the classical case follows from F 1 (δ s ) = G δ s and (ν k π)(δ s ) R for ν M p (G).

16 Distance to Random Agreement in the classical case follows from F 1 (δ s ) = G δ s and (ν k π)(δ s ) R for ν M p (G). Using the map F : F (G) CG an (unnormalised) Haar integral ĥ : CG C on CG may be defined by: ĥ(f(a)) = ε(a) ; e.g. ĥ = ε F 1, and used to define a two-norm on CG: µ CG 2 = ĥ( µ 2 ).

17 Distance to Random Consider the two-norm on F (G): a F (G) 2 = a 2. G Van Daele (2006) proved a Plancherel Theorem: a F (G) 2 = F(a) CG 2,

18 Distance to Random Consider the two-norm on F (G): a F (G) 2 = a 2. G Van Daele (2006) proved a Plancherel Theorem: a F (G) 2 = F(a) CG 2, and using a standard non-commutative L p -space Cauchy Schwarz Inequality: ν k π 1 2 F 1 (ν k π) F (G) 2 = 1 2 ν k π CG 2. The Upper Bound Lemma expresses the square of ν k π CG 2 as a sum over non-trivial representations.

19 Distance to Random There is also a supremum-presentation for lower bounds: ν k π = 1 2 sup φ F (G): φ F (G) 1 ν k (φ) π(φ).

20 Distance to Random There is also a supremum-presentation for lower bounds: ν k π = 1 2 sup φ F (G): φ F (G) 1 ν k (φ) π(φ). The guise of the classical definition favoured by probabilists ν k π TV = sup ν k (S) π(s), S G may be adapted to the quantum case. Define a subset S G by a projection 1 S ; and 1 G = 1 F (G). Then φ = 21 S 1 G is a suitable test function and ν k π ν k (1 S ) π(1 S ).

21 Diaconis-Van Daele Theory Definition: (Simeng Wang (2014)) The Fourier Transform of ν M p (G) is a linear map: ν α Irr(G) L(V α ), where the Fourier Transform of ν at the representation κ α is given by: ν Vα =: ν(α) = (I Vα ν) κ α.

22 Diaconis-Van Daele Theory Definition: (Simeng Wang (2014)) The Fourier Transform of ν M p (G) is a linear map: ν α Irr(G) L(V α ), where the Fourier Transform of ν at the representation κ α is given by: ν Vα =: ν(α) = (I Vα ν) κ α. The map F : F (G) CG also allows a Fourier Transform of a F (G) at the representation κ α to be defined: â(α) := F(a)(α).

23 Diaconis-Van Daele Theory Using the results of Van Daele (concerning the map F), 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 the quantum ν(κ α ).

24 Diaconis-Van Daele Theory Using the results of Van Daele (concerning the map F), 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 the quantum ν(κ α ). 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)

25 Upper Bound Lemma Leaning heavily on the Kac condition and the traciality of G, 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].

26 A Walk on Sekine Quantum Group The Sekine Quantum Group (1996) KP n of order 2n 2 may be realised on F (KP n ) = i,j Z n Ce (i,j) M n (C), with a suitably defined comultiplication. The quantum group KP 2 is commonly mistaken for the Kac-Paljutkin quantum group of order 8. In fact KP 2 = D4.

27 A Walk on Sekine Quantum Group The Sekine Quantum Group (1996) KP n of order 2n 2 may be realised on F (KP n ) = i,j Z n Ce (i,j) M n (C), with a suitably defined comultiplication. The quantum group KP 2 is commonly mistaken for the Kac-Paljutkin quantum group of order 8. In fact KP 2 = D4. Restrict to n odd and consider the random walk driven by ν = 1 4 (e(0,1) + e (1,0) + E 11 + E 12 + E 21 + E 22 ) M p (KP n ).

28 A Walk on Sekine Quantum Group The Sekine Quantum Group (1996) KP n of order 2n 2 may be realised on F (KP n ) = i,j Z n Ce (i,j) M n (C), with a suitably defined comultiplication. The quantum group KP 2 is commonly mistaken for the Kac-Paljutkin quantum group of order 8. In fact KP 2 = D4. Restrict to n odd and consider the random walk driven by ν = 1 4 (e(0,1) + e (1,0) + E 11 + E 12 + E 21 + E 22 ) M p (KP n ). Franz & Skalski (2009) contains formulae which can be used to understand ν k.

29 A Walk on the Sekine Quantum Groups These formulae show that ν k (E ij ) = 0 if i j 0, 1; i.e. off the subdiagonal, the superdiagonal and the diagonal. Regarding the open question mentioned earlier, how can one tell if ν k π?

30 A Walk on the Sekine Quantum Groups These formulae show that ν k (E ij ) = 0 if i j 0, 1; i.e. off the subdiagonal, the superdiagonal and the diagonal. Regarding the open question mentioned earlier, how can one tell if ν k π? In theory, the Upper Bound Lemma can be used in specific cases to answer this very question and indeed in this case ν k π 2 α Irr(KP n)\τ d α Tr [ ( ν (α) ) k ν (α) k] 0. k

31 A Walk on the Sekine Quantum Groups These formulae show that ν k (E ij ) = 0 if i j 0, 1; i.e. off the subdiagonal, the superdiagonal and the diagonal. Regarding the open question mentioned earlier, how can one tell if ν k π? In theory, the Upper Bound Lemma can be used in specific cases to answer this very question and indeed in this case ν k π 2 α Irr(KP n)\τ d α Tr [ ( ν (α) ) k ν (α) k] 0. k One must control terms at 2n 1 one dimensional representations and ( n 2) two dimensional representations. This convergence shows that, with respect to the dual basis: ν k π = 1 2n 2 e (i,j) + 1 2n I n. i,j Z n

32 A Walk on the Sekine Quantum Groups Upper Bound: For k = n αn2 with α 1 and n 7 ν k π e απ2.

33 A Walk on the Sekine Quantum Groups Upper Bound: For k = n αn2 with α 1 and n 7 It should be possible to reduce α. ν k π e απ2. The upper bound for the walk on is dominated by (n 1)/2 representations in particular. The upper bound for these terms: 2 n 1 2 v=1 ( cos 4k 2 πv ) 4e π2 (2k 1)/n 2 ; n is quite sharp. The lower bound ( ) 1 2n ν k 1 CE n+1, 2, n+1 2 in particular might deserve further analysis.

34 Some Questions Irreducibility is harder than the classical case (where not concentrated on a subgroup is enough). Can anything be said about aperiodicity in the quantum case? (U. Franz). Can a lower bound for the studied random walk on KP n, comparable with the upper bound for O(n 2 ) steps, be found? Using a matrix element as a test element, independent of n, crudely, ν k π 1 ( 3 ) k. 2 4 Find a random walk on the family of Sekine quantum groups with a driving probability exhibiting the cut-off phenomenon. Perhaps with ν n M p (KP n ) exhibiting n-dependence. Extend the Upper Bound Lemma to compact quantum groups (with the Kac Property) driven by states ν M p (G) of the form F(a). In the classical case, conjugate-invariant driving probabilities lead to a somewhat-tractable upper bound. Find convergence rates for quantum analogues of classical random walks; e.g. the random transposition shuffle.

35 References 1. Diaconis & Shahshahani (1981), Generating a Random Permutation with Random Transpositions 2. Diaconis (1988), Group Representations in Probability and Statistics 3. Sekine (1996), An Example of Finite Dimensional Kac Algebras of Kac-Paljutkin Type 4. Franz & Gohm (2005), Random Walks on Finite Quantum Groups 5. Van Daele (2006), The Fourier Transform in Quantum Group Theory 6. Franz & Skalski (2009), On Idempotent States on Quantum Groups 7. Simeng Wang (2014), L p -Improving Convolution Operators on Finite Quantum Groups