Some Unresolved and Unexplored Aspects of Random Walks on ... · Some Unresolved and Unexplored...

Some Unresolved and Unexplored Aspects ofRandom Walks on Quantum Groups

J.P. McCarthy

Cork Institute of Technology

14 mai 2019Seminaire d’Analyse Fonctionnelle, Besancon

The slides are dense, and unsuitable for note taking: slidesavailable now at jpmccarthymaths.com

The Classical ProblemWhat card shuffles are ergodic?

Markov (1906), Borel (1940)

Shuffles (σi ∈ S52) are chosen according to a fixed probabilitydistribution ν ∈ Mp(S52). Ergodic card shuffles have the propertythat

limk→∞

P [σk · · ·σ1 · σ0 = σ] =1

That is the distribution of σk · · ·σ1 · σ0 converges to the uniformdistribution.This generalises to the study of random walks on finite groups:

ξ0 → ξ1 → ξ2 → · · · → ξk .

Given a finite group, G , and elements si ∈ G chosen according to afixed probability distribution (ν ∈ Mp(G )), the random variables ξiare given by:

ξi := si · · · s1 · s0.

limk→∞

P [σk · · ·σ1 · σ0 = σ] =1

ξ0 → ξ1 → ξ2 → · · · → ξk .

ξi := si · · · s1 · s0.

limk→∞

P [σk · · ·σ1 · σ0 = σ] =1

That is the distribution of σk · · ·σ1 · σ0 converges to the uniformdistribution.

This generalises to the study of random walks on finite groups:

ξ0 → ξ1 → ξ2 → · · · → ξk .

ξi := si · · · s1 · s0.

limk→∞

P [σk · · ·σ1 · σ0 = σ] =1

ξ0 → ξ1 → ξ2 → · · · → ξk .

ξi := si · · · s1 · s0.

limk→∞

P [σk · · ·σ1 · σ0 = σ] =1

ξ0 → ξ1 → ξ2 → · · · → ξk .

ξi := si · · · s1 · s0.

Distance to Random

Theorem (Folklore)

If the driving probability ν is not supported on a subgroup nor acoset of a normal subgroup then the random walk is ergodic.

If the random walk starts at the identity, ξ0 = e, the distribution ofξk is given by

ν ? · · · ? ν ? ν︸︷︷︸k copies

=: ν?k ,

whereµ ? ν({s}) =

∑t∈G

µ({st−1})ν({t}).

Denote by π ∈ Mp(G ) the uniform or random distribution. Thedistance to random is measured by the total variation distance:

‖ν?k − π‖TV = supS⊂G|ν?k(S)− π(S)|

Distance to Random

Theorem (Folklore)

ν ? · · · ? ν ? ν︸︷︷︸k copies

=: ν?k ,

whereµ ? ν({s}) =

∑t∈G

µ({st−1})ν({t}).

Distance to Random

Theorem (Folklore)

ν ? · · · ? ν ? ν︸︷︷︸k copies

=: ν?k ,

whereµ ? ν({s}) =

∑t∈G

µ({st−1})ν({t}).

Finite Quantum Groups.. ahistorically

Define a finite group as an object in the category FinSet, togetherwith morphisms m, e, and −1, with the group axioms given byappropriate commutative diagrams.

Consider the composition offunctors

GC−→ CG D−→ (CG )∗ ∼= F (G ),

the algebra of functions on G . Apply the functor composition tothe morphisms m, e, and −1 to give maps ∆ (comultiplication), ε(counit), and S (antipode). The image of the commutativediagrams expressing associativity, identity, and inverses, arecommutative diagrams called coassociativity, the counital property,and the antipodal property:

(∆⊗ IF (G)) ◦∆ = (IF (G) ⊗∆) ◦∆

(ε⊗ IF (G)) ◦∆ = IF (G) = (IF (G) ⊗ ε) ◦∆

M ◦ (S ⊗ IF (G)) ◦∆ = ηF (G) ◦ ε = M ◦ (IF (G) ⊗ S) ◦∆

Define a finite group as an object in the category FinSet, togetherwith morphisms m, e, and −1, with the group axioms given byappropriate commutative diagrams. Consider the composition offunctors

GC−→ CG D−→ (CG )∗ ∼= F (G ),

the algebra of functions on G .

Apply the functor composition tothe morphisms m, e, and −1 to give maps ∆ (comultiplication), ε(counit), and S (antipode). The image of the commutativediagrams expressing associativity, identity, and inverses, arecommutative diagrams called coassociativity, the counital property,and the antipodal property:

(∆⊗ IF (G)) ◦∆ = (IF (G) ⊗∆) ◦∆

(ε⊗ IF (G)) ◦∆ = IF (G) = (IF (G) ⊗ ε) ◦∆

M ◦ (S ⊗ IF (G)) ◦∆ = ηF (G) ◦ ε = M ◦ (IF (G) ⊗ S) ◦∆

GC−→ CG D−→ (CG )∗ ∼= F (G ),

the algebra of functions on G . Apply the functor composition tothe morphisms m, e, and −1 to give maps ∆ (comultiplication), ε(counit), and S (antipode).

The image of the commutativediagrams expressing associativity, identity, and inverses, arecommutative diagrams called coassociativity, the counital property,and the antipodal property:

(∆⊗ IF (G)) ◦∆ = (IF (G) ⊗∆) ◦∆

(ε⊗ IF (G)) ◦∆ = IF (G) = (IF (G) ⊗ ε) ◦∆

M ◦ (S ⊗ IF (G)) ◦∆ = ηF (G) ◦ ε = M ◦ (IF (G) ⊗ S) ◦∆

GC−→ CG D−→ (CG )∗ ∼= F (G ),

the algebra of functions on G . Apply the functor composition tothe morphisms m, e, and −1 to give maps ∆ (comultiplication), ε(counit), and S (antipode). The image of the commutativediagrams expressing associativity, identity, and inverses, arecommutative diagrams called coassociativity, the counital property,and the antipodal property:

(∆⊗ IF (G)) ◦∆ = (IF (G) ⊗∆) ◦∆

(ε⊗ IF (G)) ◦∆ = IF (G) = (IF (G) ⊗ ε) ◦∆

M ◦ (S ⊗ IF (G)) ◦∆ = ηF (G) ◦ ε = M ◦ (IF (G) ⊗ S) ◦∆

Finite Quantum GroupsThe comultiplication ∆ is a *-homomorphism, and F (G ) aC∗-algebra.

A finite dimensional C∗-algebra A with a *-homomorphism∆ : A→ A⊗ A, and maps ε, and S , that satisfies the aboverelations, but is not necessarily commutative is thus considered thealgebra of functions on a finite quantum group.

DefinitionAn algebra of functions on a finite quantum group G , is a unitalC∗-algebra A =: F (G ) with a *-homomorphism ∆, a counit ε, andan antipode S , satisfying coassociativity, etc..

We will also refer to compact quantum groups via their algebra ofcontinuous functions, C (G ). Their definition may be motivated bythe theorem that

semigroup & cancellation ⇒ group .

Random Walks on Finite Quantum Groups [FG05]A probability ν on a classical group gives rise to a state on F (G ):f 7→

∑t∈G f (t) ν({t}) =: ν(f ).

Therefore the quantum probabilistic identifications are made for a(compact) quantum group G :

ν ∈ Mp(G )←→ a state ν on C (G );

(distribution of) random walk on G ←→ ν?k (ν?ν = (ν⊗ν)◦∆)

Note that for classical f ∈ F (G ):

π(f ) =∑t∈G

f (t)π({t}) =1

|G |∑t∈G

f (t) = f ,

the average — over all points t ∈ G — of f .

Note that a (compact) quantum group also has a “uniformdistribution”. Given by the Haar state,

∫G , it is also invariant in

the sense that∫G ?ν =

∫G = ν ?

∫G for all ν ∈ Mp(G ).

∑t∈G f (t) ν({t}) =: ν(f ).

(distribution of) random walk on G ←→ ν?k

(ν?ν = (ν⊗ν)◦∆)

π(f ) =∑t∈G

f (t)π({t}) =1

|G |∑t∈G

f (t) = f ,

∫G = ν ?

∑t∈G f (t) ν({t}) =: ν(f ).

π(f ) =∑t∈G

f (t)π({t}) =1

|G |∑t∈G

f (t) = f ,

∫G = ν ?

∑t∈G f (t) ν({t}) =: ν(f ).

π(f ) =∑t∈G

f (t)π({t}) =1

|G |∑t∈G

f (t) = f ,

∫G = ν ?

∑t∈G f (t) ν({t}) =: ν(f ).

π(f ) =∑t∈G

f (t)π({t}) =1

|G |∑t∈G

f (t) = f ,

∫G = ν ?

Personal Context

I Began PhD studies in October 2010.

I Began teaching at CIT in February 2011.

I Very busy and so

I Still teaching at CIT, a “technological university”; still tryingto pursue research.

Personal Context

I Very busy and so

Personal Context

I Very busy and so

Personal Context

I Very busy and so

Diaconis–Shahshahani Upper Bound Lemma

Let ν ∈ Mp(G ) be a probability on a finite quantum group G withHaar state

∫G .

Using the quantum identification subsets ←→ projections, thedistance between probabilities ν, µ ∈ Mp(G ) is given by the totalvariation distance:

‖ν − µ‖ := supp∈F (G) proj.

|ν(p)− µ(p)| =1

2‖aν − aµ‖1,

where aν ∈ F (G ) is the density of ν ∈ MP(G ):

ν(b) =

∫Gaνb ; (b ∈ F (G )).

Of primary interest are random walks which ergodic; that convergeto random in the sense that ν?k → π :=

∫G .

|ν(p)− µ(p)|

2‖aν − aµ‖1,

ν(b) =

∫Gaνb ; (b ∈ F (G )).

∫G .

|ν(p)− µ(p)| =1

2‖aν − aµ‖1,

ν(b) =

∫Gaνb ; (b ∈ F (G )).

∫G .

|ν(p)− µ(p)| =1

2‖aν − aµ‖1,

ν(b) =

∫Gaνb ; (b ∈ F (G )).

∫G .

Diaconis–Shahshahani Upper Bound LemmaLet Irr(G ) be an index set for a family of pairwise-inequivalentirreducible unitary representations of a (finite) quantum group G .Representations of G are corepresentations of F (G ),κα : Vα → Vα ⊗ F (G ), and dα := dimVα.

The Fourier Transform

F (G )→⊕

α∈Irr(G)

L(V α) , aν 7→ ν,

defined for each α ∈ Irr(G ):

ν(α) :=(IVα ⊗ ν

)◦ κα.

Theorem ([McC17])

‖ν?k − π‖2 ≤ 1

∑α∈Irr(G)\{τ}

dα Tr[(ν (α)∗)

k(ν (α))k

Diaconis–Shahshahani Upper Bound LemmaLet Irr(G ) be an index set for a family of pairwise-inequivalentirreducible unitary representations of a (finite) quantum group G .Representations of G are corepresentations of F (G ),κα : Vα → Vα ⊗ F (G ), and dα := dimVα. The Fourier Transform

F (G )→⊕

α∈Irr(G)

L(V α) , aν 7→ ν,

)◦ κα.

Theorem ([McC17])

‖ν?k − π‖2 ≤ 1

∑α∈Irr(G)\{τ}

dα Tr[(ν (α)∗)

k(ν (α))k

Diaconis–Shahshahani Upper Bound LemmaLet Irr(G ) be an index set for a family of pairwise-inequivalentirreducible unitary representations of a (finite) quantum group G .Representations of G are corepresentations of F (G ),κα : Vα → Vα ⊗ F (G ), and dα := dimVα. The Fourier Transform

F (G )→⊕

α∈Irr(G)

L(V α) , aν 7→ ν,

)◦ κα.

Theorem ([McC17])

‖ν?k − π‖2 ≤ 1

∑α∈Irr(G)\{τ}

dα Tr[(ν (α)∗)

k(ν (α))k

Diaconis–Shahshahani Upper Bound LemmaAmaury Freslon [AF18A] extended the upper bound lemma tocompact quantum groups of Kac type (S2 = IC(G)). For the totalvariation distance, the projections are in L∞(G ), and the drivingprobability ν ∈ Mp(G ) must therefore extend to L∞(G ), and alsohave an L1-density aν ∈ L1(G ).

Once these analytic concerns are taken account of, theDiaconis–Shahshahani Upper Bound Lemma for compact quantumgroups is the same as its finite counterpart (and let it be said theclassical counterpart which inspired the generalisation).

Lower Bounds for the distance to random can be found bychoosing a suitable test projection, or a suitable test function viathe presentation∥∥∥ν?k − π∥∥∥ =

2sup‖φ‖∞≤1

∣∣∣ν?k(φ)− π(φ)∣∣∣ .

2sup‖φ‖∞≤1

∣∣∣ν?k(φ)− π(φ)∣∣∣ .

2sup‖φ‖∞≤1

∣∣∣ν?k(φ)− π(φ)∣∣∣ .

Applications

The Upper Bound Lemma has been used to analyse random walkson

I the dual symmetric group, Sn [McC17]

I Sekine quantum groups, Yn [IB18,McC17,McC19]

I the Kac–Paljutkin quantum group, G0 [IB18]

I free orthogonal quantum groups, O+N [AF18A]

I free symmetric quantum groups, S+N [AF18A]

I the quantum automorphism group of (MN(C), tr) [AF18A]

I free unitary groups, U+N [AF18B]

I free wreath products Γ o∗ S+N , including quantum reflection

groups Hs +N [AF18B]

I duals of discrete groups, Γ, including for Γ = FN the freegroup on N generators, [AF18C]

Applications

Aspects as yet Unresolved and Unexplored in the QuantumSetting

Some things to look at:

I Spectral Analysis of the Stochastic Operator for ReversibleWalks

I Ergodicity

I Irreducibility

I A Sufficient Condition for Convergence?

I Periodicity

I Shannon Entropy

I Non-Kac Case?

I Homogeneous Spaces

Some “low-hanging fruit”; some hard problems.

Spectral Analysis for Reversible Walks

Let ν ∈ Mp(G ) define a random walk on a (compact) quantumgroup G . Define the stochastic operator Pν : C (G )→ C (G ) by

Pν = (ν ⊗ IC(G)) ◦∆.

For µ ∈ Mp(G ), PTν µ = ν ? µ. A standard treatment of (finite)

classical Markov chains is to do spectral analysis on stochasticoperators P ∈ L(F (X )). Let λ? be the second largest (inmagnitude) eigenvalue of Pν .Two classical results that might bestraightforward propositions for random walks on a finite quantumgroup are

‖ν?k − π‖ ≤ C λk? ,

and (reversible/symmetric walks ν = ν ◦ S)

‖ν?k − π‖2 ≤ |G | − 1

4λ2k? .

Pν = (ν ⊗ IC(G)) ◦∆.

classical Markov chains is to do spectral analysis on stochasticoperators P ∈ L(F (X )).

Let λ? be the second largest (inmagnitude) eigenvalue of Pν .Two classical results that might bestraightforward propositions for random walks on a finite quantumgroup are

‖ν?k − π‖ ≤ C λk? ,

‖ν?k − π‖2 ≤ |G | − 1

4λ2k? .

Pν = (ν ⊗ IC(G)) ◦∆.

classical Markov chains is to do spectral analysis on stochasticoperators P ∈ L(F (X )). Let λ? be the second largest (inmagnitude) eigenvalue of Pν .

Two classical results that might bestraightforward propositions for random walks on a finite quantumgroup are

‖ν?k − π‖ ≤ C λk? ,

‖ν?k − π‖2 ≤ |G | − 1

4λ2k? .

Pν = (ν ⊗ IC(G)) ◦∆.

classical Markov chains is to do spectral analysis on stochasticoperators P ∈ L(F (X )). Let λ? be the second largest (inmagnitude) eigenvalue of Pν .Two classical results that might bestraightforward propositions for random walks on a finite quantumgroup are

‖ν?k − π‖ ≤ C λk? ,

‖ν?k − π‖2 ≤ |G | − 1

4λ2k? .

Total Variation Distance Monotonic Decreasing?

Let ν ∈ Mp(G ) define a random walk on a (finite) quantum groupG . Define

d(k) := ‖ν?k − π‖

=CQG: [AF18A]

2‖aν?k − 1G‖1.

Note that, classically:

d(k + 1) =1

2‖PT

ν (ν?k − π)‖`1 ≤ ‖PTν ‖1 · d(k),

and Pν is doubly stochastic and so d(k) is monotonic decreasing.The one norm, classically, is also a Riesz norm on F (G )s.a., andthus the operator norm is determined by the positive cone. Theone norm, in the quantum case, is not a Riesz norm (a theorem ofS. Sherman). Showing that ‖Pν‖1 = 1 in the quantum case wouldshow that d(k) is monotonic decreasing...

d(k) := ‖ν?k − π‖ =CQG: [AF18A]

2‖aν?k − 1G‖1.

d(k + 1) =1

2‖PT

ν (ν?k − π)‖`1 ≤ ‖PTν ‖1 · d(k),

d(k) := ‖ν?k − π‖ =CQG: [AF18A]

2‖aν?k − 1G‖1.

d(k + 1) =1

2‖PT

ν (ν?k − π)‖`1 ≤ ‖PTν ‖1 · d(k),

and Pν is doubly stochastic and so d(k) is monotonic decreasing.

The one norm, classically, is also a Riesz norm on F (G )s.a., andthus the operator norm is determined by the positive cone. Theone norm, in the quantum case, is not a Riesz norm (a theorem ofS. Sherman). Showing that ‖Pν‖1 = 1 in the quantum case wouldshow that d(k) is monotonic decreasing...

d(k) := ‖ν?k − π‖ =CQG: [AF18A]

2‖aν?k − 1G‖1.

d(k + 1) =1

2‖PT

ν (ν?k − π)‖`1 ≤ ‖PTν ‖1 · d(k),

and Pν is doubly stochastic and so d(k) is monotonic decreasing.The one norm, classically, is also a Riesz norm on F (G )s.a., andthus the operator norm is determined by the positive cone. Theone norm, in the quantum case, is not a Riesz norm (a theorem ofS. Sherman).

Showing that ‖Pν‖1 = 1 in the quantum case wouldshow that d(k) is monotonic decreasing...

d(k) := ‖ν?k − π‖ =CQG: [AF18A]

2‖aν?k − 1G‖1.

d(k + 1) =1

2‖PT

ν (ν?k − π)‖`1 ≤ ‖PTν ‖1 · d(k),

Total Variation Distance Monotonic Decreasing?On the bus this morning, I realised for b ∈ F (G ), and ν with adensity:

‖Pνb‖1 = ‖S(aν) ?A b‖1 ≤SW‖S(aν)‖1‖b‖1 = ‖b‖1 •

In the (finite) quantum case, where

s(k) := ‖ν?k − π‖∞ := ‖aν?k − 1G‖∞,

s(k) is monotonic decreasing. Where F : aν 7→ ν, and ?A theconvolution in F (G ):

s(k + 1) = ‖PF(S(aν))(a?Ak − 1G )‖∞ ≤ ‖PF(S(aν))‖∞ · s(k),

and as F (G ) together with ‖ · ‖∞ is a C∗-algebra, and PF(S(aν))unital and positive, the norm is given by ‖PF(S(aν))(1G )‖∞ = 1.

The Upper Bound Lemma might possibly help show thatt(k) := ‖ν?k − π‖2 is monotonic decreasing

s(k) := ‖ν?k − π‖∞ := ‖aν?k − 1G‖∞,

s(k) is monotonic decreasing.

Where F : aν 7→ ν, and ?A theconvolution in F (G ):

s(k) := ‖ν?k − π‖∞ := ‖aν?k − 1G‖∞,

Ergodicity

A classical random walk on a finite group G is ergodic if it is bothirreducible and aperiodic. A random walk is irreducible if and onlyif ν ∈ Mp(G ) is not concentrated on a proper subgroup,

andaperiodic if and only if ν is not concentrated on a coset of a propernormal subgroup.

For a random walk driven by ν, the convolution powers ν?k → π ifand only if the walk is ergodic.

Of course, if the walk is not irreducible but aperiodic, theconvolution powers converge to the uniform distribution/Haarmeasure on a subgroup H.

Things are not so straightforward for random walks on quantumgroups.

Ergodicity

A classical random walk on a finite group G is ergodic if it is bothirreducible and aperiodic. A random walk is irreducible if and onlyif ν ∈ Mp(G ) is not concentrated on a proper subgroup, andaperiodic if and only if ν is not concentrated on a coset of a propernormal subgroup.

Ergodicity

Already for the Kac-Paljutkin Quantum Group G0 of order eight,there are two idempotent states that are not the Haar state on anyquantum subgroup [AP96]. It is an important problem to findnecessary and sufficient conditions on ν for the random walk drivenby ν to be ergodic as was already noted in [FS08].

Baraquin [IB18] uses the Upper Bound Lemma to find necessaryand sufficient conditions for ergodicity for random walks on theKac-Paljutkin quantum group G0 and the Sekine quantum groupsYn, arising from the Fourier Transform (F(aν) = ν) of linearcombinations of irreducible characters.

These conditions are given by inequalities between the absolutevalues of the coefficients of F−1(ν) in thecharacter-linear-expansion, and the dimension of the representationof the associated character.

Ergodicity

Irreducibility

Define the support of a ν ∈ Mp(G ) as the smallest projection pνsuch that ν(pν) = 1. Perhaps the appropriate definition forirreducibility (finite) is

∀ projections p ∈ F (G ), ∃ k ∈ N : ν?k(p) 6= 0.

Consider a finite quantum group G and a subgroup H given by asurjective unital *-homomorphism πH : F (G )→ F (H). Note thatF (G ) is a multi-matrix algebra so that

F (G ) ∼= ker πH ⊕ ImπH ∼= ker πH ⊕ F (H).

This gives an embedding

ı : F (H) ↪→ ker πH ⊕ F (H) ⊂ F (G ); a 7→ 0⊕ a.

If pν ≤ 1H := ıπH(1G ) for some quantum subgroup H of G , thenν(1H) = ν(ıπH(1G )) = 1, so that ν is supported on H.

Irreducibility

ı : F (H) ↪→ ker πH ⊕ F (H) ⊂ F (G ); a 7→ 0⊕ a.

Irreducibility

ı : F (H) ↪→ ker πH ⊕ F (H) ⊂ F (G ); a 7→ 0⊕ a.

Irreducibility

ı : F (H) ↪→ ker πH ⊕ F (H) ⊂ F (G ); a 7→ 0⊕ a.

IrreducibilityIs it the case that ν?k remains supported on H? Classically this istrivially true just by considering the random variable picture.

Itmay be a theorem (or theorem-definition?) that ν supported on H⇔ νıπH = ν. Thereafter, with ν, µ supported on H, together withπH ıπH = πH :

(µ ? ν)ıπH = ((µıπH)⊗ (νıπH)) ◦∆ ◦ ıπH= (µı⊗ νı)(πH ◦ πH)∆ ◦ ıπH= (µı⊗ νı)(∆H ◦ πH ◦ ı ◦ πH)

= (µı⊗ νı)(∆H ◦ πH)

= (µı⊗ νı) ◦ (πH ◦ πH) ◦∆

= (µıπH ⊗ νıπH) ◦∆

= µ ? ν

Franz & Skalski [FS09] show that idempotent states on finitequantum groups correspond to pre-subgroups. Perhapsirreducibility might be equivalent to not concentrated on apre-subgroup?

IrreducibilityIs it the case that ν?k remains supported on H? Classically this istrivially true just by considering the random variable picture. Itmay be a theorem (or theorem-definition?) that ν supported on H⇔ νıπH = ν.

Thereafter, with ν, µ supported on H, together withπH ıπH = πH :

= (µı⊗ νı)(∆H ◦ πH)

= (µı⊗ νı) ◦ (πH ◦ πH) ◦∆

= µ ? ν

IrreducibilityIs it the case that ν?k remains supported on H? Classically this istrivially true just by considering the random variable picture. Itmay be a theorem (or theorem-definition?) that ν supported on H⇔ νıπH = ν. Thereafter, with ν, µ supported on H, together withπH ıπH = πH :

= (µı⊗ νı)(∆H ◦ πH)

= (µı⊗ νı) ◦ (πH ◦ πH) ◦∆

= µ ? ν

IrreducibilityIs it the case that ν?k remains supported on H? Classically this istrivially true just by considering the random variable picture. Itmay be a theorem (or theorem-definition?) that ν supported on H⇔ νıπH = ν. Thereafter, with ν, µ supported on H, together withπH ıπH = πH :

= (µı⊗ νı)(∆H ◦ πH)

= (µı⊗ νı) ◦ (πH ◦ πH) ◦∆

= µ ? ν

A Sufficient Condition for Convergence?

In the classical case, e ∈ supp ν implies that ν?k converges (to theHaar state on the smallest subgroup of G on which ν is supported).

Consider the Sekine Quantum groups [YS96], a family of order 2n2

finite quantum groups with algebra of functions:

F (Yn) =

⊕i ,j∈Zn

Ce(i ,j)

⊕Mn(C).

The counit is given by the coefficient of the e(0,0) one-dimensionalfactor. Consider the basis of CYn dual to the F (Yn) basis

{e(i ,j) : i , j ∈ Zn} ∪ {Eij : i , j = 1, 2, . . . , n}

denoted by

{e(i ,j) : i , j ∈ Zn} ∪ {E ij : i , j = 1, 2, . . . , n}.

In the classical case, e ∈ supp ν implies that ν?k converges (to theHaar state on the smallest subgroup of G on which ν is supported).Consider the Sekine Quantum groups [YS96], a family of order 2n2

F (Yn) =

⊕i ,j∈Zn

Ce(i ,j)

⊕Mn(C).

The counit is given by the coefficient of the e(0,0) one-dimensionalfactor.

Consider the basis of CYn dual to the F (Yn) basis

{e(i ,j) : i , j ∈ Zn} ∪ {Eij : i , j = 1, 2, . . . , n}

denoted by

{e(i ,j) : i , j ∈ Zn} ∪ {E ij : i , j = 1, 2, . . . , n}.

In the classical case, e ∈ supp ν implies that ν?k converges (to theHaar state on the smallest subgroup of G on which ν is supported).Consider the Sekine Quantum groups [YS96], a family of order 2n2

F (Yn) =

⊕i ,j∈Zn

Ce(i ,j)

⊕Mn(C).

The counit is given by the coefficient of the e(0,0) one-dimensionalfactor. Consider the basis of CYn dual to the F (Yn) basis

{e(i ,j) : i , j ∈ Zn} ∪ {Eij : i , j = 1, 2, . . . , n}

denoted by

{e(i ,j) : i , j ∈ Zn} ∪ {E ij : i , j = 1, 2, . . . , n}.

Zhang [HZ18] proves that for a random walk on Yn driven by

ν =∑i ,j∈Zn

α(i ,j)e(i ,j) +

n∑r ,s=1

κrsErs ,

that α(0,0) > 0 implies that ν?k converges (although notnecessarily to the Haar state on a subgroup).

Could this classically true result be true generally in the quantumcase? For finite quantum groups, at least, there must be aone-dimensional factor such that

F (G ) = Ce1 ⊕ ker ε, ε(a1e1 ⊕ b) = a1,

so in the finite case, considering the dual basis, the conjecture isthat if ν = α1e

1 + · · · , that α1 6= 0 implies convergence of theconvolution powers.

ν =∑i ,j∈Zn

α(i ,j)e(i ,j) +

n∑r ,s=1

κrsErs ,

Could this classically true result be true generally in the quantumcase?

For finite quantum groups, at least, there must be aone-dimensional factor such that

F (G ) = Ce1 ⊕ ker ε, ε(a1e1 ⊕ b) = a1,

ν =∑i ,j∈Zn

α(i ,j)e(i ,j) +

n∑r ,s=1

κrsErs ,

Could this classically true result be true generally in the quantumcase? For finite quantum groups, at least, there must be aone-dimensional factor such that

F (G ) = Ce1 ⊕ ker ε, ε(a1e1 ⊕ b) = a1,

PeriodicityIn the classical case, if ν is supported on a coset gH of a propernormal subgroup, then ν?k is supported on gkH and so cannotconverge. Does the same hold in the quantum case?

Consider the Kac-Paljutkin quantum group of order eight [KP66],with algebra of functions F (G0) = C4 ⊕M2(C) with basis{e1, e2, e3, e4,E11,E12,E21,E22} and elements of the dual basisgiven by e i (e1 = ε) and E ij . It is the case that

(e2)?k =

{e2, if k odd

e1, if k even.

Is it the case that e2 is concentrated on the coset of a propernormal subgroup?

In the classical case, the random walk is periodic if and only if ν isconcentrated on such a coset. Is it the case in the quantum casethat concentrated on such a coset implies periodicity?

Consider the Kac-Paljutkin quantum group of order eight [KP66],with algebra of functions F (G0) = C4 ⊕M2(C) with basis{e1, e2, e3, e4,E11,E12,E21,E22} and elements of the dual basisgiven by e i (e1 = ε) and E ij .

It is the case that

(e2)?k =

{e2, if k odd

e1, if k even.

(e2)?k =

{e2, if k odd

e1, if k even.

(e2)?k =

{e2, if k odd

e1, if k even.

Diameter of Finite Quantum Group?

In the classical case volume and diameter bounds can be used toanalyse convergence rates.

A naıve approach to the (finite) case would look as follows. Let pkbe the support of ν?k , and define V (k) := dim ran pk be thevolume, and D the diameter to be the minimum k such thatV (k) = |G | := dimF (G ).

One would then say that the random walk has (A, d) moderategrowth if

V (k) ≥ |G |A

, 1 ≤ k ≤ ∆.

The classical study of random walks uses these ideas to producebounds on the distance to random. Typically such random walksdo not exhibit the cut-off phenomenon.

V (k) ≥ |G |A

, 1 ≤ k ≤ ∆.

V (k) ≥ |G |A

, 1 ≤ k ≤ ∆.

V (k) ≥ |G |A

, 1 ≤ k ≤ ∆.

Convolution FactorisationsConsider the group Sn and symmetric probabilities νi ∈ Mp(Sn)uniform on {(i , i), . . . , (i , n)}. The uniform/random/Haardistribution on Sn has a convolution factorisation

π = νn−1 ? · · · ? ν2 ? ν1.

Urban [RU97] poses the classical problem, given a support S ⊂ G ,does there exist a finite number of convolutions of symmetricprobabilities νi supported on S such that π has a convolutionfactorisation π = νm ? · · · ? ν1?

It may be deduced — even in the quantum case — that if π has aconvolution factorisation, then, of course,

∑α∈Irr(G)\{τ}

dα Tr

[(1∏

νi (α)∗

)(1∏

νi (α)

)]= 0,

and, on the other hand that∏1

i=m νi (α) 6= 0 implies that there isnot a convolution factorisation.

π = νn−1 ? · · · ? ν2 ? ν1.

∑α∈Irr(G)\{τ}

dα Tr

[(1∏

νi (α)∗

)(1∏

νi (α)

)]= 0,

π = νn−1 ? · · · ? ν2 ? ν1.

∑α∈Irr(G)\{τ}

dα Tr

[(1∏

νi (α)∗

)(1∏

νi (α)

)]= 0,

π = νn−1 ? · · · ? ν2 ? ν1.

∑α∈Irr(G)\{τ}

dα Tr

[(1∏

νi (α)∗

)(1∏

νi (α)

)]= 0,

Shannon Entropy

Classically, the Relative Shannon Entropy ‘distance’

S(ν, π) =

∫Gaν ln(aν)

has the property that S(δg , π) = ln |G |, S(π, π) = 0, and S(ν?k , π)is monotonic decreasing in k. There is also a relation betweenS(ν?k , π) and k S(ν, π) (that I cannot quite write down).

Considering S as a distance to random is somewhat moving awayfrom the Diaconis–Shahshahani approach, but it is used in thestudy of random walks on groups, and Crann and Kalanter [CK14]amongst others have worked on entropy in the context of quantumgroups.

Shannon Entropy

S(ν, π) =

∫Gaν ln(aν)

Considering S as a distance to random is somewhat moving awayfrom the Diaconis–Shahshahani approach, but it is used in thestudy of random walks on groups,

and Crann and Kalanter [CK14]amongst others have worked on entropy in the context of quantumgroups.

Shannon Entropy

S(ν, π) =

∫Gaν ln(aν)

Considering S as a distance to random is somewhat moving awayfrom the Diaconis–Shahshahani approach, but it is used in thestudy of random walks on groups, and Crann and Kalanter [CK14]amongst others have worked on entropy in the context of quantumgroups.

Shannon Entropy

Thus there is probably some scope to prove some results using thismachinery.

This might connect — if mirroring the classical approach toentropy — with the world of Cesaro Means:

νk =1

k∑s=1

ν?s ,

already important in the study of compact quantum groups.

It might also be interesting to see what Fourier Theory can say inlight of calculations such as:

Pν(νk) =k + 1

k· νk+1 −

k· ν.

Shannon Entropy

νk =1

k∑s=1

ν?s ,

Pν(νk) =k + 1

k· νk+1 −

k· ν.

Shannon Entropy

νk =1

k∑s=1

ν?s ,

Pν(νk) =k + 1

k· νk+1 −

k· ν.

Non-Kac Case (Freslon)?Crucial to the work thus far is the result that

‖ν − π‖ =1

2‖aν − 1G‖1.

There is a proof for finite G [McC19], and [AF18A] has the moregeneral result for compact Kac G .

AF believes the result is truebut has not tried to prove it and suggests that the Jordandecomposition may be useful.

If the previous question is solved, then one can consider randomwalks on compact quantum groups which are not of Kac type. TheUpper Bound Lemma will then involve matrices Qα measuring themodular theory of the Haar state and some (but not all) dimensionsin the formulas must be replaced by quantum dimensions.

For estimates, the main problem here is to define explicit centralstates since there is no Haar-state preserving conditionalexpectation onto the central algebra (where the analysis istractable). However, there are tools from monoidal equivalence todo this.

‖ν − π‖ =1

2‖aν − 1G‖1.

There is a proof for finite G [McC19], and [AF18A] has the moregeneral result for compact Kac G . AF believes the result is truebut has not tried to prove it and suggests that the Jordandecomposition may be useful.

‖ν − π‖ =1

2‖aν − 1G‖1.

‖ν − π‖ =1

2‖aν − 1G‖1.

Homogeneous SpacesLet G

αy X be a transitive action of a finite group on a finite setX . Fix an x0 ∈ X and consider the isotropy subgroup N ⊂ G .

The standard isomorphism X 3 gx0ϕ7−→ gN ∈ G/N respects the

action. A probability ν ∈ Mp(G ) induces a probability ν ∈ Mp(X )by

ν(xi ) = ν(gix0) := ν(ϕ(gix0)) = ν(giN).

A random walk on G given by ξ1, ξ2, . . . induces a Markov chainon X given by ξ1x0, ξ2x0, . . . .

The action Gαy X induces a representation ρ of G on F (X ) by

ρ(g)f (x) = f (α(g−1, x)). This representation splits into a directsum of irreducible representations ρβ indexed by Irr(ρ).

Diaconis [D88] exploits representation theory to prove for a rightN-invariant ν ∈ Mp(G ), π the uniform distribution on X , an upperbound lemma for this Markov chain on X , and shows that when(G ,N) is a Gelfand pair, that the Fourier analysis becomes ‘simple’.

action.

A probability ν ∈ Mp(G ) induces a probability ν ∈ Mp(X )by

Diaconis [D88] exploits representation theory to prove for a rightN-invariant ν ∈ Mp(G ), π the uniform distribution on X , an upperbound lemma for this Markov chain on X ,

and shows that when(G ,N) is a Gelfand pair, that the Fourier analysis becomes ‘simple’.

References

[IB18] Baraquin, Random Walks on Finite Quantum Groups

[CK14] Crann & Kalantar, An uncertainty principle for unimodularquantum groups

[D88] Diaconis, Group Representations in Probability and Statistics

[FG05] Franz & Gohm, Random Walks on Finite Quantum Groups

[FS08] Franz & Skalski, On ergodic properties of convolutionoperators associated with compact quantum groups

[FS09] Franz & Skalski, A new characterisation of idempotent stateson finite and compact quantum groups

[AF18A] Freslon, Cut-off phenomenon for random walks on freeorthogonal quantum groups

[AF18B] Freslon, Quantum reflections, random walks and cut-off

[AF18C] Freslon, Positive definite functions and cut-off for discretegroups

More References

[KP66] Kac & Paljutkin, Finite Group Rings

[McC17] McCarthy, Random Walks on Finite Quantum Groups:Diaconis–Shahshahani Theory for Quantum Groups

[McC19] McCarthy, Diaconis–Shahshahani Upper Bound Lemma forQuantum Groups

[AP96] Pal, A counterexample on idempotent states on a compactquantum group

[YS96] Sekine, An Example of Finite Dimensional Kac Algebras ofKac-Paljutkin Type

[RU97] Urban, Some Remarks on the Random Walk on Finite Groups

[HZ18] Zhang, Idempotent States on Sekine Quantum Groups

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