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Periodicity of Random Walks on Finite Quantum Groups

September 12, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research Updates | Leave a comment

In the case of a finite classical group G, we can show that if we have i.i.d. random variables \zeta_i\sim\nu\in M_p(G), that if \text{supp }\nu\subset Ng, for Ng a coset of a proper normal subgroup N\rhd G, that the random walk on G driven by \nu, the random variables:

\xi_k=\zeta_k\cdots \zeta_1,

exhibits a periodicity because

\xi_k\in Ng^{k}.

This shows that a necessary condition for ergodicity of a random walk on a finite classical group G driven by \nu\in M_p(G) is that the support of \nu not be concentrated on the coset of a proper normal subgroup.

I had hoped that something similar might hold for the case of random walks on finite quantum groups but alas I think I have found a barrier.

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Publication

February 25, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups, Journal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

Abstract

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group \widehat{S_n} as well as on the truly quantum groups of Sekine.

General Mathematical Audience Talk: The Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

September 1, 2018 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

Abstract Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.

The Cut-Off Phenomenon in Random Walks on Compact Quantum Groups

February 13, 2018 in Quantum Groups, Research, Research Updates | 1 comment

Amaury Freslon has put a pre-print on the arXiv, Cut-off phenomenon for random walks on free orthogonal quantum groups, that answers so many of these questions, some of which appeared as natural further problems in my PhD thesis.

It really is a fantastic paper and I am delighted to see my PhD work cited: it appears that while I may have taken some of the low hanging fruit, Amaury has really extended these ideas and has developed some fantastic examples: all beyond my current tools.

This pre-print gives me great impetus to draft a pre-print of my PhD work, hopefully for publication. I am committed to improving my results and presentation, and Amaury’s paper certainly provides some inspiration is this direction.

As things stand I do not have to tools to develop results as good as Amaury’s. Therefore I am trying to develop my understanding of compact quantum groups and their representation theory. Afterwards I can hopefully study some of the remaining further problems mentioned in the thesis.

As suggested by Uwe Franz, representation theoretic methods (such as presented by Diaconis (1988) for the classical case), might be useful for analysing random walks on quantum homogeneous spaces.

Talk: The Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

May 21, 2017 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | 1 comment

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

Abstract A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.

PhD — Finished!

May 9, 2017 in General, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

After a long time I have finally completed my PhD studies when I handed in my hardbound thesis (a copy of which you can see here).

It was a very long road but thankfully now the pressure is lifted and I can enjoy my study of quantum groups and random walks thereon for many years to come.

Quantum Diaconis-Shahshahani Lemma

May 21, 2015 in General, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Let \mathbb{G} be a finite quantum group described by A=\mathcal{C}(\mathbb{G}) with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) S^2=I_A. Under the assumption of finiteness, there is a unique Haar state, h:A\rightarrow \mathbb{C} on A.

Representation Theory

A representation of \mathbb{G} is a linear map \kappa:V\rightarrow V\otimes A that satisfies

\left(\kappa\otimes I_A\right)\circ\kappa =\left(I_V\otimes \Delta\right)\circ \kappa\text{\qquad and \qquad}\left(I_V\otimes\varepsilon\right)\circ \kappa=I_V.

The dimension of \kappa is given by \dim\,V. If V has basis \{e_i\} then we can define the matrix elements of \kappa by

\displaystyle\kappa\left(e_j\right)=\sum_i e_i\otimes\rho_{ij}.

One property of these that we will use it that \varepsilon\left(\rho_{ij}\right)=\delta_{i,j}.

Two representations \kappa_1:V_1\rightarrow V_1\otimes A and \kappa_2:V_2\rightarrow V_2\otimes A are said to be equivalent, \kappa_1\equiv \kappa_2, if there is an invertible intertwiner between them. An intertwiner between \kappa_1 and \kappa_2 is a map T\in L\left(V_1,V_2\right) such that

\displaystyle\kappa_2\circ T=\left(T\otimes I_A\right)\circ \kappa_1.

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if \{\kappa_\alpha\}_{\alpha} is a maximal family of pairwise inequivalent irreducible representation that \{\rho_{ij}^\alpha\}_{\alpha,i,j} is a basis of A. When we refer to “the matrix elements” we always refer to such a family. We define the span of \{\rho_{ij}\} as \mathcal{C}\left(\kappa\right), the space of matrix elements of \kappa.

Given a representation \kappa, we define its conjugate, \overline{\kappa}:\overline{V}\rightarrow\overline{V}\otimes A, where \overline{V} is the conjugate vector space of V, by

\displaystyle\overline{\kappa}\left(\bar{e_j}\right)=\sum_i \bar{e_i}\otimes\rho_{ij}^*,

so that the matrix elements of \overline{\kappa} are \{\rho_{ij}^*\}.

Timmermann shows that the matrix elements have the following orthogonality relations:

  • If \alpha and \beta are inequivalent then h\left(a^*b\right)=0, for all a\in \mathcal{C}\left(\kappa_\alpha\right) and b\in\mathcal{C}\left(\kappa_\beta\right).
  • If \kappa is such that the conjugate, \overline{\kappa}, is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

\displaystyle h\left(\rho_{ij}^*\rho_{kl}\right)=\frac{\delta_{i,k}\delta_{j,l}}{d_\alpha}.

This second relation is more complicated without the S^2=I_A assumption and refers to the entries and trace of an intertwiner F from \kappa to the coreprepresention with matrix elements \{S^2\left(\rho_{ij}\right)\}. If S^2=I_A, then this intertwiner is simply the identity on V and so the the entries \left[F\right]_{ij}=\delta_{i,j} and the trace is d=\dim V.

Denote by \text{Irr}(\mathbb{G}) the set of unitary equivalence classes of irreducible unitary representations of \mathbb{G}. For each \alpha\in\text{Irr}(\mathbb{G}), let \kappa_\alpha:V_{\alpha}\rightarrow V_{\alpha}\otimes A be a representative of the class \alpha where V_\alpha is the finite dimensional vector space on which \kappa_\alpha acts.

Diaconis-Van Daele Fourier Theory

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PhD Progression Report

December 2, 2011 in Research, Research Updates | Leave a comment

The following runs a thread through what I’ve looked at over the past year: Progression Report.

Research Update 9

January 25, 2011 in Research, Research Updates | Leave a comment

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I managed to get through two sections last week: Compact Hilbert Space Operators and The Spectral Theorem. I also have 9 of 12 chapter 2 exercises completed. I have been writing my study up here and this is proving fruitful on three counts:

  1. I can put questions in red for my supervisor to see
  2. I am not happy putting up something on this page that I haven’t justified to myself. This means I have to fill in some extra steps (in blue)
  3. I should have a nice set of notes to peruse should I need them

Unfortunately this week will be mostly concerned with preparing lectures for two modules that I will be lecturing in CIT:

MATH6014

MATH6037

Research Update 8

January 17, 2011 in Research, Research Updates | Leave a comment

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

Before the Christmas break I finished off the chapter 1 exercises.

Chapter 2: C*-Algebras and Hilbert Space Operators.

2.1 C*-Algebras

Initially we defined a C*-algebra, A, as a complete normed algebra, together with a conjugate-linear involution * that satisfies the C*-equation:

\|a^*a\|=\|a\|^2, \forall\,a\in A

Self-adjoint or Hermitian elements are defined to have the property a^*=a. As a consequence of this, and the C*-equation, the spectral radius of a self-adjoint element, \nu(a), is equal to its norm, \|a\|. As a corollary of this, of all the norms that can be put on the *-algebra, only one makes it into a C*-algebra – i.e. satisfying the C*-equation.

In the previous chapter we have seen that an algebra, A, can be unitised to form a new algebra, \tilde{A}, which contains A as a subspace. In general, the norm got by extending the norm on A to a norm on \tilde{A} does not make \tilde{A} into a C*-algebra. However Theorem 2.1.6 shows that there does exist a (unique) norm on \tilde{A} making it a C*-algebra. In many examples we may now assume that a general C*-algebra is unital – replacing it with the unique unitisation, \tilde{A}, if necessary.

One such result which depends on this fact is that the the spectrum of a self-adjoint element is real.

A central result in this chapter is that all abelian C*-algebras are C_0(X), for some locally compact Hausdorff space, X. In fact X is the character space \Phi(A) (as with Belton, this is via the Gelfand transformation). This identification allows the development of the powerful functional calculus. Briefly, if a is a normal element of a C*-algebra A, (a^*a=aa^*), and z is the inclusion map from \sigma(a)\rightarrow \mathbb{C}, then there exists a unique *-homomorphism \varphi:C(\sigma(a))\rightarrow A such that \varphi(z)=a. This unique *-homorphism is called the functional calculus at a. This particular section ended with the Belton result that if X is a compact Hausdorff space, \Phi(C(X))\cong X (via x\mapsto \delta_x).

2.2 Positive Elements of C*-Algebras

This section introduces a partial order on A_{\text{SA}} (the set of self-adjoint elements of A). Namely, an element a\in A_{\text{SA}} is positive if \sigma(a)\subset \mathbb{R}^+. The partial order is defined in the obvious way.

As a consequence of the Gelfand transformation and the functional calculus, we can show that positive elements of a C*-algebra possess unique positive square roots.  Another prominent result is that for an arbitrary element a\in A, a^*a is positive.

2.3 Operators and Sesquilinear Forms

As a first move, we prove that bounded operators on Hilbert spaces have adjoints. Next projections are examined and partial isometries are examined. This leads onto the polar decomposition theorem. Namely, if T is a continuous linear operator on a Hilbert space H, there exists a unique partial isometry S such that T=S|T|; where |T|=(T^*T)^{1/2}. The rest of the section focusses on the connection between operators and sesquilinear forms.

2.4 Compact Hilbert Space Operators

At first this chapter looks at some of the basic properties of these objects – e.g. if T is compact so are |T| and T^*. Thus K(H) is self-adjoint and thus a C*-algebra (it is a closed ideal in B(H)). We see that normal compact operators are diagonalisable.

We look at the finite rank operators, F(H) and see that they are dense in K(H). Next the operator x\otimes y is examined:

(x\otimes y)(z)=\langle z,y\rangle x

These are rank-one, and the x\otimes x are rank-one projections if x is a unit vector. This leads on to the fact that F(H) is linearly spanned by these rank-one projections.

This is a synopsis of what I covered up until recently (up to p.56). As an experiment I am attempting to do my study of Murphy by way of fully presenting the details on this webpage. I am unsure of whether or not this is too time consuming. Presently I am on page 63 and I will have to cover the rest of the chapter material (10 pages) in one day or similar if I am going to consider this tactic feasible.