Let be a finite quantum group described by 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.) . Under the assumption of finiteness, there is a unique Haar state, on .

# Representation Theory

A *representation* of is a linear map that satisfies

The dimension of is given by . If has basis then we can define the *matrix elements* of by

One property of these that we will use it that .

Two representations and are said to be *equivalent*, , if there is an invertible *intertwiner* between them. An intertwiner between and is a map such that

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

Timmermann shows that if is a maximal family of pairwise inequivalent irreducible representation that is a basis of . When we refer to “the matrix elements” we always refer to such a family. We define the span of as , the *space of matrix elements of* .

Given a representation , we define its *conjugate*, , where is the conjugate vector space of , by

so that the matrix elements of are .

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

- If and are inequivalent then for all and .
- If is such that the conjugate, , is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

This second relation is more complicated without the assumption and refers to the entries and trace of an intertwiner from to the coreprepresention with matrix elements . If , then this intertwiner is simply the identity on and so the the entries and the trace is .

Denote by the set of unitary equivalence classes of irreducible unitary representations of . For each , let be a representative of the class where is the finite dimensional vector space on which acts.

# Diaconis-Van Daele Fourier Theory

We define a map by

We define

I think in the finite dimensional case that but I don’t think I necessarily need this.

Van Daele shows that this map allows us to give the structure of a quantum group with Haar state given by

He also proves a Plancherel theorem which says that

where the star involution in is given by

Given a representation (), following Wang we define the *Fourier Transform of at the representation * as the map by

## Diaconis-Van Daele Inversion Theorem

**Let be the counit of and **.* Then we have*

* where the sum is over the irreducible representations of .*

*Proof*: Both sides are linear in so it suffices to check for . The left-hand side reads

To calculate the right-hand-side, we calculate for a given representation the trace of . Let and calculate

This is zero unless . If then we have

How much of is sent to :

Now times this trace is and so we are done

Now there are two convolution theorems at play. The first is Van Daele’s:

## Van Daele’s Convolution Theorem

**For all** *we have *

* where is a rather messy convolution in and is the nice convolution got from :*

We also have something we can salvage from Timmermann (third line) which combined with this yields:

## Diaconis-Van Daele Convolution Theorem

**For a representation of ** and *we have*

*Proof:*

## Van Daele Plancherel Theorem

**For all** , *we have*

As it happens, I should point out that there is a Van Daele Inversion Theorem which says that

where (and so an element of the dual goes in the ) and the antipode on is given by

I haven’t used this… yet.

## Lemma

**Where the sum is over irreducible and unitary representations,**

*Proof:* The proof uses the convolution theorem of Van Daele and the definition of to find

Now use the Diaconis-Van Daele Inversion Theorem

Two more results that will be used in the final proof.

## Proposition

** Suppose that is a state**, *then where is the trivial representation, , we have .*

*Proof*:

## Proposition

** Suppose that is a non-trivial and irreducible representation**, *then* .

*Proof*: A calculation:

Note that is the matrix element of the trivial representation and is not equivalent to the trivial representation. Therefore, by the first orthogonality relation

and so as required

Note in particular that . One more result and definition before we prove the upper bound lemma.

## Proposition

**If is unitary**, *then the map, , given by is a -homomorphism.*

* Proof*: Let us call the map by : we want to show that , where the first involution is in and the second is in . First take . We have

Now looking at the other quantity:

Considering that the involution in is the conjugate-transpose this is enough to show the result

## Definition

Define as the norm on got from the inner product on given by

i.e.

## Quantum Diaconis-Shahshahani (Upper Bound) Lemma

**Suppose that is a state and a natural number.** *Then*

*where the sum is over all non-trivial, irreducible representations.*

*Proof*: Writing

Now using the above lemma, and the fact that the map is a -homomorphism, we have that this is given by

where the sum is over all irreducible representations.

Now note that

If , the trivial representation, then this yields zero as both terms are one. If is non-trivial, then and we have:

To get the result, apply the Diaconis-Van Daele Convolution Theorem times

# What is Next?

The norm that we are measuring with is not necessarily the most natural. I want/need to find a norm, , on that is perhaps natural in that *lower bounds *on are available. Presumably it will be the case that

,

where is some constant dependent on or perhaps even .

After this I want to look at this stuff in the cocommutative case…. and the classical case.

Also I heard that there might be a family of quantum groups (of ‘Kac-Paljutkin’-type) on which I could study families of random walks…

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