In Group Representations in Probability and Statistics, Diaconis presents his celebrated Upper Bound Lemma for a random walk on a finite group driven by . It states that

,

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

In this post, we begin this study by looking a the (co)-representations of a quantum group . The first thing to do is to write down a satisfactory definition of a representation of a group which we can quantise later. In the chapter on Diaconi-Fourier Theory here we defined a representation of a group as a group homomorphism

While this was perfectly adequate for when we are working with finite groups, it might not be as transparently quantisable. Instead we define a representation of a group as an action

.

such that the map , is linear.

If we reverse arrows here we have a definition for a co-representation. A *co-representation* of a quantum group is a linear map:

with the property that

, and

.

This means that co-multiplication is a co-representation of a quantum group. We can call the dimension of the the *dimension *of the co-representation. Now we might try and define a sub-co-representation. First a sub-representation is a restriction of a representation to a stable subspace . This works absolutely a fine: a *sub-co-representation of a co-representation *is a co-representation such that and

for all .

At this stage I might like to put an inner product on and additionally assume that an inner product can be put on these vector spaces. For a finite quantum group the following defines an inner product and thus makes into a Hilbert space:

,

where and is the Haar state. Now we look at the form

.

and see if this can help provide us find some sub-co-representations. We calculate

.

At this point we can look to An Invitation to Quantum Groups and Duality which develops the theory needed to calculate this sum.

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