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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