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

$\displaystyle \|\nu^{\star k}-\pi\|^2\leq \frac{1}{4}\sum d_\rho \text{Tr }(\widehat{\nu(\rho)}^k(\widehat{\nu(\rho)}^\ast)^k)$,

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

In this post, we begin this study by looking a the (co)-representations of a quantum group $A$. 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

$\rho:G\rightarrow GL(V)$

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

$\Phi:V\times G\rightarrow V$.

such that the map $\rho(g):V\rightarrow V$$\rho(g)x=\Phi(x,g)$ is linear.

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

$\chi:V\rightarrow V\otimes A$

with the property that

$(I\otimes \Delta)\circ \chi=(\chi\otimes I)\circ \chi$, and

$(I \otimes\varepsilon)\circ\chi=I$.

This means that co-multiplication is a co-representation of a quantum group. We can call the dimension of $V$ 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 $\Phi:G\times V\rightarrow V$ to a stable subspace $W\subset V$. This works absolutely a fine: a sub-co-representation of a co-representation $\chi$ is a co-representation $\chi_o:W\rightarrow G\otimes W$ such that $W\subset V$ and

$\chi(w)=\chi_0(w)$ for all $w\in W$.

At this stage I might like to put an inner product on $A$ 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 $A$ into a Hilbert space:

$\langle a,b\rangle=\eta(a^\ast b)$,

where $a,\,b\in A$ and $\eta$ is the Haar state. Now we look at the form

$\langle u,v\rangle_\chi=\langle\chi(u),\chi(v) \rangle_{A\otimes V}$.

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

$\displaystyle \langle u,v\rangle_\chi=\left\langle\sum_i a_i\otimes u_i,\sum_j b_j\otimes v_j\right\rangle=\sum_{i,j}\langle a_i,b_j\rangle\langle u_i,v_j\rangle$

$=\sum_{i,j}\eta(a_i^\ast b_j)\langle u_i,v_j\rangle.$.

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