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.

Advertisement