## Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by is *ergodic* if the convolution powers converge to the Haar state .

The classical theorem for finite groups:

#### Ergodic Theorem for Random Walks on Finite Groups

*A random walk on a finite group driven by a probability is ergodic if and only if is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.*

Not concentrated on a proper subgroup gives *irreducibility*. A random walk is *irreducible *if for all , there exists such that .

Not concentrated on the coset of a proper normal subgroup gives *aperiodicity. *Something which should be equivalent to aperiodicity is if

is equal to one (perhaps via invariance ).

If is concentrated on the coset a proper normal subgroup , specifically on , then we have periodicity (), and , the order of .

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

#### Ergodic Theorem for Random Walks on Finite Quantum Groups

*A random walk on a finite quantum group driven by a state is ergodic if and only if “X”.*

## Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to *group-like-projections,* which I will say correspond to *quasi-subgroups *following Kasprzak & Sołtan.

I have shown that if is concentrated on a proper quasi-subgroup , in the sense that for a group-like-projection , that so are the . The analogue of irreducible is that for all projections in , there exists such that . If is concentrated on a quasi-subgroup , then for all , , where .

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

,

converge to an idempotent state . If for all then the also, so that (as the Haar state is faithful). I was able to prove that is supported on the quasi-subgroup given by the idempotent .

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.

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