On the Symmetries of the Deletion Channel

In this paper, we consider a class of symmetry groups associated to communication channels, which can informally be viewed as the transformations of the set of inputs that ``commute'' with the action of the channel. These groups were first studied by Polyanskiy in (IEEEToIT 2013). We show the simple result that the input distribution that attains the maximum mutual information for a given channel is a ``fixed point'' of its group. We conjecture (and give empirical evidence) that the channel group of the deletion channel is extremely small (it contains a number of elements constant in the blocklength). We prove a special case of this conjecture. This serves as some formal justification for why the analysis of the binary deletion channel has proved much more difficult than its memoryless counterparts.