I have a probability distribution over $n$-bit strings $p_{X^n}$. This is the optimal input that achieves the capacity of $W_{Y|X}^{\otimes n}$, which is $n$ iid copies of a channel $W_{Y|X}$ acting on each bit.
Does the symmetry of the problem give us some general form for $p_{X^n}$? I think it holds that $p_{X^n}$ should only have support on strings that are permutation invariant. Is this true and is this also the most general thing I can say?