Each essential communicating class $C_1,...,C_k$ (of a finite state MC) has an unique stationary distribution $\pi_{C_l}$ for $1\leq l\leq k$(Levin and Yuval corollary 1.17). We can then define: \begin{equation*} \pi_l(i) = \begin{cases} \pi_{C_l}(i)&\text{if $i\in C_l$}\\ 0 &\text{otherwise} \end{cases} \end{equation*} for $1\leq l\leq k$
It is clear that each $\pi_l$ is stationary. Is it correct to state that they are linearly independent? If that is not true, would it be correct to state that $\pi_l$ cannot possibly be a convex combination of $\pi_1,...,\pi_{l-1},\pi_{l+1},...,\pi_k$?
Yes, this is immediate from the fact that the $C_l$ are disjoint from each other. So if you have a linear combination $f=\sum a_l\pi_l$ of the $\pi_l$, then for each $i$, $f(i)$ is just $a_l\pi_l(i)$ for the unique $l$ such that $i\in C_l$ (or $0$ if $i$ is not in any $C_l$). The only way $f$ can be $0$ is thus if $a_l=0$ for all $l$.