We study the problem of indecomposability of translations of simple modules in the principal block of BGG category $\mathcal{ O}$ for $\mathfrak{ sl}_n$, as conjectured in \cite{ KiM}. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for $n\leq 6$. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results.
Categories: math.RT