A hereditary theorem for rigid and pseudorigid components of Lusztig's nilpotent varieties in type $A$

For irreducible components $C_{\sigma }$ of Lusztig’s nilpotent varieties of type $A$ with graded dimension $(1,2,\dots ,n,n-1,\dots ,1)$ arising from permutations $\sigma \in S_n$, we characterize the property that $C_{\sigma }\oplus C_{\sigma }$ is an irreducible component combinatorially in terms of $\sigma$. The permutations that occur, which we call \emph{pseudosmooth}, are described by a recursion on direct sums and deleting suitable corners, whose terminal cases are the permutations obtained from $$3412,4231,35142,42513,45312,426153,463152,526413$$ by inflating the entries into consecutive decreasing blocks. The main new input is a hereditary property valid for decompositions of multisegments with disjoint extreme points.