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

Abstract

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 \[ 3412,\quad 4231,\quad 35142,\quad 42513,\quad 45312,\quad 426153,\quad 463152,\quad 526413 \] 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.