Type A Nilpotent Hessenberg varieties with Hessenberg function \(h(i)\le i+1\)
Published in arxiv.org/abs/2607.25520, 2026
We study the type A nilpotent Hessenberg varieties associated with Hessenberg functions that satisfy \(h(i)\le i+1\). We call these the generalized parabolic Peterson varieties. We show that such varieties can be decomposed into the union of specific generalized parabolic Peterson varieties \(\operatorname{Pet}_{\lambda,\alpha}\), such that \(\lambda\) is an integer partition, \(\alpha\) is an integer composition, and \(\alpha\) is dominated by \(\lambda\). We prove that when \(\alpha\) is dominated by \(\lambda\), the cardinality of the maximal dimensional components of \(\operatorname{Pet}_{\lambda,\alpha}\) equals the Kostka number \(\mathcal{K}_{\lambda\alpha}\), and its dimension is determined only by \(\lambda\) and the length of \(\alpha\). We provide a recursive formula for the Poincaré polynomial of \(\operatorname{Pet}_{\lambda,\alpha}\).