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Regular nilpotent Hessenberg varieties form an important family ofsubvarieties of the flag variety, which are often singular and sometimes notnormal varieties. Like Schubert varieties, they contain distinguished pointscalled permutation flags. In this paper, we give a combinatorialcharacterization for a permutation flag of a regular nilpotent Hessenbergvariety to be a singular point. We also apply this result to characterizeregular nilpotent Hessenberg varieties which are normal algebraic varieties.

On singularity and normality of regular nilpotent Hessenberg varieties