The second result provides an explicit irreducible decomposition of the representation for parabolic Hessenberg varieties [Definition 4.11] via Young's rule (see [JK, Chapter 2]).
Proposition 4.3 (Young's Rule [JK]) Let [[tau].sup.[lambda]] denote the character of the trivial representation for the Young subgroup [S.sub.[lambda]].
Using Young's rule we obtain the irreducible decomposition of the ordinary cohomology for all parabolic regular semisimple Hessenberg varieties.