Restrictions of the dot action representation

Abstract

Regular semisimple Hessenberg varieties are subvarieties of flag varieties defined for any reductive group. Their cohomology carries a representation of the associated Weyl group, known as the dot action representation. In type $A$, these representations can be computed using the chromatic quasisymmetric functions of naturally labeled unit interval graphs. We prove that restricting the dot action representation of any Weyl group to a type $A$ parabolic subgroup yields a representation isomorphic to a direct sum of type $A$ dot action representations. This work is motivated by the conjecture that an analogous property holds for all parabolic subgroups. In particular, for classical Weyl groups of types $B/C$ and $D$, we use chromatic symmetric functions to provide a streamlined, combinatorial formula for the restriction to a specific maximal type $A$ parabolic subgroup. As an application, we prove a conjecture of Lesnevich regarding the representations associated with ideals of types $B$ and $C$ identified under the standard poset isomorphism. Specifically, we show that the corresponding dot action representations are isomorphic if and only if the ideals determine the same reflection sets in their respective root systems. Furthermore, we show that a Catalan number counts the total number of such ideals.