Which Schubert Varieties are Hessenberg Varieties?

TL;DR

This paper proves that all Schubert varieties in the full flag variety are Hessenberg varieties but not all are adjoint Hessenberg varieties, especially in types A and C, with many not being isomorphic to any.

Contribution

It establishes that every Schubert variety is a general Hessenberg variety and provides criteria showing most are not adjoint Hessenberg varieties in certain types.

Findings

All Schubert varieties are general Hessenberg varieties.

Most Schubert varieties in types A and C are not adjoint Hessenberg varieties.

Some Schubert varieties in type A are not isomorphic to any adjoint Hessenberg variety.

Abstract

After proving that every Schubert variety in the full flag variety of a complex reductive group $G$ is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of $G$ increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.