Recursive Formula for the Equations of Hessenberg Varieties
arXiv:2607.12261
The paper develops a recursive formula for the equations that define Hessenberg varieties inside flag varieties, using a partial order on Hessenberg functions, and provides a geometric proof of affine pavings and dimension counts for these varieties.
Abstract
Hessenberg varieties are subvarieties of the flag variety, defined by containment conditions on flags with respect to a linear operator. The study of these varieties lies in the intersection of algebraic geometry, combinatorics, and representation theory. In this paper, we develop an algebro-geometric procedure for determining the closed subvariety structure of a Hessenberg variety in the flag variety for any linear operator and Hessenberg function , by imposing a partial order on the Hessenberg functions and analyzing the relation of the corresponding Hessenberg varieties. In particular, we give a concrete recursive formula for determining all equations cutting out a given Hessenberg variety in each Schubert cell. As an application, we provide an alternative geometric proof of Tymoczko's results on the existence of affine pavings of a given Hessenberg variety and on the dimension count of its cells.
24 pages, 8 figures, 1 table