A Hessenberg generalization of the Garsia-Procesi basis for the cohomology ring of Springer varieties

Aba Mbirika

The electronic journal of combinatorics, Tome 17 (2010)

Résumé

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.

Détail

DOI : 10.37236/425

Classification : 14M15, 14F25, 14L35
Mots-clés : flag varieties, Hessenberg variety, Springer variety, Hessenberg function, Young diagram, cohomology