A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

Abstract

This manuscript is a contributed chapter in the forthcoming CRC Press volume, titled the Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety. The book, as a whole, is aimed at a diverse audience of researchers and graduate students seeking an expository introduction to the area. In our chapter, we give an overview of some of the past research on the cohomology rings of regular nilpotent Hessenberg varieties, with no claim to being exhaustive. For the purposes of this manuscript, we focus mainly on the case of Lie type A, with some brief remarks on the general Lie types. We end the chapter with a selection of topics currently active in this area.

An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties

“…We also prove this for types F 4 and E 6 by using Maple. Note that [11] gives a monomial basis for the cohomology rings of type A regular nilpotent Hessenberg varieties which is different from our basis. The paper is organized as follows.…”

Characters and Chromatic Symmetric Functions

“…Corollary 11. Let P be a unit interval order on [n] labeled as in (16) and let w ∈ S n be the corresponding 312-avoiding permutation as in (17). Then we have…”

“…in each standard expansion of X inc(P ),q are given by θ q (inc(P )) = θ q ( C w (q)) for the 312avoiding permutation w related to P as in (17). For interpretations of some of these coef- (For progress on this conjecture, see [10], [17] and references there. )…”

“…Proposition 39. Let P be an n-element unit interval order, labeled as in ( 16) and corresponding by (17) to the 312-avoiding permutation w ∈ S n . Then we have…”

“…Problem 42. Let P be an n-element unit interval order labeled as in ( 16), and let w(P ) be the corresponding 312-avoiding permutation as in (17). Find functions δ 1 , δ 2 so that where U in the final sum is a P -permutation.…”