A survey of recent developments on Hessenberg varieties

Abstract

This article surveys recent developments on Hessenberg varieties, emphasizing some of the rich connections of their cohomology and combinatorics. In particular, we will see how hyperplane arrangements, representations of symmetric groups, and Stanley's chromatic symmetric functions are related to the cohomology rings of Hessenberg varieties. We also include several other topics on Hessenberg varieties to cover recent developments.

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

“…, m 4 ) = (1, 0, 0, 2), then we have ℓ := h − m = (2, 5, 4, 5). Hence, the procedure is as follows: 3) , where ℓ (1) = (2, 5, 4, 5), ℓ (2) = (4, 3, 4),…”

“…, m ′ 4 ) = (1, 2, 0, 2), then we have ℓ ′ := h − m ′ = (2, 3, 4, 5). Hence, the procedure is as follows: 3) , where ℓ ′(1) = (2, 3, 4, 5), ℓ ′(2) = (2, 3, 4), ℓ ′(3) = (2, 3), and ℓ ′( 4) is not defined. The product v…”

“…Their topology makes connections with other research areas such as the logarithmic derivation modules in hyperplane arrangements and the Stanley's chromatic symmetric functions in graph theory (see e.g. the survey article [3]).…”

Which Schubert Varieties are Hessenberg Varieties?

“…Hessenberg varieties have been studied by applied mathematicians, combinatorialists, geometers, representation theorists, and topologists. See [AH19] for a survey of some recent developments. Our goal is to understand better the structure of these varieties, and in particular what restrictions on such structure exist.…”

Uniform bases for ideal arrangements

“…This subject is relatively new, and it has been found that geometry, combinatorics, and representation theory interact nicely on Hessenberg varieties (cf. [2]). As one of the interactions, the cohomology ring of a regular nilpotent Hessenberg variety can be described in terms of the logarithmic derivation module of the ideal arrangement ( [4]).…”