Uniform bases for ideal arrangements

Abstract

In this paper we introduce and study uniform bases for the ideal arrangements. In particular, explicit uniform bases are presented on each Lie type. Combining the explicit uniform bases with the work of Abe-Horiguchi-Masuda-Murai-Sato, we also obtain explicit presentations of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types. Contents 1. Introduction 1 2. Ideal arrangements 3 3. Uniform bases 6 4. Main theorem 10 5. The invertible matrices associated with uniform bases 13 6. Uniform bases for the ideal arrangements in a root subsystem 50 7. The cohomology rings of regular nilpotent Hessenberg varieties 53 References 56

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

“…In this section we state the main theorems in each type. We first recall one-to-one correspondence between the set of lower ideals and the set of Hessenberg functions considered in [8]. To do that, we need a "good" decomposition of the set of positive roots.…”

“…We will take Hessenberg functions in stead of lower ideals for each type as explained in [8]. Also, in what follows, we denote the regular nilpotent Hessenberg variety by Hess(N, h) where h is the Hessenberg function associated with a lower ideal I and Hess(N, h) means Hess(N, I).…”

“…Cn and the set of Hessenberg functions h for type C n are in one-to-one correspondence which sends I to h I in (5.3). Similarly to types A and B, a Hessenberg function h can be expressed pictorially by drawing a configuration of shaded boxes and the corresponding lower ideal associated with h consists of positive roots in shaded boxes of h (see [4,Section 10.4] or [8,Section 5.3]).…”

“…We explain polynomials f Cn i,j given in [8] which are originally given in [4] in a slightly different form. For 1 ≤ i ≤ j ≤ 2n + 1 − i, we define…”

“…More specifically, the cohomology rings of regular nilpotent Hessenberg varieties can be described in terms of the logarithmic derivation modules of ideal arrangements. From this point of view, [4] also gave explicit presentations of the cohomology rings of regular nilpotent Hessenberg varieties in types A, B, C, G. Recently, the authors gave in [8] explicit presentations of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types from the point of view of hyperplane arrangements.…”