← Research introduction

Type A₃ moment graph explorer

Flag(ℂ⁴) · Hessenberg × Opposite Schubert

w ↔ w(i,j), α = tw(i) − tw(j). Edges that swap the same pair of values share one color and are exactly parallel in three-dimensional space.

Current intersection
h = 4444 · Ω₁₂₃₄ · 72 edges
Is regular?
Yes
Is connected?
Yes
No vertex exceeds mₕ − ℓₕ(1234) = 6 − 0 = 6.
Drag rotate · Wheel zoom · Click vertex label
Use the ← → keys to select among the visible vertices.
Convention: ΩwT = {v ∈ 𝔖₄ ∣ w ≤ v in Bruhat order}
24 vertices · 72 edges

14 Dyck paths

Hessenberg function

Keep precisely the edges whose position pair satisfies i < j ≤ h(i).

h = 4444

24 Bruhat intervals

Opposite Schubert Ωw

Keep the fixed points v ≥ w and the edges joining them.

Pale yellow marks the permutations satisfying w⁻¹(w(j) + 1) ≤ h(j) whenever w(j) ≤ 3.

w = 1234

The three-dimensional coordinates come from the inverse-permutation permutahedron. Thus the Hessenberg condition acts on positions, while root colors and parallel directions are determined by the swapped values.