Deck data of a periodic star model #
From Proposition 4.1 (periodic-mesh-calculations.tex): for a
covering π : ℤΔ → Q of a finite quiver by a star model, the deck group is
generated by a transformation of height form g (v, t) = (ρ v, t + c₀) with
c₀ > 0, and the fibres of π are the g-orbits. We restate this in the form
used by the gauge and trace arguments: π ∘ g = π, and two vertices with the
same image differ by a power of g or its inverse.
theorem
MultiplicativeBasisDeformations.StarDeck.exists_ne_same_image
{t : ℕ}
{L : Fin t → ℕ}
{V : Type u}
[Quiver V]
[∀ (x y : V), Subsingleton (x ⟶ y)]
{Q : TranslationQuiver V}
(hQ : Q.IsStable)
(π : (starTree L).ZVert ⥤q V)
(hτ : ∀ (Y : (starTree L).ZVert), π.obj ((starTree L).ztau Y) = hQ.tauE (π.obj Y))
[Finite V]
:
A covering of a finite quiver by ℤΔ identifies two distinct vertices.
theorem
MultiplicativeBasisDeformations.StarDeck.deck_data
{t : ℕ}
{L : Fin t → ℕ}
{V : Type u}
[Quiver V]
[∀ (x y : V), Subsingleton (x ⟶ y)]
{Q : TranslationQuiver V}
(hQ : Q.IsStable)
(π : (starTree L).ZVert ⥤q V)
(hπ : π.IsCovering)
(hτ : ∀ (Y : (starTree L).ZVert), π.obj ((starTree L).ztau Y) = hQ.tauE (π.obj Y))
[Finite V]
(ht : 0 < t)
(hcl : t = 1 ∨ t = 3)
:
∃ (ρ : StarV L ≃ StarV L) (hρ : ∀ (a b : StarV L), (starTree L).G.Adj (ρ a) (ρ b) ↔ (starTree L).G.Adj a b) (c0 : ℕ)
(hρp : ∀ (v : StarV L), (starTree L).par (ρ v) = (starTree L).par v + ↑↑c0),
0 < c0 ∧ (∀ (a b : StarV L), starAdj a b ↔ starAdj (ρ a) (ρ b)) ∧ (∀ (Z : (starTree L).ZVert), π.obj (((starTree L).zAct ρ hρ (↑c0) hρp).obj Z) = π.obj Z) ∧ ∀ (X Y : (starTree L).ZVert),
π.obj X = π.obj Y → ∃ (n : ℕ), Y = (StarGauge.gZ ρ hρ c0 hρp)^[n] X ∨ X = (StarGauge.gZ ρ hρ c0 hρp)^[n] Y
The deck data of a periodic star model (Proposition 4.1(3)).