Antipodal transformation and finite-type Garside structure

The antipodal map is an involutive category automorphism by F1. Deligne (1.26)(i) is precisely the naturality identity $g{\mathrm{\Delta }}_B={\mathrm{\Delta }}_A\varphi (g)$ for $g:A\to B$. Given $a,b$ starting at $A$, (1.26)(ii) applied with $g=1_A$ makes each divide a power of $\mathrm{\Delta }$ starting at $A$; a sufficiently large power is a common right multiple. Here powers mean successive components, whose sources alternate under $\varphi$. Thus Both gallery divisibility orders are lattices applies and supplies both lattices.

Every minimal class $u(A,B)$ divides ${\mathrm{\Delta }}_A$: the hyperplanes separating $A$ from $B$ and those separating $B$ from $-A$ partition all hyperplanes, so concatenating corresponding minimal paths is minimal from $A$ to $-A$. Conversely any prefix of ${\mathrm{\Delta }}_A$ crosses no hyperplane twice, since the complete gallery crosses each once and the congruence preserves length. It is therefore minimal. In particular each atom is simple.

Smallness, cancellation and atomicity were proved in F1. Combined with naturality, the atom condition and both lattices, these are all conditions of the categorical Garside definition. There is at most one simple per ordered pair of chambers, and there are finitely many chambers; the structure is finite type.