The reflected models of type A in characteristic two (Appendix B) #
characteristic-two-mesh-calculations.tex: "For the reflected model
ℤA_{2m+1}/⟨τ^s ρ⟩, the brick hypothesis forces s > m. Otherwise, retaining the central
vertex and the first s vertices of each arm gives the reflected A_{2s+1} model. The
degree-2s paths just constructed go from each vertex to its reflection and increase
the repetition coordinate by s. They become nonzero radical endomorphisms at every
object of this quotient. This contradicts the existence of a brick. In the original
model a path from τv to v has degree d = 2q with s ∣ q − 1. But q ≤ m < s, so a
nonzero such path of degree at least three is impossible."
We re-root at the middle vertex. The twisted traces of words of length at least m + 1
vanish (x^{m+1} = 0 on an arm), and a row of word degree j has j = 1 + n s; so
for s ≥ m all classes vanish. For s < m, the deletion to A_{2s+1} and the top paths
of LineTop.line_return contradict the brick hypothesis (not_brick_star_to, the
version of not_brick_star with paths ending at the deck image).
Long words on two equal arms vanish: x₁ = −x₀ and x₀^{L+2} = 0.
Exclusion of a period by deletion, with paths to the deck image. Let Del be a
ρ-stable set of vertices not containing the centre, and n c₀ = d > 0. If each
retained vertex v carries a path of degree d to ρⁿ v that is nonzero in Π
modulo Del, the brick hypothesis fails.
Positions under the substar embedding #
Reflected type A under the brick hypothesis (any characteristic): the
derivatives are surjective.