Propositions 7.9 and 12.8 with characters in W(P) #
prop:family-criterion and prop:polynomial-mesh-comparison, for a basic connected
representation-finite algebra A in characteristic two, with the residual characters
χ_i = w(p_i) − ν_{v_i} in the weight lattice L = W(P) of the universal mesh weight w
(Lemma 7.4), as in the source. The rows are those of Proposition 6.1 (residual_dual), of length
at least three (shorter rows carry no cokernel class); their characters are pairwise distinct and
part of an integer basis of W(P) (Proposition 7.5), and have degree len p_i − 2
(rowsWP). The family statements are the generic ones of FamilyCriterionGen.lean and
PolynomialMesh.lean for this weight.
Rows of length below three carry no cokernel class.
The rows of Proposition 6.1 with their characters in W(P).
The coordinates of an integer basis on chosen basis vectors.
Instances For
Proposition 7.9 (prop:family-criterion) for a basic connected representation-finite
algebra A in characteristic two, with the residual characters in W(P): for the rows of
Proposition 6.1 and the universal mesh weight, every family μ_i, i ∈ I₀, of normalized
relative cocycles of character χ_i ∈ W(P) with nonzero classes, associative, whose binary fibres
have the basis of kP as a normed multiplicative ray basis, gives A as a binary fibre with a
normed multiplicative ray basis presenting P (FamilyCriterionStmt for B = A).
Proposition 12.8 (prop:polynomial-mesh-comparison) for a basic connected
representation-finite algebra in characteristic two, with the residual characters in W(P):
for the rows of Proposition 6.1 (of length at least three) and the universal mesh weight, every
family μ_i indexed by I = I₀ with the hypotheses of Proposition 7.9 has polynomial lifts with
retained mesh coefficients a_i t_i, a_i ≠ 0, presenting every fibre, whose modules exhaust
the indecomposables of the fibre (PolyMeshStmtG).