Proposition 12.8 for A₀ and for a representation-finite algebra #
prop:polynomial-mesh-comparison (deformation-refinements.tex): "Let m + Σ_{i∈I} t_i μ_i
be an associative family satisfying the hypotheses of Proposition 7.9, such as
eq. selected-binary-family, now indexed by I = I₀. There are polynomial lifts […] for which the
retained mesh coefficients are F_i(t) = a_i t_i, a_i ∈ k^*. After every specialization to
k, these modules exhaust the indecomposables of the fibre algebra."
In the setting of Proposition 7.9 (familyCriterion_A0): the rows R_i and dual mesh weights of
Proposition 6.1 (rows of length at least three, the degree 2s + 2 of Proposition 6.1), the
characters χ_i (rowChar), and a family μ_i indexed by all rows with the hypotheses of
Proposition 7.9 (normalized relative, of character χ_i, not a coboundary, associative family).
The internal degree len R_i − 2 of μ_i is derived from its character
(RaySys.Sys.hasDeg₂_of_char), the coordinates χ_i ↦ e_i are the dual weights, and the
generic polynomialMeshComparison applies (polynomialMeshComparison_A0); for A₀ of a
representation-finite algebra the mesh hypotheses are the Auslander–Reiten ones
(IndSkeleton.polynomialMeshComparison_ar).
The dual-weight coordinates ℤ^ι × ℤ → ℤ^ι.
Instances For
The conclusion of Proposition 12.8 for the family μ with the rows Rw and characters
rowChar.
Instances For
Proposition 12.8 on A₀, in the setting of Proposition 7.9.
Characters in any lattice #
The conclusion of Proposition 12.8 for the family μ with the rows Rw and characters
rowChar.
Instances For
Proposition 12.8 on A₀, in the setting of Proposition 7.9.
Proposition 12.8 (prop:polynomial-mesh-comparison) for A₀ of a basic
representation-finite algebra A: for the rows and dual weights of Proposition 6.1 and a family
μ_i indexed by all rows with the hypotheses of Proposition 7.9, the conclusions of
polynomialMeshComparison hold (polynomial lifts, retained mesh coefficients a_i t_i with
a_i ≠ 0, the mesh presentation of every fibre, and exhaustion of its indecomposables).