Theorems · Theorem · nonassociative algebras

RootPairing.chainBotCoeff_if_one_zero

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N} [inst_5 : Finite ι]
  [inst_6 : CharZero R] [inst_7 : IsDomain R] {i j : ι} [inst_8 : P.IsNotG2],
  P.root i + P.root j ∈ Set.range ⇑P.root → RootPairing.chainBotCoeff i j = if P.pairingIn ℤ i j = 0 then 1 else 0

For a reduced, crystallographic, irreducible root pairing other than 𝔤₂, if the sum of two roots is a root, the bottom chain coefficient is either one or zero according to whether they are perpendicular. To see that this lemma fails for 𝔤₂, let α (short) and β (long) be a base. Then the roots α and α + β provide a counterexample.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.G2

Cited by

2 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFiniteCharZeroIsDomainRootPairing.IsNotG2