Theorems · Theorem · nonassociative algebras

RootPairing.chainBotCoeff_mul_chainTopCoeff

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : CharZero R]
  [inst_2 : IsDomain R] [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : AddCommGroup N] [inst_6 : Module R N]
  {P : RootPairing ι R M N} [inst_7 : Finite ι] [inst_8 : P.IsCrystallographic] {b : P.Base} {i j k l m : ι}
  [P.IsReduced] [P.IsIrreducible],
  i ∈ b.support →
    j ∈ b.support →
      i ≠ j →
        P.root k + P.root i = P.root l →
          P.root k - P.root j = P.root m →
            P.root k + P.root i - P.root j ∈ Set.range ⇑P.root →
              (RootPairing.chainBotCoeff i m + 1) * (RootPairing.chainTopCoeff j k + 1) =
                (RootPairing.chainTopCoeff j l + 1) * (RootPairing.chainBotCoeff i k + 1)

This is Lemma 2.6 from [Geck](Geck2017).

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas

Cited by

1 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCharZeroIsDomainAddCommGroupModuleAddCommGroupModuleFiniteRootPairing.IsCrystallographicRootPairing.IsReducedRootPairing.IsIrreducible