Theorems · Theorem · nonassociative algebras

RootPairing.isSimpleModule_weylGroupRootRep_iff

∀ {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) [Nontrivial M],
  IsSimpleModule (MonoidAlgebra R ↥P.weylGroup) P.weylGroupRootRep.asModule ↔
    ∀ (q : Submodule R M), (∀ (i : ι), q ∈ Module.End.invtSubmodule ↑(P.reflection i)) → q ≠ ⊥ → q = ⊤

Defined in

Mathlib.LinearAlgebra.RootSystem.Irreducible

Cited by

1 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleNontrivial