Existence of a 779247-dim irreducible e₈-representation with 40 tensor-square isotypic components

e8_irrep_tensor_square_decomp

Submitter: Kim Morrison.

Notes: e₈ is the largest exceptional Lie algebra (dim 248), defined here by the Serre construction (Mathlib's `LieAlgebra.e₈`). The formal statement is existential: it asks for some 779247-dimensional irreducible representation whose tensor square has 40 isotypic components. In the intended solution, one constructs such a witness using the highest-weight representation with highest weight ω₁ + ω₈ (the sum of the first and last fundamental weights, in Bourbaki labelling). The U(g)-module structure on V ⊗ V used in the statement comes from lifting the Lie action via the universal enveloping algebra.

Source: Classical: representation theory of the exceptional Lie algebra e₈.

Informal solution: Construct V as V(ω₁+ω₈), the unique 779247-dim irrep of e₈. Tensor square (LiE-verified) has 40 distinct simple summand isomorphism classes (155 components with multiplicity), with the smallest being the trivial 1-dim, the 248-dim adjoint (mult 2), the 3875-dim V(ω₁), and so on up to the 85,424,220,000-dim summand. This is used to produce the existential witness required by the formal statement.

theorem declaration uses `sorry`e8_irrep_tensor_square_decomp :
    ∃ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V)
      (_ : LieRingModule (LieAlgebra.e₈ ℂ) V) (_ : LieModule ℂ (LieAlgebra.e₈ ℂ) V),
      Module.finrank ℂ V = 779247 ∧
      LieModule.IsIrreducible ℂ (LieAlgebra.e₈ ℂ) V ∧
      (isotypicComponents (UniversalEnvelopingAlgebra ℂ (LieAlgebra.e₈ ℂ))
        (V ⊗[ℂ] V)).ncard = 40 := by⊢ ∃ V x x_1 x_2,
  ∃ (x_3 : LieModule ℂ (LieAlgebra.e₈ ℂ) V),
    Module.finrank ℂ V = 779247 ∧
      LieModule.IsIrreducible ℂ (LieAlgebra.e₈ ℂ) V ∧
        (isotypicComponents (UniversalEnvelopingAlgebra ℂ (LieAlgebra.e₈ ℂ)) (V ⊗[ℂ] V)).ncard = 40
  sorryAll goals completed! 🐙

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