Class-function properties of the colour and Jacobi–Trudi

characters

colourChar α and jtChar μ are invariant under conjugation, and under inversion (a permutation is conjugate to its inverse, having the same cycle type).

theorem RS.fibreCard_comp_perm {n N : ℕ} (g : Fin n → Fin N) (τ : Equiv.Perm (Fin n)) (j : Fin N) : fibreCard (g ∘ ⇑τ) j = fibreCard g j

Fibre sizes are invariant under precomposition with a permutation.

theorem RS.colourChar_conj {n N : ℕ} (α : Fin N → ℕ) (π τ : Equiv.Perm (Fin n)) : colourChar α (τ * π * τ⁻¹) = colourChar α π

The colour character is a class function.

theorem RS.colourChar_inv {n N : ℕ} (α : Fin N → ℕ) (π : Equiv.Perm (Fin n)) : colourChar α π⁻¹ = colourChar α π

The colour character is invariant under inversion.

theorem RS.jtChar_inv (μ : YoungDiagram) (π : Equiv.Perm (Fin μ.card)) : jtChar μ π⁻¹ = jtChar μ π

The Jacobi–Trudi character is invariant under inversion.