Ch 4.5 Wreath products Shu Xiao Li York University June 25, 2015 - - PowerPoint PPT Presentation

Ch 4.5 Wreath products

Shu Xiao Li

York University

June 25, 2015

Consider finite group Γ, and PSH A(Sn[Γ]) with basis irreducible

• characters. Let Irr(Γ) = (ρ1, . . . , ρd) be irreducible representations
• f Γ, and χλ be irreducible representations of Sn, indexed by

partitions λ in Parn. Define a representation χλ,ρ of Sn[Γ] such that σ ∈ Sn, γ ∈ Γn acts on χλ ⊗ (ρ⊗n) as σ(u ⊗ (v1 ⊗ · · · ⊗ vn)) = σ(u) ⊗ (vσ−1(1) ⊗ · · · ⊗ vσ−1(n)) γ(u ⊗ (v1 ⊗ · · · ⊗ vn)) = u ⊗ (γ1v1 ⊗ · · · γnvn).

Theorem 4.36

The irreducible CSn[Γ]-modules are induced characters χλ = IndSn[Γ]

Sdeg(λ)[Γ]

• χλ(1),ρ1⊗···⊗χλ(d),ρd
• as λ runs through all functions

λ : Irr(Γ) → Par ρi → λ(i). with d

i=1 |λ(i)| = n.

Furthermore, one has a PSH-isomorphism A(S[Γ]) → Λ⊗d χλ → sλ(1) ⊗ · · · ⊗ sλ(d).

Theorem 3.12 tells A(S[Γ]) ∼ = ⊗ρ∈CA(S[Γ])(ρ) where C is the primitive elements in Σ and A(ρ) ⊂ A is the Z-span of Σ(ρ) = {σ ∈ Σ | there exists n ≥ 0 with (σ, ρn) = 0}. From the definition of coproduct, it is clear that ρ ∈ Irr(Sn[Γ]) is primitive if and only if n = 1 i.e. ρ ∈ Irr(Γ) = {ρ1, . . . , ρd}.

From i) and ii), we obtain Φ : R(Sn) → R(Sn[Γ]) and Ψ : R(Sn[Γ]) → R(Sn) such that Φ(χ) = χ ⊗ (ρ⊗n) and Ψ(α) = HomCΓn(ρ⊗n, α). From iii), iv), and self-duality, we could prove Φ, Ψ are PSH-morphisms, Ψ ◦ Φ = id and Im(Φ) = R(Sn[Γ])(ρ). Therefore, taking direct sum of all n ≥ 0, we obtain an isomorphism between A(S[Γ])(ρ) and A(S), which is isomorphic to Λ.

The End