Download Representation Theory of the Symmetric Groups : the by Tullio Ceccherini-Silberstein PDF

By Tullio Ceccherini-Silberstein

A self-contained creation to the illustration idea of the symmetric teams, together with an exhaustive exposition of the Okounkov-Vershik approach.

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Extra info for Representation Theory of the Symmetric Groups : the Okounkov-Vershik Approach, Character Formulas, and Partition Algebras

Example text

Moreover if v ∈ V K then Ev = 1 1 σ (k)v = v = v. |K| k∈K |K| k∈K Therefore E is a projection from V onto V K . It is orthogonal because σ is unitary: E∗ = 1 1 σ (k)∗ = σ (k −1 ) = E. 6 dimV K = 1 |K| χ ResK σ , ιK G L(K) . 19) and the previous lemma we get dimV K = tr 1 1 1 G σ (k) = χ σ (k) = χ ResK σ , ιK |K| k∈K |K| k∈K |K| L(K) . 7 (Orthogonality relations for the characters of irreducible representations) Let (σ, V ) and (ρ, W ) be two irreducible representations of G. Then 1 χσ , χρ |G| L(G) = 1 0 if σ ∼ ρ if σ ∼ ρ.

21) Indeed f (g)δg = f = g∈G f (g)λ(g)δ1G = λ(f )δ1G . 16 The characters χ ρ , ρ ∈ G form an orthogonal basis for the space of central functions of G. In particular, |G| equals the number of conjugacy classes of G. 7) we have that {χ ρ : ρ ∈ G} is an orthogonal system for the space of central functions in L(G). We now show that it is complete. Suppose that f is central and that f, χ ρ L(G) = 0 for all ρ ∈ G. (vi)). 21) we deduce that f ≡ 0. Denoting by C the set of all conjugacy classes of G, we have that the characteristic functions 1C , C ∈ C, form another basis for the space of central functions.

14 (Fourier inversion formula) For every f ∈ L(G) we have f (g) = 1 |G| dρ tr[ρ(g)∗ ρ(f )]. 15 (Plancherel formula) For f1 , f2 ∈ L(G) we have f1 , f2 L(G) = 1 |G| dρ tr[ρ(f1 )ρ(f2 )∗ ]. 16 (Characterization of central functions) A function f ∈ L(G) is central if and only if the Fourier transform ρ(f ) is a multiple of the identity IW for all irreducible representations (ρ, W ) ∈ G. With the choice of an orthonormal basis {v1ρ , v2ρ , . . 39)) consisting of all f ∈ L(G) whose Fourier transform ⊕ρ∈G ρ(f ) is a diagonal operator in the given basis.

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