4.4 Symmetric Groups Farid Aliniaeifard York University - PowerPoint PPT Presentation

4.4 Symmetric Groups Farid Aliniaeifard York University <http://math.yorku.ca/~faridanf/> June 18, 2015

Overview 3.3 Λ is the unique indecomposable PSH 4.4 Symmetric Groups

3.3 Λ is the unique indecomposable PSH ◮ Goal: If A has only one primitive elements in its PSH-basis Σ, then A must be isomorphic as a PSH to the ring of Symmetric functions Λ, after rescale the grading of A . PSH-isomorphism: ′ betweeen two PSH’s A and A ′ having φ A PSH-morphism A → A ′ is a graded Hopf Algebra morphism for which PSH-bases Σ and Σ ′ and Σ = Σ ′ it will be called a ′ . If A = A φ ( N )Σ ⊂ N Σ PSH-endomorphism. If φ is an isomorphism and restrict to a ′ , it will be called a PSH-isomorphism; if it is both bijection Σ → Σ a PSH-isomorphism and an endomorphism, it is a PSH-automorphism.

Theorem: Let A be a PSH with a PSH-basis Σ containing only one primitive ρ , and assume that the grading has been rescaled so that ρ has degree 1. Then, after renaming ρ = e 1 = h 1 , one can find unique sequences { h n } n =0 , 1 , 2 ,... and { e n } n =0 , 1 , 2 ,... of elements of Σ having the following properties: ◮ (a) h 0 = e 0 = 1, and h 1 = e 1 := ρ has ρ 2 a sum of two elements of Σ, namely ρ 2 = h 2 + e 2 . ◮ (b) For all n = 0 , 1 , 2 , . . . , there exit unique elements h n , e n in A n ∩ Σ that satisfy h ⊥ e ⊥ 2 e n = 0 and 2 h n = 0 with h 2 , e 2 being the two elements of Σ introduced in (a).

◮ (c) For k = 0 , 1 , 2 , . . . , n one has h ⊥ k h n = h n − k and σ ⊥ h n = 0 for σ ∈ Σ \ { h 0 , h 1 , . . . , h n } e ⊥ k e n = e n − k and σ ⊥ e n = 0 for σ ∈ Σ \ { e 0 , e 1 , . . . , e n } . In particular, e ⊥ k h n = 0 = e ⊥ k h n for k ≥ 2. ◮ (d) Their coproducts are � ∆( h n ) = h i ⊗ h j i + j = n � ∆( e n ) = e i ⊗ e j . i + j = n

◮ (e) The elements h n , e n in A satisfy the same relation � ( − 1) i e i h j = δ o , n i + j = n as their coproduct in Λ, along with the property that A = Z [ h 1 , h 2 , . . . ] = Z [ e 1 , e 2 , . . . ] ◮ (f) There is exactly one nontrivial automorphism A ω → A as a PSH, swapping h n ↔ e n . ◮ (g) There are exactly two PSH-isomorphisms A → Λ. γ : A → Λ γω : A → Λ h n �→ h n ( X ) and e n �→ h n ( X ) e n �→ e n ( X ) h n �→ e n ( X )

⊥ Recall: Given a Hopf algebra A of finite type, and its (graded) dual A ◦ , let ( ., . ) = ( ., . ) A be the paring ( f , a ) = f ( a ) for f ∈ A ◦ f ⊥ and a ∈ A . Then define for each f in A ◦ an operator A → A as follows: for ∈ A with ∆( a ) = � a 1 ⊗ a 2 , let � f ⊥ ( a ) = ( f , a 1 ) a 2 .

4.4 Symmetric Groups Notations: ◮ Consider the tower of symmetric groups G n = S n and A = A ( G ∗ ) =: A ( S ). ◮ Denote by 1 S n , sgn S n the trivial character and sign character of S n ◮ For a partition λ of n , denote by 1 S λ , sgn S λ the trivial and sign character restrict to Younge subgroup S λ = S λ 1 × S λ 2 × . . . ◮ denote by 1 λ the class function which is the characteristic function for the S n -conjugacy class of permutation of cycle type λ ◮ z λ := m 1 ! . m 2 ! . . . if λ = (1 m 1 ! , 2 m 2 , . . . ) with multiplicity m i for the part i

Theorem Irreducible complex characters { χ λ } of S n are indexed by partition λ in Par n , and one has a PSH-isomorphism, the Frobenius characteristic map , A = A ( S n ) ch → Λ that for n ≥ 0 and λ ∈ Par n sends 1 S n �→ h n sgn S n �→ e n χ λ �→ s λ Ind S n S λ 1 S λ �→ h λ Ind S n S λ sgn S λ �→ e λ 1 λ �→ p λ z λ (where ch is extended to a C -linear map A C �→ Λ C ) and for n ≥ 1 sends 1 ( n ) �→ p n n

Proof. ◮ A with m := Ind i + j : A i ⊗ A j → A i + j i , j ∆ := ⊕ i + j = n Res i + j : A n → ⊕ i + j = n A i ⊗ A j i , j is a PSH with PSH-basis Σ = ⊔ n ≥ 0 Irr ( S n ) . ◮ The unique irreducible character ρ = 1 S 1 of S 1 is the only element of C = Σ ∩ P , P is the set of primitive elements. ◮ Thus Theorem (g) tells us that there are two PSH-isomorphisms A → Λ, each of which sends Σ to the PSH-basis of Schur functions { s λ } for Λ.

◮ we can pin down one of the two isomorphisms to call ch , by insisting that it map the two characters 1 S 2 and sgn S 2 in Irr ( S 2 ) to h 2 and e 2 . ◮ 1 ⊥ S 2 annihilates sgn S n and sgn S 2 annihilates 1 S n ◮ Theorem (b) ⇒ ch (1 S n ) = h n and ch ( sgn S n ) = e n . ◮ By induction products Ind S n Ind S n S λ 1 S λ �→ h λ S λ sgn S λ �→ e λ ◮ A C is a Hopf algebra and has the C -bilinear form ( ., . ) C . ◮ 1 ( n ) is a primitive element in A C ⇒ ch (1 ( n ) ) is a scalar multiple of p n (Corollary 3.9).

◮ To find the scalar: p n = m n ⇒ ( h n , p n ) Λ = ( h n , m n ) Λ = 1, while ch − 1 ( h n ) = 1 S n , we have (1 S n , 1 ( n ) ) = 1 n !( n − 1)! = 1 n ⇒ ch (1 ( n ) ) = p n n . ◮ Exercise 4.28(d) ⇒ ch (1 λ ) = p λ z λ

References Grinberg and Reiner Hopf Algerba in Combinatorics Bruce Sagan (2000) The Symmetric Group

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