Fixed-point subalgebra of quiver Hecke algebras for a quiver - PowerPoint PPT Presentation

Fixed-point subalgebra of quiver Hecke algebras for a quiver automorphism and application to the Hecke algebra of G ( r , p , n ) Salim Rostam Laboratoire de mathématiques de Versailles (LMV) Université de Versailles Saint-Quentin (UVSQ) Nikolaus conference 2016

Motivations Let n , e , p ∈ N ∗ with e ≥ 2. Let q , ζ be some elements of a field F of respective order e , p . Let Λ = (Λ i ) i be a Z / e Z -tuple of non-negative integers and set r := p � i Λ i . The Ariki–Koike algebra H Λ n ( q , ζ ) is a Hecke algebra of the complex reflection group G ( r , 1 , n ). It is a F -algebra generated by S , T 1 , . . . , T n − 1 , the “cyclotomic relation” being: � � Λ i = 0 . ( S − ζ j q i ) i ∈ Z / e Z j ∈ Z / p Z

Motivations Let n , e , p ∈ N ∗ with e ≥ 2. Let q , ζ be some elements of a field F of respective order e , p . Let Λ = (Λ i ) i be a Z / e Z -tuple of non-negative integers and set r := p � i Λ i . The Ariki–Koike algebra H Λ n ( q , ζ ) is a Hecke algebra of the complex reflection group G ( r , 1 , n ). It is a F -algebra generated by S , T 1 , . . . , T n − 1 , the “cyclotomic relation” being: � � Λ i = 0 . ( S − ζ j q i ) i ∈ Z / e Z j ∈ Z / p Z There is an automorphism σ H of H Λ n ( q , ζ ) of order p given by: σ H ( S ) := ζ S , ∀ a , σ H ( T a ) := T a . n ( q , ζ ) σ H of fixed points is a Hecke algebra of The subalgebra H Λ G ( r , p , n ).

Cyclotomic quiver Hecke algebra Let Γ be a quiver (= oriented graph) with vertex set K . The quiver Hecke algebra R n (Γ) is generated over F by: e ( k ) for k ∈ K n , y 1 , . . . , y n , ψ 1 , . . . , ψ n − 1 , together with some relations. Exemple of relation For k ∈ K n such that k a Γ → k a +1 then ψ 2 a e ( k ) = ( y a +1 − y a ) e ( k ).

Cyclotomic quiver Hecke algebra Let Γ be a quiver (= oriented graph) with vertex set K . The quiver Hecke algebra R n (Γ) is generated over F by: e ( k ) for k ∈ K n , y 1 , . . . , y n , ψ 1 , . . . , ψ n − 1 , together with some relations. Exemple of relation For k ∈ K n such that k a Γ → k a +1 then ψ 2 a e ( k ) = ( y a +1 − y a ) e ( k ). For Λ = (Λ k ) k ∈ K ∈ N K , the cyclotomic quiver Hecke algebra R Λ n (Γ) is the quotient of R n (Γ) by the following relations: Λ k 1 ∀ k ∈ K n , y e ( k ) = 0 . 1

Graded isomorphism theorem Theorem (Brundan–Kleshchev, Rouquier) The Ariki–Koike algebra H Λ n ( q , ζ ) is isomorphic over F to the cyclotomic quiver Hecke algebra R Λ n (Γ e , p ) , where Γ e , p is given by: 0 0 0 1 1 1 e − 1 e − 1 e − 1 . . . . . . . . . ... 2 ... 2 ... 2 . . . . . . . . . � �� � p p ′ := gcd ( p , e ) copies Remark The integer p ′ is the smallest integer m ≥ 1 such that ζ m ∈ � q � .

Graded isomorphism theorem Theorem (Brundan–Kleshchev, Rouquier) The Ariki–Koike algebra H Λ n ( q , ζ ) is isomorphic over F to the cyclotomic quiver Hecke algebra R Λ n (Γ e , p ) , where Γ e , p is given by: 0 0 0 1 1 1 e − 1 e − 1 e − 1 . . . . . . . . . ... 2 ... 2 ... 2 . . . . . . . . . � �� � p p ′ := gcd ( p , e ) copies Remark The integer p ′ is the smallest integer m ≥ 1 such that ζ m ∈ � q � . Our aim is to find an isomorphism Φ : H Λ n ( q , ζ ) → R Λ n (Γ e , p ) such that we get a “nice” automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ e , p ).

Fixed-point quiver Hecke subalgebra Let σ : K → K a bijection of finite order p such that: ∀ k , k ′ ∈ K , k → k ′ = ⇒ σ ( k ) → σ ( k ′ ) .

Fixed-point quiver Hecke subalgebra Let σ : K → K a bijection of finite order p such that: ∀ k , k ′ ∈ K , k → k ′ = ⇒ σ ( k ) → σ ( k ′ ) . Theorem The map σ induces a well-defined automorphism of R n (Γ) by: ∀ k ∈ K n , σ ( e ( k )) := e ( σ ( k )) , σ ( y a ) := y a , ∀ a ∈ { 1 , . . . , n } , ∀ a ∈ { 1 , . . . , n − 1 } , σ ( ψ a ) := ψ a . Definition We set: R n (Γ) σ := { h ∈ R n (Γ) : σ ( h ) = h } .

Fixed-point cyclotomic quiver Hecke subalgebra Theorem We can give a presentation of R n (Γ) σ in terms of the following generators: e ( γ ) := e ( k ) + e ( σ ( k )) + · · · + e ( σ p − 1 ( k )) for γ = [ k ] ∈ K n / � σ � , y 1 , . . . , y n , ψ 1 , . . . , ψ n − 1 . Exemple of relation If γ ∈ K n / � σ � verifies “ γ a → γ a +1 ” then ψ 2 a e ( γ ) = ( y a +1 − y a ) e ( γ ).

Fixed-point cyclotomic quiver Hecke subalgebra Theorem We can give a presentation of R n (Γ) σ in terms of the following generators: e ( γ ) := e ( k ) + e ( σ ( k )) + · · · + e ( σ p − 1 ( k )) for γ = [ k ] ∈ K n / � σ � , y 1 , . . . , y n , ψ 1 , . . . , ψ n − 1 . Exemple of relation If γ ∈ K n / � σ � verifies “ γ a → γ a +1 ” then ψ 2 a e ( γ ) = ( y a +1 − y a ) e ( γ ). We now assume that Λ k = Λ σ ( k ) for all k ∈ K . Theorem The automorphism σ induces an automorphism of R Λ n (Γ) . Moreover: n (Γ) σ ≃ R n (Γ) σ �� � Λ γ 1 e ( γ ) = 0 : γ ∈ K n / � σ � R Λ y . 1

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation translation

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation translation translation

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation + translation translation rotation by η

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation + translation translation rotation by η

Application to the Hecke algebra of G ( r , p , n ) Theorem We can choose Φ such that the automorphism Φ ◦ σ H ◦ Φ − 1 of R Λ n (Γ) comes from the following bijection of the vertices (case e := 6 , p := 9 and p ′ = gcd (9 , 6) = 3 ) : 9 0 0 0 5 1 5 1 5 1 4 2 4 2 4 2 3 3 3 translation + translation translation rotation by η where η ∈ Z / e Z is determined by the equality ζ p ′ = q η .