Universal Deformation Rings and Fusion David Meyer University of - PowerPoint PPT Presentation

Universal Deformation Rings and Fusion David Meyer University of Iowa Maurice Auslander Distinguished Lectures and International Conference David Meyer Universal Deformation Rings and Fusion

Universal Deformation Rings Let Γ be a finite group, V an absolutely irreducible F p Γ-module. By Mazur’s work, V has a well-defined universal deformation ring R (Γ , V ) which is universal with respect to all lifts of V over complete local commutative Noetherian rings with residue field F p . Theorem By Mazur, if dim F p ( H 1 (Γ , Hom F p ( V , V ))) = r and dim F p ( H 2 (Γ , Hom F p ( V , V ))) = s , then: R (Γ , V ) ∼ = Z p � t 1 , t 2 , ... t r � / I where r is minimal and I is an ideal whose minimal numbers of generators is bounded above by s. David Meyer Universal Deformation Rings and Fusion

General Setting p a prime G a finite p ′ -group Γ an extension of G by N := Z / p Z × Z / p Z Assume that F p is a splitting field of G . We have a short exact sequence 0 → Z / p Z × Z / p Z → Γ → G ∼ = Γ / N → 1 Z / p Z × Z / p Z is a 2-dimensional F p representation of G denoted by 1 φ . Let V be a 2-dimensional irreducible F p G -module inflated to Γ. 2 Question What is the relationship between the fusion of N = Z / p Z × Z / p Z in Γ and H 2 (Γ , Hom F p ( V , V )), resp. R (Γ , V )? David Meyer Universal Deformation Rings and Fusion

Cohomology Let ˜ φ denote the contragredient of φ and let W ˜ φ (resp. W det ◦ ( ˜ φ ) ) denote the F p Γ-module associated to ˜ φ (resp. det ◦ ( ˜ φ )). Theorem Using the above notation, φ ⊗ V ∗ ⊗ V ) ⊕ ( W det ◦ ( ˜ φ ) ⊗ V ∗ ⊗ V )] Γ / N . H 2 (Γ , Hom F p ( V , V )) ∼ = [( W ˜ This result provides a way of using character theory to compute the first and second cohomology group of Γ with coefficients in Hom F p ( V , V ). To prove the theorem we need the following result. Lemma Using the above notation, for all i ≥ 1, H i ( N , V ∗ ⊗ V ) ∼ = V ∗ ⊗ V ⊗ H i ( N , F p ) as F p Γ / N -modules, and H i (Γ , V ∗ ⊗ V ) ∼ = H 0 (Γ / N , H i ( N , V ∗ ⊗ V )) ∼ = [ H i ( N , V ∗ ⊗ V )] G . David Meyer Universal Deformation Rings and Fusion

Cohomology Proof of the Theorem . By the lemma, H 2 ( N , Hom F p ( V , V )) ∼ = Hom F p ( V , V ) ⊗ H 2 ( N , Z / p Z ) as F p Γ / N -modules. ι → C ∗ → 1, where p → C ∗ Consider the Kummer sequence 1 → µ p − − → C ∗ denotes the map given by z p p C ∗ → z p . We consider this sequence − − as a sequence of Z N -modules with trivial N -action. Applying the functor Hom Z Γ / N ( Z , − ) we obtain the long exact sequence δ ι ∗ p ∗ δ ι ∗ → H 1 ( N , µ p ) → H 1 ( N , C ∗ ) → H 1 ( N , C ∗ ) → H 2 ( N , µ p ) ... − − − − − → p ∗ ι ∗ δ H 2 ( N , C ∗ ) → H 2 ( N , C ∗ ) → H 3 ( N , µ p ) − − − → ... p ∗ Since N is elementary abelian, H i ( N , C ∗ ) → H i ( N , C ∗ ) is trivial. Thus, − we get the short exact sequence of F p Γ / N -modules δ ι ∗ 0 → H 1 ( N , C ∗ ) → H 2 ( N , Z / p Z ) → H 2 ( N , C ∗ ) → 0. − − David Meyer Universal Deformation Rings and Fusion

Cohomology Applying the functor Hom F p ( V , V ) ⊗ − , and taking fixed points, we obtain H 2 (Γ , Hom F p ( V , V )) ∼ = [ H 1 ( N , C ∗ ) ⊗ Hom F p ( V , V )] Γ / N ⊕ [ H 2 ( N , C ∗ ) ⊗ Hom F p ( V , V )] Γ / N . Therefore, our result follows once we show that H 1 ( N , C ∗ ) ∼ = W ˜ φ and H 2 ( N , C ∗ ) ∼ φ as F p Γ / N - modules. = W det ◦ ˜ Since N is an elementary abelian p -group which acts trivially on C ∗ , H 1 ( N , C ∗ ) = Hom ( N , C ∗ ) ∼ = Hom F p ( N , F p ) as F p G -modules, which implies H 1 ( N , C ∗ ) ∼ = W ˜ φ . It remains to determine the Γ / N -module structure of H 2 ( N , C ∗ ). Our result follows after a quick computation, using that H 2 ( N , C ∗ ) = N ∧ N . � David Meyer Universal Deformation Rings and Fusion

Cohomology So we have shown Theorem φ ⊗ V ∗ ⊗ V ) ⊕ ( W det ◦ ( ˜ φ ) ⊗ V ∗ ⊗ V )] Γ / N . H 2 (Γ , Hom F p ( V , V )) ∼ = [( W ˜ Additionally, we have Corollary Under the same hypotheses φ ⊗ V ∗ ⊗ V ) Γ / N (a) H 1 (Γ , Hom F p ( V , V )) = ( W ˜ (b) H 1 (Γ , Hom F p ( V , V )) is a summand of H 2 (Γ , Hom F p ( V , V )) (c) dim F p ( H 1 (Γ , Hom F p ( V , V ))) ≤ dim F p ( H 2 (Γ , Hom F p ( V , V ))) David Meyer Universal Deformation Rings and Fusion

Definitions Let N , Γ , G , φ be as above. For every irreducible F p G -module V , let d i V = 1 dim F p ( H i (Γ , Hom F p ( V , V )) for i=1,2. Note that this number depends on φ . We say an irreducible F p G -module V 0 is cohomologically maximal 2 for φ if d 2 V 0 is maximal among all d 2 V . Similarly, we say an irreducible representation ρ of G over F p is cohomologically maximal for φ if ρ corresponds to a F p G -module with this property. We call the orbits of the action φ of G on N the fusion orbits of φ . 3 For all m ≥ 1, let F φ, m be the number of fusion orbits of φ with cardinality m . Then, the sequence { F φ, m } m ≥ 1 is called the fusion of φ . David Meyer Universal Deformation Rings and Fusion

Main Result We now consider the case where G is dihedral of order 2 n . Let Rep 2 ( G ) be a complete set of representatives of isomorphism classes of all 2-dimensional representations of G over F p and let Irr 2 ( G ) ⊂ Rep 2 ( G ) be the subset of isomorphism classes of irreducible 2-dimensional representations. For ρ in Irr 2 ( G ), let V ρ be an irreducible F p G -module with representation ρ . Main Theorem Assuming the above notation, there exists a subset Ω of Irr 2 ( G ), and a map of sets T : Irr 2 ( G ) → Rep 2 ( G ) such that the following is true. (a) If n is odd, then Ω = Irr 2 ( G ) and T is a bijection. If n is even, Ω = Irr 2 ( G ) ∩ T ( Irr 2 ( G )) and for all ψ in Ω, | T − 1 ( ψ ) | = 2. (b) If φ ∈ Ω, then the fusion of φ is uniquely determined by the set { ker ( ρ ) | ρ ∈ Irr 2 ( G ) is cohomologically maximal for φ } = { ker ( ρ ) | ρ ∈ Irr 2 ( G ) with R (Γ , V ρ ) ≇ Z p } . David Meyer Universal Deformation Rings and Fusion

Main Result So for φ in Ω, the fusion can be detected by the set { ker ( ρ ) | ρ ∈ Irr 2 ( G ) is cohomologically maximal for φ } = { ker ( ρ ) | ρ ∈ Irr 2 ( G ) with R (Γ , V ρ ) ≇ Z p } . Moreover, we have the following. Proposition Let G = D 2 n . If n is odd, φ 1 , φ 2 ∈ Ω. Then φ 1 and φ 2 have the same fusion if and 1 only if T − 1 ( φ 1 ) and T − 1 ( φ 2 ) have the same kernel. If n is even, φ 1 , φ 2 ∈ Ω . Then φ 1 and φ 2 have the same fusion if and 2 only if { kernel of ψ : ψ ∈ T − 1 ( φ 1 ) } = { kernel of ψ : ψ ∈ T − 1 ( φ 2 ) } . If φ is in Ω, then V = V ψ is cohomologically maximal for φ if and 3 only if T ( ψ ) = φ . David Meyer Universal Deformation Rings and Fusion

Cohomology for G = D 2 n For G = D 2 n , all 2-dim. irreducible representations are of the form: ω i � � � � 0 0 1 θ i θ i For i = 1 < n r − → , s − → , 2 , ω a primitve ω − i 1 0 0 nth root of unity in F ∗ p , p ≡ 1 mod( n ). Note: θ i = Ind G � r � ( χ i ), where χ i is the one-dimensional representation of � r � with χ i ( r ) = ω i . We define the map T : Irr 2 ( G ) to Rep 2 ( G ) by T ( θ i ) = T ( Ind G � r � ( χ i )) = Ind G � r � ( χ 2 i ). David Meyer Universal Deformation Rings and Fusion

Cohomology for G = D 2 n Proposition Let G = D 2 n . Let Ω be as before. If n is odd, then T : Irr 2 ( G ) → Irr 2 ( G ) = Ω , T a bijection, and for 1 any φ irreducible, there exists a unique ψ = T − 1 ( φ ) irreducible with d 2 V ψ = 2. For all other V , d 2 V = 1. So V ψ is cohomologically maximal for φ . For n even, then T : Irr 2 ( G ) → Rep 1 ( G ), and for any φ in Ω, there 2 exist exactly two V ψ irreducible with d 2 V ψ = 2. For all other V , d 2 V = 1. Thus, there are precisely two ψ that are cohomologically maximal for φ . These exceptional representations are exactly the elements of T − 1 ( { φ } ). David Meyer Universal Deformation Rings and Fusion

Universal Deformation Rings Using a result of Bleher, Chinburg, de Smit we can show the following. Proposition Let G = D 2 n . Let R = R(Γ , V ), let φ be in Ω. Then, for V in Irr 2 ( G ), the universal deformation ring � Z p if V is not cohomologically maximal for φ R = Z p [[ t ]] / ( t 2 , tp ) if V is cohomologically maximal for φ Additionally, R(Γ , V ) = Z p [[ t ]] / ( t 2 , tp ) if and only if d 2 V is equal to two. V is cohomologically maximal for φ if and only if d 1 V = 1 and d 2 V = 2. Otherwise, d 1 V = 0 and d 2 V = 1. David Meyer Universal Deformation Rings and Fusion

Fusion for Dihedral Groups We compute the fusion for N = Z / p Z × Z / p Z in Γ. Γ / N = D 2 n , p ≡ 1 (mod n ), φ = θ i . Fusion Elements of N are fused if and only if they are in the same φ orbit. The cardinality of the orbits are as follows:  1 , if ( x , y ) = 0   ∗ × F p ∗ , y / x ∈ � ω i � | Orbit (( x , y )) | = n / gcd( i , n ) , if ( x , y ) ∈ F p  2 n / gcd( i , n ) , otherwise  Thus, fusion is determined by gcd( i , n ). David Meyer Universal Deformation Rings and Fusion

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