A McKay correspondence for reflections groups joint work with - PowerPoint PPT Presentation

A McKay correspondence for reflections groups joint work with Ragnar-Olaf Buchweitz and Colin Ingalls Eleonore Faber University of Michigan Auslander Conference, Woods Hole 2016 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Kleinian singularities Focus on n = 2, and k = C . Then Theorem (F . Klein, 1884) Let Γ ⊆ SL 2 ( C ) be a finite group. Then the quotient singularity X = C 2 / Γ = Spec ( S Γ ) , i.e., the orbit space of Γ acting on C 2 , is of the form X = Spec ( C [ x , y , z ] / ( f )) , Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Kleinian singularities Focus on n = 2, and k = C . Then Theorem (F . Klein, 1884) Let Γ ⊆ SL 2 ( C ) be a finite group. Then the quotient singularity X = C 2 / Γ = Spec ( S Γ ) , i.e., the orbit space of Γ acting on C 2 , is of the form X = Spec ( C [ x , y , z ] / ( f )) , where f is of type A n : z 2 + y 2 + x n + 1 , D n : z 2 + x ( y 2 + x n − 2 ) for n ≥ 4 , E 6 : z 2 + x 3 + y 4 , E 7 : z 2 + x ( x 2 + y 3 ) , E 8 : z 2 + x 3 + y 5 . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence A 1 and A 2 – the cone and the cusp x 2 + y 2 − z 2 = 0 z 2 + y 2 − x 3 = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence A 3 and A 4 z 2 + y 2 − x 4 = 0 z 2 + y 2 − x 5 = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence A 5 and A 6 z 2 + y 2 − x 6 = 0 z 2 + y 2 − x 7 = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence D 4 : z 2 + x ( y 2 − x 2 ) = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence D 5 and D 6 z 2 + x ( y 2 − x 3 ) = 0 z 2 + x ( y 2 − x 4 ) = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence D 7 and D 8 z 2 + x ( y 2 − x 5 ) = 0 z 2 + x ( y 2 − x 6 ) = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence z 2 + x 3 + y 4 = 0 E 6 : Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence z 2 + x ( x 2 + y 3 ) = 0 E 7 : Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence z 2 + x 3 + y 5 = 0 E 8 : Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Dual resolution graphs Let X be a normal surface singularity and let π : � X − → X be its minimal resolution, with exceptional curves � i E i . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Dual resolution graphs Let X be a normal surface singularity and let π : � X − → X be its minimal resolution, with exceptional curves � i E i . Form a graph with vertices: i ← → E i edges: i − j ← → E i ∩ E j � = ∅ . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Dual resolution graphs Let X be a normal surface singularity and let π : � X − → X be its minimal resolution, with exceptional curves � i E i . Form a graph with vertices: i ← → E i edges: i − j ← → E i ∩ E j � = ∅ . Theorem (Du Val) The dual resolution resolution graphs of the Kleinian singularities are Coxeter–Dynkin diagrams of type ADE. Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Example: x 2 + y 2 = z 2 π − − − − → Dual resolution graph of type A 1 : • Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Example: z 2 + x ( y 2 − x 2 ) = 0 π − − − − → Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Example: z 2 + x ( y 2 − x 2 ) = 0 π − − − − → Dual resolution graph of type D 4 : • • • • Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence McKay correspondence Let Γ ⊆ SL 2 ( C ) be a finite group with irreducible representations ρ 0 , . . . ρ m : ρ 0 = trivial representation, ρ 1 = c = canonical representation Γ ֒ → GL 2 ( C ) . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence McKay correspondence Let Γ ⊆ SL 2 ( C ) be a finite group with irreducible representations ρ 0 , . . . ρ m : ρ 0 = trivial representation, ρ 1 = c = canonical representation Γ ֒ → GL 2 ( C ) . Form a graph: vertices: i ← → ρ i m ij arrows: i − → j iff ρ j appears with multiplicity m ij in the tensor product represenation c ⊗ ρ i Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence McKay correspondence Let Γ ⊆ SL 2 ( C ) be a finite group with irreducible representations ρ 0 , . . . ρ m : ρ 0 = trivial representation, ρ 1 = c = canonical representation Γ ֒ → GL 2 ( C ) . Form a graph: vertices: i ← → ρ i m ij arrows: i − → j iff ρ j appears with multiplicity m ij in the tensor product represenation c ⊗ ρ i Observation (J. McKay, 1979): These graphs are extended Coxeter Dynkin diagrams of type ADE (with arrows in both directions). Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Example: D 4 The group Γ is generated by � 1 � � i � � 0 � � 0 � 0 0 1 i ± , ± , ± , ± . 0 1 0 − i − 1 0 i 0 Five irreps ρ i , four one-dimensional and one two-dimensional ρ 1 = c . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence Example: D 4 The group Γ is generated by � 1 � � i � � 0 � � 0 � 0 0 1 i ± , ± , ± , ± . 0 1 0 − i − 1 0 i 0 Five irreps ρ i , four one-dimensional and one two-dimensional ρ 1 = c . The McKay graph: ρ 4 ρ 0 ρ 1 ρ 3 ρ 2 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Classical McKay correspondence McKay correspondence Thus for n = 2 and Γ ∈ SL 2 ( C ) : Have 1-1 correspondence between exceptional curves E i on the minimal resolution of C 2 / Γ . irreducible representations of Γ (mod the trivial representation). indecomposable projective Γ ∗ S = End R S -modules (modulo the trivial module). indecomposable CM -modules over R (modulo R itself). [This follows from Herzog’s theorem , which says that add R ( S ) = CM ( R ) .] Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

McKay for reflection groups Theorem (Buchweitz–F–Ingalls) If G ⊆ GL 2 ( C ) is a reflection group, let z = � s ∈ reflections ( G ) l s be the hyperplane arrangement and set ∆ = z 2 . � g ∈ G g, ¯ 1 A = A / AeA and T = S G . Let further A = G ∗ S, e = | G | Then A ∼ ¯ = End T / ∆ ( S / z ) is a NCR of T / ∆ , that is, gldim ¯ A = 2 and S / z is in CM ( T / ∆) . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

McKay for reflection groups Theorem (Buchweitz–F–Ingalls) If G ⊆ GL 2 ( C ) is a reflection group, let z = � s ∈ reflections ( G ) l s be the hyperplane arrangement and set ∆ = z 2 . � g ∈ G g, ¯ 1 A = A / AeA and T = S G . Let further A = G ∗ S, e = | G | Then A ∼ ¯ = End T / ∆ ( S / z ) is a NCR of T / ∆ , that is, gldim ¯ A = 2 and S / z is in CM ( T / ∆) . In particular: add T / ∆ ( S / z ) = CM ( T / ∆) , i.e., S / z is a CM -representation generator. Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Higher dimension The swallowtail: ∆ of S 4 16 x 4 z − 4 x 3 y 2 − 128 x 2 z 2 + 144 xy 2 z − 27 y 4 + 256 z 3 = 0 Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Higher dimension The swallowtail: ∆ of S 4 16 x 4 z − 4 x 3 y 2 − 128 x 2 z 2 + 144 xy 2 z − 27 y 4 + 256 z 3 = 0 = T / ∆ ⊕ � T / ∆ ⊕ syz ( � Here S / z ∼ T / ∆) ⊕ M 2 2 , 0 . Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016

Questions Questions What are the R -direct summands of S / z ? Can one describe the R -direct summands of S / z for some specific groups, e.g., S n ? What about the geometry? Eleonore Faber (University of Michigan) McKay for reflections Woods Hole 2016