In algebraic geometry, Procesi bundles are vector bundles of rank n ! {\displaystyle n!} on certain symplectic resolutions of quotient singularities, particularly on the Hilbert scheme of n {\displaystyle n} points in the complex plane. They play a fundamental role in geometric representation theory and were crucial in Mark Haiman's proof of the n! theorem and Macdonald positivity conjecture, and were named after Italian mathematician Claudio Procesi.
Definition Let H n = Hilb n ( C 2 ) {\displaystyle H_{n}={\text{Hilb}}^{n}(\mathbb {C} ^{2})} denote the Hilbert scheme of n points in the complex plane C 2 {\displaystyle \mathbb {C} ^{2}} , which provides a resolution of singularities of the quotient ( C 2 ) n / S n {\displaystyle (\mathbb {C} ^{2})^{n}/S_{n}} , where S n {\displaystyle S_{n}} is the symmetric group of degree n {\displaystyle n} . A Procesi bundle P {\displaystyle {\mathcal {P}}} on H n {\displaystyle H_{n}} is a C × {\displaystyle \mathbb {C} ^{\times }} -equivariant vector bundle of rank n ! {\displaystyle n!} together with an isomorphism End ( P ) ≅ C [ x , y ] # S n {\displaystyle {\text{End}}({\mathcal {P}})\cong \mathbb {C} [x,y]\#S_{n}} (where C [ x , y ] # S n {\displaystyle \mathbb {C} [x,y]\#S_{n}} is the smash product algebra) of C [ V ] S n {\displaystyle \mathbb {C} [V]^{S_{n}}} -algebras, such that Ext i ( P , P ) = 0 {\displaystyle {\text{Ext}}^{i}({\mathcal {P}},{\mathcal {P}})=0} for all i > 0 {\displaystyle i>0} . The isomorphism ensures that each fiber of P {\displaystyle {\mathcal {P}}} is naturally the regular representation of S n {\displaystyle S_{n}} . More generally, for a finite subgroup Γ ⊂ SL 2 ( C ) {\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {C} )} and its wreath product Γ n = S n ⋉ Γ n {\displaystyle \Gamma _{n}=S_{n}\ltimes \Gamma ^{n}} , Procesi bundles can be defined on symplectic resolutions of C 2 n / Γ n {\displaystyle \mathbb {C} ^{2n}/\Gamma _{n}} .
… excerpt ends here. Continue reading the full article.
