Minimal free resolutions of orbit closures of quivers Andr as - PowerPoint PPT Presentation

Minimal free resolutions of orbit closures of quivers Andr´ as Cristian L˝ orincz Humboldt University Joint work with Jerzy Weyman Free Resolutions and Representation Theory, ICERM, August 2020 Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Basics k is a field of characterisitic 0. Mat( m , n ) denotes the space of m × n matrices with entries in k . Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Basics k is a field of characterisitic 0. Mat( m , n ) denotes the space of m × n matrices with entries in k . A quiver Q is an oriented graph with vertices Q 0 and arrows Q 1 . Given a dimension vector α ∈ Z Q 0 ≥ 0 , we work in the representation variety � a � ha rep( Q , α ) = Mat( α ha , α ta ) , ta a ∈ Q 1 An element M ∈ rep( Q , α ) is called a representation of Q (with dimension vector α ). Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Basics k is a field of characterisitic 0. Mat( m , n ) denotes the space of m × n matrices with entries in k . A quiver Q is an oriented graph with vertices Q 0 and arrows Q 1 . Given a dimension vector α ∈ Z Q 0 ≥ 0 , we work in the representation variety � a � ha rep( Q , α ) = Mat( α ha , α ta ) , ta a ∈ Q 1 An element M ∈ rep( Q , α ) is called a representation of Q (with dimension vector α ). The action of the base change group � GL ( α ) = GL ( α x ) x ∈ Q 0 acts on rep( Q , α ) by g · M = ( g ha M a g − 1 ta ) a ∈ Q 1 , where g = ( g x ) x ∈ Q 0 ∈ GL ( α ) and M = ( M a ) a ∈ Q 1 ∈ rep( Q , α ). Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Under the action of GL ( α ) on rep( Q , α ), orbits correspond to isomorphism classes of modules over the path algebra k Q . Example a � α 2 α 1 When Q = A 2 , the orbits correspond to α 2 × α 1 matrices of fixed rank r . Orbit closures are precisely the determinantal varieties. Their defining ideals are generated (minimally) by the ( r + 1) × ( r + 1) minors of the α 2 × α 1 generic matrix of variables. Their minimal free resolutions were determined by Lascoux. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Under the action of GL ( α ) on rep( Q , α ), orbits correspond to isomorphism classes of modules over the path algebra k Q . Example a � α 2 α 1 When Q = A 2 , the orbits correspond to α 2 × α 1 matrices of fixed rank r . Orbit closures are precisely the determinantal varieties. Their defining ideals are generated (minimally) by the ( r + 1) × ( r + 1) minors of the α 2 × α 1 generic matrix of variables. Their minimal free resolutions were determined by Lascoux. Example a b � α 2 � α 3 α 1 When Q is the equioriented A 3 , the orbits correspond to pairs of matrices ( A , B ) such that rank A , rank B and rank BA are fixed. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Dynkin quivers The Euler form of a quiver is a quadratic map E Q : Z Q 0 → Z given by � α 2 � E Q ( α ) = x − α ta α ha . x ∈ Q 0 a ∈ Q 1 Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Dynkin quivers The Euler form of a quiver is a quadratic map E Q : Z Q 0 → Z given by � α 2 � E Q ( α ) = x − α ta α ha . x ∈ Q 0 a ∈ Q 1 Theorem The algebra k Q has finitely many indecomposable modules ⇐ ⇒ Q is a disjoint union of Dynkin quivers ⇐ ⇒ E Q is positive definite ⇐ ⇒ rep( Q , α ) has finitely many GL ( α ) -orbits for all α . Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Dynkin quivers The Euler form of a quiver is a quadratic map E Q : Z Q 0 → Z given by � α 2 � E Q ( α ) = x − α ta α ha . x ∈ Q 0 a ∈ Q 1 Theorem The algebra k Q has finitely many indecomposable modules ⇐ ⇒ Q is a disjoint union of Dynkin quivers ⇐ ⇒ E Q is positive definite ⇐ ⇒ rep( Q , α ) has finitely many GL ( α ) -orbits for all α . Furthermore, Q is of tame representation type ⇐ ⇒ Q is a disjoint union of extended Dynkin and Dynkin quivers ⇐ ⇒ E Q is positive semi-definite. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

1-step representations For two dimension vectors β ⊂ α we put � Grass( β, α ) := Grass( β x , α x ) . x ∈ Q 0 Consider the subbundle Z ( Q , β ⊂ α ) ⊂ rep( Q , α ) × Grass( β, α ) consisting of points ( V , { R x } ) such that the collection of subspaces { R x } x ∈ Q 0 forms a subrepresentation of V . If the projection onto the first factor q : Z ( Q , β ⊂ α ) → rep( Q , α ) is an orbit closure O V (this is always the case for Dynkin quivers), then we call V a 1-step representation. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. If M , N are representations with dense orbits such that with Ext 1 Q ( M , N ) = 0, then V = M ⊕ N is 1-step and q − 1 ( V ) is irreducible. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. If M , N are representations with dense orbits such that with Ext 1 Q ( M , N ) = 0, then V = M ⊕ N is 1-step and q − 1 ( V ) is irreducible. If Q is Dynkin then the irreducible components of the complement of the dense orbit in rep( Q , α ) are 1-step. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. If M , N are representations with dense orbits such that with Ext 1 Q ( M , N ) = 0, then V = M ⊕ N is 1-step and q − 1 ( V ) is irreducible. If Q is Dynkin then the irreducible components of the complement of the dense orbit in rep( Q , α ) are 1-step. Assume Q is Dynkin and write α = β + γ . The representation V whose orbit closure is the image of Z ( β ⊂ α ) is precisely the generic extension of the generic representations in rep( Q , β ) and rep( Q , γ ). This can be found algorithmically. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. If M , N are representations with dense orbits such that with Ext 1 Q ( M , N ) = 0, then V = M ⊕ N is 1-step and q − 1 ( V ) is irreducible. If Q is Dynkin then the irreducible components of the complement of the dense orbit in rep( Q , α ) are 1-step. Assume Q is Dynkin and write α = β + γ . The representation V whose orbit closure is the image of Z ( β ⊂ α ) is precisely the generic extension of the generic representations in rep( Q , β ) and rep( Q , γ ). This can be found algorithmically. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Examples Example For A 2 , non-equioriented A 3 , all representations are 1-step. If M , N are representations with dense orbits such that with Ext 1 Q ( M , N ) = 0, then V = M ⊕ N is 1-step and q − 1 ( V ) is irreducible. If Q is Dynkin then the irreducible components of the complement of the dense orbit in rep( Q , α ) are 1-step. Assume Q is Dynkin and write α = β + γ . The representation V whose orbit closure is the image of Z ( β ⊂ α ) is precisely the generic extension of the generic representations in rep( Q , β ) and rep( Q , γ ). This can be found algorithmically. Let ξ denote the dual of the quotient bundle (rep( Q , α ) × Grass( β, α )) / Z ( Q , β ⊂ α ). If R x (resp. Q x ) denotes the tautological subbundle (resp. factorbundle) on Gr( β x , α x ), for a ∈ Q 1 R ta ⊗ Q ∗ x ∈ Q 0 , then ξ = � ha Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Kempf-Weyman geometric technique Put A = k [rep( Q , α )]. One can construct a complex F • with terms i + j � H j (Gr( β, α ) , � F i = ξ ) ⊗ A ( − i − j ) . j ≥ 0 Since ξ is semi-simple, the terms F i can be computed (as representations of general linear groups) using the Cauchy formula, Littlewood-Richardson rule and Borel-Weil-Bott theorem. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers

Kempf-Weyman geometric technique Put A = k [rep( Q , α )]. One can construct a complex F • with terms i + j � H j (Gr( β, α ) , � F i = ξ ) ⊗ A ( − i − j ) . j ≥ 0 Since ξ is semi-simple, the terms F i can be computed (as representations of general linear groups) using the Cauchy formula, Littlewood-Richardson rule and Borel-Weil-Bott theorem. Theorem Assume V is a 1-step representation. (a) If F i = 0 for all i < 0 , and the fiber q − 1 ( V ) is connected, then F • is a minimal free resolution of the normalization of O V , and the normalization has rational singularities. (b) If F i = 0 for all i < 0 and F 0 = A, then O V is normal and it has rational singularities. Andr´ as Cristian L˝ orincz Minimal free resolutions of orbit closures of quivers