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Minimal free resolutions of orbit closures of quivers

Andr´ as Cristian L˝

• rincz

Humboldt University

Joint work with Jerzy Weyman Free Resolutions and Representation Theory, ICERM, August 2020

Andr´ as Cristian L˝

• rincz

Minimal free resolutions of orbit closures of quivers

Basics

k is a field of characterisitic 0. Mat(m, n) denotes the space

• f m × n matrices with entries in k.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Basics

k is a field of characterisitic 0. Mat(m, n) denotes the space

• f m × n matrices with entries in k.

A quiver Q is an oriented graph with vertices Q0 and arrows

• Q1. Given a dimension vector α ∈ ZQ0

≥0, we work in the

representation variety rep(Q, α) =

• a∈Q1

Mat(αha, αta), ta

a

ha

An element M ∈ rep(Q, α) is called a representation of Q (with dimension vector α).

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Basics

k is a field of characterisitic 0. Mat(m, n) denotes the space

• f m × n matrices with entries in k.

A quiver Q is an oriented graph with vertices Q0 and arrows

• Q1. Given a dimension vector α ∈ ZQ0

≥0, we work in the

representation variety rep(Q, α) =

• a∈Q1

Mat(αha, αta), ta

a

ha

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) =

• x∈Q0

GL(αx) acts on rep(Q, α) by g · M = (ghaMag−1

ta )a∈Q1,

where g = (gx)x∈Q0 ∈ GL(α) and M = (Ma)a∈Q1 ∈ rep(Q, α).

Andr´ as Cristian L˝

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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 kQ. Example α1

a

α2

When Q = A2, the orbits correspond to α2 × α1 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 α2 × α1 generic matrix of variables. Their minimal free resolutions were determined by Lascoux.

Andr´ as Cristian L˝

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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 kQ. Example α1

a

α2

When Q = A2, the orbits correspond to α2 × α1 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 α2 × α1 generic matrix of variables. Their minimal free resolutions were determined by Lascoux. Example α1

a

α2

b

α3

When Q is the equioriented A3, the orbits correspond to pairs of matrices (A, B) such that rank A, rank B and rank BA are fixed.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Dynkin quivers

The Euler form of a quiver is a quadratic map EQ : ZQ0 → Z given by EQ(α) =

• x∈Q0

α2

x −

• a∈Q1

αtaαha.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Dynkin quivers

The Euler form of a quiver is a quadratic map EQ : ZQ0 → Z given by EQ(α) =

• x∈Q0

α2

x −

• a∈Q1

αtaαha. Theorem The algebra kQ has finitely many indecomposable modules ⇐ ⇒ Q is a disjoint union of Dynkin quivers ⇐ ⇒ EQ is positive definite ⇐ ⇒ rep(Q, α) has finitely many GL(α)-orbits for all α.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Dynkin quivers

The Euler form of a quiver is a quadratic map EQ : ZQ0 → Z given by EQ(α) =

• x∈Q0

α2

x −

• a∈Q1

αtaαha. Theorem The algebra kQ has finitely many indecomposable modules ⇐ ⇒ Q is a disjoint union of Dynkin quivers ⇐ ⇒ EQ 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 ⇐ ⇒ EQ is positive semi-definite.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

1-step representations

For two dimension vectors β ⊂ α we put Grass(β, α) :=

• x∈Q0

Grass(βx, αx). Consider the subbundle Z(Q, β ⊂ α) ⊂ rep(Q, α) × Grass(β, α) consisting of points (V , {Rx}) such that the collection of subspaces {Rx}x∈Q0 forms a subrepresentation of V . If the projection onto the first factor q : Z(Q, β ⊂ α) → rep(Q, α) is an orbit closure OV (this is always the case for Dynkin quivers), then we call V a 1-step representation.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step. If M, N are representations with dense orbits such that with Ext1

Q(M, N) = 0, then V = M ⊕ N is 1-step and q−1(V ) is

irreducible.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step. If M, N are representations with dense orbits such that with Ext1

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˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step. If M, N are representations with dense orbits such that with Ext1

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˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step. If M, N are representations with dense orbits such that with Ext1

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˝

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Minimal free resolutions of orbit closures of quivers

Examples

Example For A2, non-equioriented A3, all representations are 1-step. If M, N are representations with dense orbits such that with Ext1

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 Rx (resp. Qx) denotes the tautological subbundle (resp. factorbundle) on Gr(βx, αx), for x ∈ Q0, then ξ =

a∈Q1 Rta ⊗ Q∗ ha

Andr´ as Cristian L˝

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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 Fi =

• j≥0

Hj(Gr(β, α),

i+j

• ξ) ⊗ A(−i − j).

Since ξ is semi-simple, the terms Fi 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˝

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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 Fi =

• j≥0

Hj(Gr(β, α),

i+j

• ξ) ⊗ A(−i − j).

Since ξ is semi-simple, the terms Fi 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 Fi = 0 for all i < 0, and the fiber q−1(V ) is connected, then F• is a minimal free resolution of the normalization of OV , and the normalization has rational singularities. (b) If Fi = 0 for all i < 0 and F0 = A, then OV is normal and it has rational singularities.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Minimal free resolutions of quiver loci

In [L–Weyman ’19], we show that the Euler form EQ controls directly the vanishing behavior of Fi for i ≤ 0. Theorem (L–Weyman ’19) Let Q be a Dynkin quiver, and V a 1-step representation. Then OV is normal, has rational singularities, and the complex F• gives the minimal free resolution of its defining ideal.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Minimal free resolutions of quiver loci

In [L–Weyman ’19], we show that the Euler form EQ controls directly the vanishing behavior of Fi for i ≤ 0. Theorem (L–Weyman ’19) Let Q be a Dynkin quiver, and V a 1-step representation. Then OV is normal, has rational singularities, and the complex F• gives the minimal free resolution of its defining ideal. Theorem (L–Weyman ’19) Let Q be an extended Dynkin quiver, V a 1-step representation and assume that the fiber q−1(V ) is connected. Then the normalization of OV has rational singularities, and the complex F• gives the minimal free resolution of the normalization. Some particular cases of the above were proved by [Sutar ’15].

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Defining ideals for type A quivers

An orbit closure of a representation of the equioriented quiver of type An is isomorphic to the opposite cell in a Schubert variety of a partial flag variety SL(n)/P.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Defining ideals for type A quivers

An orbit closure of a representation of the equioriented quiver of type An is isomorphic to the opposite cell in a Schubert variety of a partial flag variety SL(n)/P. In [L–Weyman ’19], an explicit set of minimal free generators of the defining ideals of 1-step representations for type A quivers is provided, in the form of some minors of block matrices of generic variables. Theorem (L-Weyman ’19) Let Q be a quiver of type An and Y ∈ Rep(Q, α) be an arbitrary

• representation. Then OY can be written as a (scheme-theoretic)

intersection of n − 1 orbit closures of 1-step representations.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Equioriented type A3 quiver

For the non-equioriented A3 quiver, all representations are 1-step. Minimal free resolutions of their orbit closures were computed first by [Sutar ’13]. From now on, Q is the equioriented A3 quiver α1

a

α2

b

α3

Consider the subbundle Z Z ⊂ rep(Q, α) × Flag(r1, r2, α2) × Grass(r, α3) consisting of elements of the form (V , R1 ⊂ R2, R) such that image V (a) ⊂ R1 and V (b)(R2) ⊂ R. Proposition (L. ’20) Any orbit closure in rep(Q, α) has a desingularization as subbundle Z as above via the projection q : Z → rep(Q, α).

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Equivalent problem

We recall that the irreducible (polynomial) representations of GL(W ) are parameterized by partitions λ of length dim W . λ → SλW . The construction of these representations is functorial.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Equivalent problem

We recall that the irreducible (polynomial) representations of GL(W ) are parameterized by partitions λ of length dim W . λ → SλW . The construction of these representations is functorial. Proposition (L. ’20) The problem of determining the (equivariant) terms of the minimal free resolutions for all orbit closures of the equioriented A3 quiver is equivalent to the problem of determining the cohomology (as representations of GL(n)) of the bundles on Flag(r1, r2, n) of the form SλR2 ⊗ Sµ(W /R1)∗, for all 0 ≤ r1 ≤ r2 ≤ n = dim W and partitions λ, µ.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Splitting technique

The cohomology of SλR2 ⊗ Sµ(W /R1)∗ can be approximated by computing the cohomology of the split bundles B1 = Sλ(R1 ⊕ R2/R1) ⊗ Sµ(W /R1)∗, B2 = SλR2 ⊗ Sµ(W /R2 ⊕ R2/R1)∗.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Splitting technique

The cohomology of SλR2 ⊗ Sµ(W /R1)∗ can be approximated by computing the cohomology of the split bundles B1 = Sλ(R1 ⊕ R2/R1) ⊗ Sµ(W /R1)∗, B2 = SλR2 ⊗ Sµ(W /R2 ⊕ R2/R1)∗. Example Let X = Flag(1, 2, 3), take the bundle S(3,2)R2 ⊗ S(3,1)(W /R1)∗. The only non-zero cohomology groups of Bi are H2(X, B1) = H3(X, B1) = S(1,1,−1)W ; H2(X, B2) = H3(X, B2) = S(1,0,0)W . This implies Hi X, S(3,2)R2 ⊗ S(3,1)(W /R1)∗ = 0, for all i ≥ 0.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Schur complexes

Let 0 → A → B → C → 0 be an exact sequence of vector spaces (or bundles). For any partition λ, we have a right resolution of the module SλA: SλB →

• |µ|=|λ|−1

|ν|=1

(SµB⊗Sν′C)⊕cλ

µ,ν · · · →

• |µ|=|λ|−i

|ν|=i

(SµB⊗Sν′C)⊕cλ

µ,ν · · · → Sλ′C. Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

Schur complexes

Let 0 → A → B → C → 0 be an exact sequence of vector spaces (or bundles). For any partition λ, we have a right resolution of the module SλA: SλB →

• |µ|=|λ|−1

|ν|=1

(SµB⊗Sν′C)⊕cλ

µ,ν · · · →

• |µ|=|λ|−i

|ν|=i

(SµB⊗Sν′C)⊕cλ

µ,ν · · · → Sλ′C.

When applied to one of the following sequences 0 → R2 → W → W /R2 → 0, or 0 → (W /R1)∗ → W ∗ → R∗

1 → 0,

the Schur complex can be used to prove that certain representation cancels out between neighboring cohomology groups of the split bundles. The hypercohomology of the Schur complexes can be analyzed in the category of homogeneous bundles over the Grassmannian.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

An algorithm

Example Let rep(Q, α) be the space of pairs of matrices (A, B), where A is 4 × 3 and B is 3 × 4. Consider the orbit closure OV given by rank A ≤ 1, rank B ≤ 1 and BA = 0. Then OV is Gorenstein. The number of irreducible GL(d)-module summands obtained in the terms F0, F1, F2, F3, F4, F5, F6 are 1, 3, 6, 17, 35, 48, 52, respectively.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers

An algorithm

Example Let rep(Q, α) be the space of pairs of matrices (A, B), where A is 4 × 3 and B is 3 × 4. Consider the orbit closure OV given by rank A ≤ 1, rank B ≤ 1 and BA = 0. Then OV is Gorenstein. The number of irreducible GL(d)-module summands obtained in the terms F0, F1, F2, F3, F4, F5, F6 are 1, 3, 6, 17, 35, 48, 52, respectively. Take the resolution of both SλR2 and Sµ(W /R1)∗ by the respective Schur complexes. Taking the total complex of their tensor product gives a resolution Tot• of SλR2 ⊗ Sµ(W /R1)∗. Proposition (L. ’20) The complex Tot• is an acyclic resolution of SλR2 ⊗ Sµ(W /R1)∗. One can compute cohomology explicitly by working in the basis of ”standardized” pairs of standard Z2-graded tableau.

Andr´ as Cristian L˝

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Minimal free resolutions of orbit closures of quivers