Tensor Complexes. Christine Berkesch Duke University

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1 Duke University Joint work with Daniel Erman, Manoj Kummini, and Steven V Sam SIAM Conference on Applied Algebraic Geometry Minisymposium on Combinatorial Structures in Algebraic Statistics October 2011

2 Overview Examples 1 Two examples where free resolutions appear in algebraic geometry. 2 Explicit construction of free resolutions from tensors. 3.

3 Free resolutions Let S = k[x 1,..., x N ] with deg(x i ) = 1. A graded free resolution over S is an acyclic complex 0 F : 0 F 0 F 1 F c 0 with F i = j Z S( j)β i,j and all i have degree 0. Further, F is minimal if (image i ) x 1,..., x N F i 1 i. 1 c

4 Example 1: Determinantal varieties S = k[x ij ], M = (x ij ) S p q, p q. I X = (q q)-minors of M S. X = Spec(S/I X ) has closed points {(p q)-matrices of rank < q} k pq. The Eagon Northcott complex resolves S/I X with pdim S (S/I X ) = codim A pq(x) = p q + 1.

5 Example 1: Determinantal varieties S = k[x ij ], M = (x ij ) S p q, p q. I X = (q q)-minors of M S. X = Spec(S/I X ) has closed points {(p q)-matrices of rank < q} k pq. The Eagon Northcott complex resolves S/I X with pdim S (S/I X ) = codim A pq(x) = p q + 1.

6 Example 2: Hyperdeterminantal hypersurfaces Let P 1 P 1 have coordinates ([u 0 : u 1 ], [v 0 : v 1 ]). Choose scalars a ijk to define multilinear forms: f 1 = a 100 u 0 v 0 + a 110 u 1 v 0 + a 101 u 0 v 1 + a 111 u 1 v 1, f 2 = a 200 u 0 v 0 + a 210 u 1 v 0 + a 201 u 0 v 1 + a 211 u 1 v 1, f 3 = a 300 u 0 v 0 + a 310 u 1 v 0 + a 301 u 0 v 1 + a 311 u 1 v 1. V = Var(f 1, f 2, f 3 ) P 1 P 1. Let be a hyperdeterminant [Cayley, GKZ]. Fact: V if and only if (a ijk ) Var( ).

7 Example 2: Hyperdeterminantal hypersurfaces Let P 1 P 1 have coordinates ([u 0 : u 1 ], [v 0 : v 1 ]). Choose scalars a ijk to define multilinear forms: f 1 = a 100 u 0 v 0 + a 110 u 1 v 0 + a 101 u 0 v 1 + a 111 u 1 v 1, f 2 = a 200 u 0 v 0 + a 210 u 1 v 0 + a 201 u 0 v 1 + a 211 u 1 v 1, f 3 = a 300 u 0 v 0 + a 310 u 1 v 0 + a 301 u 0 v 1 + a 311 u 1 v 1. V = Var(f 1, f 2, f 3 ) P 1 P 1. Let be a hyperdeterminant [Cayley, GKZ]. Fact: V if and only if (a ijk ) Var( ).

8 Hyperdeterminants from free resolutions Set S = k[x ijk ] with i {1, 2, 3} and j, k {0, 1}. Consider the complex F : S 6 S 6 ( 1) 0, with x x x x 101 x 100 x 201 x 200 x 301 x 300 = 0 x x x 301 x x x x 111 x 110 x 211 x 210 x 311 x x x x 311 Then det( ) = S Thus V if and only if F S x ijk a ijk does not give an isomorphism of k-vector spaces.

9 The framework The underlying tensors in Ex. 1: S = k[x ij ], M = (x ij ) = φ for φ = (x ij ) S p (S q ). Ex. 2: S = k[x ijk ], φ = (x ijk ) S 3 (S 2 ) (S 2 ). Fix a, b 1,..., b n N. S = Z[x i,j ] with 1 i a, J = (j 1,..., j n ) with 1 j k b k. Tensor: φ = (x i,j ) S a (S b 1) (S bn ) Construction of a tensor complex: (tensor φ, weight vector w) free resolution F(φ, w)

10 The framework The underlying tensors in Ex. 1: S = k[x ij ], M = (x ij ) = φ for φ = (x ij ) S p (S q ). Ex. 2: S = k[x ijk ], φ = (x ijk ) S 3 (S 2 ) (S 2 ). Fix a, b 1,..., b n N. S = Z[x i,j ] with 1 i a, J = (j 1,..., j n ) with 1 j k b k. Tensor: φ = (x i,j ) S a (S b 1) (S bn ) Construction of a tensor complex: (tensor φ, weight vector w) free resolution F(φ, w)

11 Main properties Theorem [B. Erman Kummini Sam]: 1 F(φ, w) is a graded minimal pure free resolution of a CM module M(φ, w). 2 F(φ, w) is uniformly minimal over Z. 3 F(φ, w) is G = GL a GL b1 GL bn -equivariant. 4 There is an explicit presentation of F(φ, w). provide a unifying view of: Koszul complexes, Buchsbaum Eisenbud matrix complexes, including the Eagon Northcott and Buchsbaum Rim complexes, Gelfand Kapranov Zelevinsky discriminantal complexes, Eisenbud Schreyer pure free resolutions.

12 Main properties Theorem [B. Erman Kummini Sam]: 1 F(φ, w) is a graded minimal pure free resolution of a CM module M(φ, w). 2 F(φ, w) is uniformly minimal over Z. 3 F(φ, w) is G = GL a GL b1 GL bn -equivariant. 4 There is an explicit presentation of F(φ, w). provide a unifying view of: Koszul complexes, Buchsbaum Eisenbud matrix complexes, including the Eagon Northcott and Buchsbaum Rim complexes, Gelfand Kapranov Zelevinsky discriminantal complexes, Eisenbud Schreyer pure free resolutions.

13 The geometric technique of Kempf Lascoux Weyman Note that A a b 1 b n = Spec(S). π : A a b 1 b n P(Z b 1) P(Z bn ) A a b 1 b n. K(φ) : a Koszul complex of multilinear forms. Tensor with O(w) and apply Rπ to obtain F(φ, w). This implies acyclicity and G-equivariance of F(φ, w).

14 Explicit differentials: the benefits of symmetry The G-action on F(φ, w) allows us to understand its differentials. After fixing distinguished bases in F (φ, w) i i 1 F(φ, w)i, we explicitly obtain the matrix i (unique up to sign). These entries are given by minors of the flattening ( φ S a (S b 1 ) (S b k ) ).

15 (4 2 2)-example Let A = Z 4, B 1 = B 2 = Z 2, and S = Z[x i,j ]. Let φ = (x i,j ) S (A B1 B 2 ) and w = (0, 0, 2). F (φ, w) : S 0 A Sym 0 B 1 Sym 2 B 2 1 S( 2) 2 A Div 0 B 1 2 B 1 Sym 0 B 2 S( 4) 4 A 2 Div 2 B1 2 B1 0 Div 0 B2 2 B2 3 Betti diagram: β(f(φ, w) ) = 6. 3 Fixing bases, 1 and 2 can be written in terms of a distinguished flattening of φ.

16 (4 2 2)-example φ : A B 1 B 2 Then as a x 1,(1,1) x 2,(1,1) x 3,(1,1) x 4,(1,1) b x 1,(1,2) x 2,(1,2) x 3,(1,2) x 4,(1,2) c x 1,(2,1) x 2,(2,1) x 3,(2,1) x 4,(2,1). d x 1,(2,2) x 2,(2,2) x 3,(2,2) x 4,(2,2) T 1 = g,,(2,0) g,,(1,1) g,,(0,2) f {1,2},{1,2}, ac;12 ad;12 + bc;12 bd;12 f {1,3},{1,2}, ac;13 ad;13 + bc;13 bd;13 f {1,4},{1,2}, ac;14 ad;14 + bc;14 bd;14. f {2,3},{1,2}, ac;23 ad;23 + bc;23 bd;23 f {2,4},{1,2}, ac;24 ad;24 + bc;24 bd;24 f {3,4},{1,2}, ac;34 ad;34 + bc;34 bd;34

17 The support of M(φ, w) Let Y (φ) denote the support of the module M(φ, w). Y (φ) is independent of w. Y (φ) is an integral subvariety of A a b 1 b n of codimension a i (b i 1). Y (φ) is G-equivariant. Theorem [BEKS]: If a = 1 + i (b i 1), then Y (φ) is set-theoretically defined by hyperdeterminant of φ φ is (a b 1 b n )-subtensor of φ. Y (φ) is a hyperdeterminantal variety.

18 The support of M(φ, w) Let Y (φ) denote the support of the module M(φ, w). Y (φ) is independent of w. Y (φ) is an integral subvariety of A a b 1 b n of codimension a i (b i 1). Y (φ) is G-equivariant. Theorem [BEKS]: If a = 1 + i (b i 1), then Y (φ) is set-theoretically defined by hyperdeterminant of φ φ is (a b 1 b n )-subtensor of φ. Y (φ) is a hyperdeterminantal variety.

19 Ties to algebraic statistics In recent work, Sturmfels and Zwiernik express hyperdeterminants in terms of cumulants. The hypersurface cut out by such an equation is a context-specific independence (CSI) split model for binary random variables. Questions: Does Y (φ) have a statistical interpretation? Can statistical tools be applied to understand the defining equations of Y (φ)?