T -equivariant tensor rank varieties and their K-theory classes

Transcription

1 T -equivariant tensor rank varieties and their K-theory classes 2014 July 18 Advisor: Professor Anders Buch, Department of Mathematics, Rutgers University

2 Overview 1 Equivariant K-theory Overview 2 Determinantal varieties Definitions Example; Thom-Porteous formula 3 Generalization: multidimensional matrices 4 Geometry of tensors Rank of a tensor Border rank 5 Known results and progress Known results 6 Progress 7 End; acknowledgements

3 Overview Initial definitions Let C denote the field of complex numbers, C = C \ {0}, T = (C ) n (complex n-torus group), and V a finite-dimensional vector space. Definition: (T -stable variety) An affine (resp. projective) variety X is T -stable if it is invariant under the action of T on the ambient affine (resp. projective) space; i.e. for all x X and t T, t.x X. Definition: (coordinate ring) Let I be the ideal of an irreducible affine variety X V. The coordinate ring of X is the quotient ring O X = C[z 1,..., z k ]/I, and is identified with the ring of polynomial functions on X. The coordinate of V is O V = C[z 1,..., z k ], where k = dim V.

4 Overview Equivariant K-theoretic classes Every T -stable variety X corresponds to a unique K-theoretic class (aka Grothendieck class) [O X ] in Rep(T ), the ring of T -representations modulo T -equivariant short exact sequences, via the formula [O X ] i ( 1) i [F i /MF i ] Rep(T ), where (F i ) is a finite graded resolution of O X by projective O V -modules and M is the ideal of 0, M = I (0) < O V. Note: Rep(T ) = Z[u 1 ±1,..., u± n 1], the ring of integer Laurent polynomials.

5 Overview Why study equivariant K-theory classes? Equivariant K-theory classes capture the geometry of intersections and unions of varieties: 1. When Y and Z intersect transversally, this reflects algebraically via the formula [O Y Z ] = [O Y ] [O Z ]. 2. Like many Euler-characteristic type invariants, K-theory classes have an inclusion-exclusion principle under suitable conditions: [O Y Z ] = [O Y ] + [O Z ] [O Y Z ]. 3. To a weak extent, we can go backwards in the above statements. This makes equivariant K-theory a valuable tool in intersection theory.

6 Definitions Definitions Let W denote the space of complex m n matrices, and let r min(m, n). Definition: (determinantal variety) The set of m n matrices of rank at most r is the determinantal variety Ω r. Determinantal varieties are determined by the vanishing of the (r + 1) (r + 1) minors of m n matrices, so they are projective varieties. Problem statement: Determine the Grothendieck class of Ω r.

7 Example; Thom-Porteous formula Example; Thom-Porteous formula Example: (W = 2 2 matrices, r = 1) Ω 1 is the set of 2 2 matrices whose 2 2 minors vanish, i.e. the set of 2 2 matrices with determinant 0. The coordinate ring of this variety is O W / ad bc, and the corresponding Grothendieck class is given by the polynomial 1 u 1 u 2 v1 1 v 2 1. In general, it is known that Grothendieck classes of determinantal varieties are given by the Thom-Porteous formula: [O r ] = G (e r) (f r)(f E) Rep(T ).

8 Problem statement In this project, we seek to generalize these notions and results to higher-dimensional matrices. Problem statement: 1. Formulate notions of the rank of a multidimensional matrix, in a way that respects a natural action of the general linear group (i.e. is invariant under generalized row/column operations ). 2. Define interesting T -stable varieties using these notions of rank (generalizing the notion of determinantal variety), compute their ideals, and compute their Grothendieck classes.

9 Tensor products The space of matrices of dimensions k 0 k 1 k p can be identified with the tensor product V 0 V 1 V p, dim V i = k i. It is also identified with various spaces of linear maps by the general and canonical isomorphism V W = Hom(V, W ). This space has basis e i0 e i1 e ip, which can be identified with the matrix with 1 in the (i 0, i 1,..., i p ) position and 0 elsewhere. There are numerous ways to define the rank of a tensor. The idea of the decomposition rank of a tensor is classical and well-studied, so it is natural to use this to define the rank of a multidimensional matrix.

10 Rank of a tensor Rank of a tensor Definition: (decomposable) Let α V 0 V p. α is said to be decomposable, or simple, if α = v 0 v p for some collection of vectors v i V i, i = 0,..., p. Every tensor is a finite linear combination of decomposable tensors. Definition: (rank) An element β V 0 V p is said to be of rank r if it can be expressed as a sum of r simple tensors, and β cannot be expressed as a sum of fewer than r simple tensors.

11 Border rank Rank of a tensor, ctd. Definition: (border rank) Let σ k denote the Zariski closure of the set {β V 0 V p : rank(β) k}. The border rank of γ V 0 V p is the minimal k such that γ σ k. Problem statement: (border rank) Understand the ideal of σ k and compute its Grothendieck class.

12 Known results Known results For 2-dimensional matrices the theory is well understood. The ideal of σ k is a classical open problem, and is not solved in general for 3-dimensional and higher matrices. For k = 2 σ k can be defined by the vanishing of minors of certain flattened matrices. Geometrically, σ k are k-secant varieties of Segre embeddings of products of projective spaces. σ k is well understood in certain small cases (e.g or matrices) and specific values of k.

13 Progress 1. Used the software package Macaulay2 to compute ideals and Grothendieck classes of some simple cases: and 2 2 3, ranks 1 and 2 2. Also attempted to compute larger cases; time and memory constraints urge other approaches a. Macaulay2 computations rely on Buchberger s algorithm to compute Gröbner bases, which has doubly exponential time complexity b. Inputs increase quickly with size and dimension of matrices; the ideal of matrices of rank at most 1 contains 321 polynomials 3. Conjectured some reductions for varieties of higher border rank 4. Formulated some other notions of rank and demonstrated equivariance and GL(n)-invariance

14 Further work 1. Prove conjectures regarding reductions and use them to compute specific Grothendieck classes via Macaulay2 2. Obtain a multidimensional analogue to the classical Thom-Porteous formula

15 End; acknowledgements Thank you for listening. Thanks to Professor Anders Buch for sponsoring this project.