T -equivariant tensor rank varieties and their K-theory classes 2014 July 18 Advisor: Professor Anders Buch, Department of Mathematics, Rutgers University
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
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.
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.
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.
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.
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 ).
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.
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.
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.
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.
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. 2 2 2 or 2 2 3 matrices) and specific values of k.
Progress 1. Used the software package Macaulay2 to compute ideals and Grothendieck classes of some simple cases: 2 2 2 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 3 3 3 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
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
End; acknowledgements Thank you for listening. Thanks to Professor Anders Buch for sponsoring this project.