 Author
 Year
 2015
 Title
 On traces of tensor representations of diagrams
 Journal
 Linear Algebra and its Applications
 Volume
 476
 Pages (fromto)
 2841
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

Let T be an (abstract) set of types, and let (unknown symbol), o : T > Z(+). A Tdiagram is a locally ordered directed graph G equipped with a function tau : V (G) > T such that each vertex v of G has indegree (unknown symbol)(tau(v)) and outdegree o(tau(v)). (A directed graph is locally ordered if at each vertex v, linear orders of the edges entering v and of the edges leaving v are specified.)
Let V be a finitedimensional Flinear space, where F is an algebraically closed field of characteristic 0. A function R on T assigning to each t is an element of T a tensor R(t) is an element of V*(circle times l(t)) circle times Vcircle times o(t) is called a tensor representation of T. The trace (or partition function) of R is the Fvalued function pR on the collection of Tdiagrams obtained by 'decorating' each vertex v of a Tdiagram G with the tensor R(tau(v)), and contracting tensors along each edge of G, while respecting the order of the edges entering v and leaving v. In this way we obtain a tensor network.
We characterize which functions on Tdiagrams are traces, and show that each trace comes from a unique 'strongly nondegenerate' tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations.
 URL
 go to publisher's site
 Language
 English
 Permalink
 http://hdl.handle.net/11245/1.483458
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