2015/01/20 by Alexander Schrijver, Schrijver, Alexander
Computer Science · Mathematics · #05C20 #14L24 #15A72 #81T #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1501.04945
openalex publication_date 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let T be a set, of \em types, and let ι,o:T→\oZ+. A \em T-diagram is a locally ordered directed graph G equipped with a function τ:V(G)→ T such that each vertex v of G has indegree ι(τ(v)) and outdegree o(τ(v)). (A directed graph is \em 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 finite-dimensional \oF-linear space, where \oF is an algebraically closed field of characteristic 0. A function R on T assigning to each t∈ T a tensor R(t)∈ V*⊗ ι(t)⊗ V⊗ o(t) is called a \em tensor representation of T. The \em trace (or \em partition function) of R is the \oF-valued function pR on the collection of T-diagrams obtained by `decorating' each vertex v of a T-diagram G with the tensor R(τ(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 \em tensor network. We characterize which functions on T-diagrams 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.