On traces of tensor representations of diagrams

TL;DR

This paper characterizes functions on diagrams that are traces of tensor representations, providing a unique correspondence with strongly nondegenerate tensor representations, applicable to various diagram types like knots and group representations.

Contribution

It offers a complete characterization of trace functions on tensor diagram representations and establishes their unique correspondence with strongly nondegenerate tensor representations.

Findings

Characterizes which functions are traces of tensor representations.

Proves each trace corresponds uniquely to a strongly nondegenerate tensor representation.

Applicable to virtual knots, chord diagrams, and group representations.

Abstract

Let $T$ be a set, of {\em types}, and let $\iota,o:T\to\oZ_+$. A {\em $T$-diagram} is a locally ordered directed graph $G$ equipped with a function $\tau:V(G)\to T$ such that each vertex $v$ of $G$ has indegree $\iota(\tau(v))$ and outdegree $o(\tau(v))$. (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 $t\in T$ a tensor $R(t)\in V^{*\otimes \iota(t)}\otimes V^{\otimes o(t)}$ is called a {\em tensor representation} of $T$. The {\em trace} (or {\em partition function}) of $R$ is the $\oF$-valued function $p_R$ on the collection of $T$-diagrams obtained by `decorating' each vertex $v$ of a $T$-diagram $G$ 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 {\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.