Trace diagrams, signed graph colorings, and matrix minors

Abstract

Trace diagrams are structured graphs with edges labeled by matrices. Each diagram has an interpretation as a particular multilinear function. We provide a rigorous combinatorial definition of these diagrams using a notion of signed graph coloring, and prove that they may be efficiently represented in terms of matrix minors. Using this viewpoint, we provide new proofs of several stan-dard determinant formulas and a new generalization of the Jacobi determinant theorem.