Vol. 3, No. 1, 2010

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Trace diagrams, signed graph colorings, and matrix minors

Steven Morse and Elisha Peterson

Vol. 3 (2010), No. 1, 33–66
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 standard determinant formulas and a new generalization of the Jacobi determinant theorem.

Keywords
trace diagrams, graph coloring, matrix minors, multilinear algebra, planar algebra, tensor diagrams, determinant, cofactor
Mathematical Subject Classification 2000
Primary: 05C15, 15A69
Secondary: 57M07, 16W22
Milestones
Received: 22 June 2009
Revised: 19 December 2009
Accepted: 27 December 2009
Published: 20 April 2010

Communicated by Kenneth S. Berenhaut
Authors
Steven Morse
United States Army
Fort Campbell, KY 42223
United States
Elisha Peterson
Department of Mathematical Sciences
United States Military Academy
West Point, NY 10996-1905
United States