Vol. 3, No. 1, 2010

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 3, 365–546
Issue 2, 183–364
Issue 1, 1–182

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
 
Other MSP Journals
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