Trace diagrams, signed graph colorings, and matrix minors

Morse, Steven; Peterson, Elisha

doi:10.2140/involve.2010.3.33

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Trace diagrams, signed graph colorings, and matrix minors

Steven Morse and Elisha Peterson

Vol. 3 (2010), No. 1, 33–66

DOI: 10.2140/involve.2010.3.33

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

Vol. 3, No. 1, 2010