Trace Diagrams

My Papers and Preprints on Diagrams

  • "Unshackling Linear Algebra from Linear Notation" , submitted. arXiv:0910.1362
  • "On a Diagrammatic Proof of the Cayley-Hamilton Theorem" , submitted. arXiv:0907.2364
  • "Computing SL(2,C) Central Functions with Spin Networks" (with Sean Lawton), to appear in Geometry Dedicatae. arXiv:0903.2372 (see also this mathematica notebook with combinatorial recurrence implementation)
  • "Trace Diagrams, Signed Graph Colorings, and Matrix Minors" (with Steve Morse), Involve 3 (2010), 33-66. arXiv:0903.1373
  • "Spin Networks and SL(2,C) Character Varieties" (with Sean Lawton), Handbook of Teichmuller Theory Volume II (Chapter 16), 2009, Athanase Papadopoulos, ed., EMS Publishing House. arXiv: math.QA/0511271
  • "Trace Diagrams, Representations, and Low-Dimensional Topology" , PhD Thesis at the University of Maryland (advised by Bill Goldman), 2006. PDF

My Talks on Diagrams

  • "Unshackling Linear Algebra from Linear Notation" , Davidson College Math Coffee, Feb. 9, 2010.
  • "Trace Diagrams and their Applications" , Olin College Research Seminar, Feb. 5, 2010.
  • "Trace Diagrams: Unshackling Linear Algebra from “Linear” Notation" , general contributed paper session, JMM, Jan 15, 2010.
  • "Trace Diagrams: Unshackling Linear Algebra from Linear Notation" , Bard College Mathematics Seminar, Nov 12, 2009. PDF Slides and PDF Handout
  • "Unshackling Linear Algebra from Linear Notation" , USMA Math Department Research Seminar, Sep 9, 2009. PDF Slides
  • "Trace Diagram Recurrences and Central Functions of SL(2,C)-Character Varieties" , AMS Special Session on Geometry, Algebra, and Topology of Character Varieties, JMM, Jan 8, 2009. PDF Slides
  • "Signed Graph Coloring, the Art of Linear Algebra, and a Theorem of Jacobi" (with Steve Morse), MathFest, July 31, 2008. PDF slides
  • "The Character Variety's New Clothes" , AMS Session on Algebra and Number Theory, JMM 2008, Jan 7, 2008. PDF slides
  • "The Art of Linear Algebra" , USMA Department of Mathematical Sciences Research Seminar, Dec 5, 2007. PDF slides and PDF handout
  • "Trace Diagrams, Spin Networks, and Spaces of Graphs" , first of two invited lectures at the 7th KAIST Geometric Topology Fair in Gyeongju, Korea, July 10, 2007. PDF slides
  • "Diagrammatic Central Functions" , second of two invited lectures at the 7th KAIST Geometric Topology Fair in Gyeongju, Korea, July 10, 2007. PDF slides
  • "Trace Diagrams, Surfaces, and Character Varieties" , Kansas State University Mathematics Colloquium, April 15, 2007. PDF slides
  • "Trace Diagrams, Surfaces, and Low-Dimensional Topology" , PhD Defense at the University of Maryland, Apr 25, 2006. PDF slides

Trace diagrams are graphs decorated by "just enough" extra structure to give each diagram an interpretation as a multilinear function. The structure allows one to make rigorous arguments about the functions underlying the diagrams using purely graphical techniques. The figure shown at right is a diagrammatic representation of a general cofactor of a matrix, obtained by crossing out a number of rows and columns in a matrix.

There are lots of related ideas out there (e.g. spin networks, birdtracks, Penrose tensor diagarms, skein theory), so the term "trace diagram" is used to distinguish a particular kind of diagram with lots of matrix labels. Diagrammatic notation is extremely powerful and can simplify many classical proofs. Some examples of this capability (for linear algebra) can be found in [1]. Some other applications include:

  • Invariant Theory: Diagrams can be used to describe functions on matrices that are invariant under simultaneous conjugation (think traces of products of matrices). Trace diagrams are very powerful for demonstrating relations among these functions.
  • Geometric Structures: Because invariant theory is vital for understanding character varieties, trace diagrams can be used to describe the moduli space of geometric structures on a surface. (This was the primary direction of my thesis [2].)

Where to Learn about Trace Diagrams

If you're interested in learning about trace diagrams, I recommend starting with the paper [3] or the talk [4], both titled "Unshackling linear algebra from linear notation". Both of these sources are intended to give you a general idea for how the diagrams work. The first chapter of [5] is also good. For a rigorous treatment that includes all the little details, see [1].

1. Steve Morse and Elisha Peterson, Trace diagrams, signed graph colorings, and matrix minors, to appear in Involve.
2. Elisha Peterson, Trace Diagrams, Representations, and Low-Dimensional Topology, PhD Thesis at the University of Maryland, PDF.
3. Elisha Peterson, Unshackling linear algebra from linear notation, available at
4. Elisha Peterson, Unshackling linear algebra from linear notation (talk), PDF.
5. G.E. Stedman, Diagram Techniques in Group Theory, available on (amazon). Stedman is also a physicist. This book overlaps a fair bit with Cvitanovic, but not completely. There is a good first chapter on vector diagrams, and some good stuff on representing Lie group representations diagrammatically.
6. Predrag Cvitanović, Group Theory: Birdtracks, Lie's, and Exceptional Groups, Princeton University Press 2009; available at or on (amazon). This is by far the most thorough reference, but unfortunately is not well-known, especially in the mathematics community. It has some fascinating results for anyone who is interested in the classification of Lie groups. Section 4.9 contains a history of diagrammatic notation which is definitely worth reading.
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