Trace diagram

In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as graphs with edges labeled by matrices. Without the matrix labels, they are equivalent to Penrose's graphical notation. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley-Hamilton Theorem, have very simple diagrammatic proofs.

Formal definition

Let V be a vector space of dimension n. An n-trace diagram is a directed graph whose edges may be labeled by elements of an n×n matrix group and whose vertices have degree 1 or n. Additionally, an order of edges at each vertex is specified.

If a trace diagram's degree 1 vertices are partitioned into a subset of "inputs" and a subset of "outputs", then it may be identified with a unique multilinear function between tensor powers of the vector space V. The degree 1 vertices correspond to the inputs and outputs of the function, while the degree n vertices correspond to anti-symmetric functions such as the determinant. If there are no degree 1 vertices, the diagram is said to be "closed" and corresponds to a constant. The direct method for computing the function is to decompose the trace diagram into smaller pieces whose functions are known. Alternately, the function may be computed by counting certain kinds of colorings of the graph.

The diagrams may be specialized for particular Lie groups by altering the definition slightly. In this context, they are sometimes called birdtracks, tensor diagrams, or Penrose graphical notation.

Properties of trace diagrams

Let "G" be the group of n×n matrices. If a trace diagram is labeled by "k" different matrices, it may be interpreted as a function from $G^k$ to an algebra of multilinear functions. This function is invariant under simultaneous conjugation, that is, the function corresponding to $(g\_1,ldots,g\_k)$ is the same as the function corresponding to $(a\; g\_1\; a^\{-1\},\; ldots,\; a\; g\_k\; a^\{-1\})$ for any invertible $ain\; G$.

Applications

Trace diagrams have primarily been used by physicists as a tool for studying Lie groups. The most common applications use representation theory to construct spin networks from trace diagrams. In mathematics, they have been used to study character varieties.

See also

*Penrose graphical notation
*multilinear map
*spin network

References

Books:

* Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990
* Group Theory: Birdtracks, Lie's, and Exceptional Groups, Predrag Cvitanović, Princeton University Press, 2008, http://birdtracks.eu/