Classification and reduction of Yutsis graphs





Project description




Dries Van Dyck

Veerle Fack

GYutsis Applet

The GYutsis applet generates a summation formula over products of 6-j coefficients for a general angular momenta recoupling coefficient (or 3n-j coefficients). For problems upto 15 j's, corresponding with 6 initial angular momenta, the generated formulas are confirmed to be optimal.

In this applet the generated formula, and thus the number of Wigner 6j-symbols and summations over dummy variables, is only shown when the graph is completely reduced, i.e. equal to a triangular delta.

Defining a problem and generating the summation formula

In the input field, indicated by the label "Braket:", one can enter a general recoupling coefficient in its mathematical standard format. Once a problem is successfully defined the "Reduce"-button becomes active. By pressing the "Reduce"-button a summation formula is generated and presented in the output field labeled "Summation formula:", together with the number of Wigner 6j-symbols and the number of summations over dummy variables in the corresponding fields.

When the applet initially starts, a problem is already presented in the input field, but not entered yet so that the user can edit or delete it in order to define his own problem. In the examples menu, one can select some special cases, which are filled in but not entered yet, allowing the user to edit them. The cases are the following:

  • W9j: the definition of the Wigner 6-j symbol.
  • Cb6: smallest problem for which the heavier heuristics perform better than the Edge Cost Heuristic.
  • C8: problem delivering a cubic cage of girth 8.
  • C9,7: problem for which the Cycle Count Heuristic performs best.
  • C9,12: problem for which the More Smaller/Less Bigger Heuristic performs 3 6-j symbols better than the Cycle Count Heuristic.

All features described above can also be selected from the menus.

Changing the heuristic of the algorithm

When no triangles or bubbles are available in the Yutsis graph, the algorithm used to generate the summation formula delegates the task of selecting an operation to an heuristic. Three heuristics are available:
Edge Cost Heuristic
this is the most simple and fastest heuristic: a cost is associated with each edge, equal to the difference in length of the two smallest cycles in which the edge participates. The cost of a cycle is defined as the minimum cost of its edges. This heuristic will interchange the edge with minimum cost out of the cycle with minimum cost. When two cycles have the same cost, the cycle for which the minimum edge cost is most reached is preferred, and if this is equal, the cycle with minimum total cost is choosen.

For small problems (upto 22 j's) this heuristic suffices, for higher problems the heavier heuristics provide shorter formulae.

More Smaller/Less Bigger Heuristic
this is the default heuristic and for most cases the best choice. This heuristic considers all possible interchanges making a girth cycle smaller. An interchange is preferred over an other if it makes more cycles of length l smaller, or if equal, makes less bigger. This criterium is repeatedly used for rising l starting at the girth until a difference found.
Cycle Count Heuristic
this heuristic also considers all possible interchanges making a girth cycle smaller, but prefers an interchange over an other if it it results in a graph with more cycles of length l. Again this criterium is used for rising l, starting at the girth minus 1, until a difference is found.

This heuristic yields shorter formulae than the Edge Cost Heuristic. For some cases it also performs better than the More Smaller/Less Bigger Heuristic, but these cases are rare.

Changing the format of the formula

The summation formula can be generated in four different output formats:
a compact, human readable format. This is the default.
for the popular typesetting system LaTeX. By default a macro is used to represent the Wigner 6-j symbol. This can be turned of by clicking on the "Use macros" checkbox under the Output-menu. Afterwards the formula will be outputted in plain LaTeX.
for the popular computer algebra package Maple. Macros are used to represent the Kronecker delta symbol, the triangular symbol and the Wigner 6-j symbol (they are in fact Maple functions).
for the Maple package Racah (package especially for Racah algebra).
In the advanced version of this applet default macros can be obtained for the LaTeX and Maple format.

Mathematical Standard Format

The mathematical standard format, e.g.
is the most known format for general recoupling coefficients.

It is not needed to specify the intermediate angular momenta:
is also accepted. In addition the root label can be dropped too:
The intermediate angular momenta/root will get labels of the form t<number>, with <number> starting at 1. Note that in this case it is forbidden to use labels of this form.

GYutsis Applet Advanced

There is also an advanced version of this applet, which gives detailed information about the reduction process, by allowing the user to:
  • execute the algorithm step by step,
  • monitor the applied rules and how they alter the graph,
  • print the graph in its current state,
  • print the girth cycles of the graph,
  • print the cycle that will be chosen in the next step.
In addition default macros can be obtained for the LaTeX and Maple format of the summation formula. This advanced version can be found at