# Graphical su(3) tensor product reduction scheme

## Input Values for Multiplets

Introduction

In order to describe physical systems in terms of irreducible Lie algebra representations, it is convenient to reduce tensor products into direct sums of irreducible representations. As opposed to analytic reduction methods, which are quite tedious to calculate by hand, the graphical method for rank 2 algebras devised by J. P. Antoine and D. Speiser in 1964 is more practical. Two representations are coupled by superposition of the center of one weight diagram with the other representation in the "landscape of irreducible representations". Summation over the multiplicities of the states, taking the sign of the various sectors of the landscape into account, yields the reduction into the direct sum of irreducible representations. Using this algorithm, one can easily calculate the reduction of higher-dimensional tensor products.
More detailed information on this method can be found here and the corresponding bachelor thesis is uploaded here.

Input Values

The representations to be coupled are each uniquely specified by two values (p,q), which correspond to the sidelengths of their weight diagram. The algorithm is optimized such that the input (p1,q1,p2,q2) is rearranged in order to always superimpose the smaller weight diagram with the landscape. The landscape itself is dynamically generated based on the input in order to further minimize computational cost. Nevertheless, the calculations increase in complexity for higher dimensional representations, resulting in longer computation times.
The result of the tensor product reduction is given below. The superimposed weight diagram (blue) and the landscape (black) are depicted further down and can be fully explored by using the scroll bars.

This program was designed to calculate any more difficult su(3) tensor product reductions the reader might encounter in the future and hopefully, it has sparked interest in this graphical method. The design of this application was kindly provided by my predecessor Patrick BÃ¼hlmann, whose so(5) applet can be found here.

Greetings, Andrin Kessler