MSc thesis defense by Emil Hoffmann Kozuch

Tree level colour factors for simple lie groups

Abstract: In this paper two methods for how to calculate contractions of Lie algebra structure constants, for simple compact Lie groups, are derived.

The two methods are both based on the pictorial representation of group elements called \textit{Birdtracks}\cite{Bird-track-book}. The easiest algebras to describe are the four families $A_n$, $B_n$, $C_n$ and $D_n$ plus $g_2$, these algebras are described by the first method which decomposes the spaces $V\otimes V$ or $V\otimes \overline{V}$ into irreducible subspaces described by their primitive invariants. This method however is not able to decompose the algebras $f_4$, $e_6$, $e_7$ and $e_8$ into solvable diagrams. These algebras are instead described using the second method, where the adjoint representation is decomposed into subgroups of the more simple algebras, which allows one to calculate contraction of structure constants. Lastly a method for how to reformulate the colour factors for \textit{tree}-level scattering amplitudes in terms of irreducible representations of the symmetric group enables us to easily compare the result of colour calculations for specific scattering calculations between different algebras.