# Kleiss-Kuijf relations The Kleiss-Kuijf relations are linear relations amongst the most naive sum over [[0354 Colour decomposition|colour-ordered amplitudes]] in the full amplitude of [[0071 Yang-Mills|Yang-Mills theory]]. One way to write them is: $A_{n}[1,\{\alpha\}, n,\{\beta\}]=(-1)^{|\beta|} \sum_{\sigma \in \mathrm{OP}\left(\{\alpha\},\left\{\beta^{T}\right\}\right)} A_{n}[1, \sigma, n],$where OP denotes ordered permutations, meaning that the ordering within $\{\alpha\}$ and $\{\beta^T\}$ are preserved in the union, e.g., $\{\alpha\} \cup\left\{\beta^{T}\right\}=\{2\} \cup\{4,3\}$ then $\sigma=\{243\},\{423\},\{432\}$. The relations are inherited from the gauge structure, and reduce the number of linearly independent terms in the colour decomposition from $(n-1)!$ to $(n-2)!$. ## Refs - originals - conjectured in [[KleissKuijf1989]] - proven in [[1999#Del Duca, Dixon, Maltoni]] - rewriting in the [[Rsc0003 ElvangHuang Scattering amplitudes]] form - [[BernCarrascoJohansson2008]] - review - [[Rsc0003 ElvangHuang Scattering amplitudes]]