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Harmonic maps of finite uniton number and their canonical elements

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group $Ω_\mathrm{alg}{G}$ of algebraic loops in $G$ and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low dimensional spin groups making use of spinor representations.