Extended solutions of the harmonic map equation in the special unitary group (Q2922441)

Harmonic maps from a Riemann surface \(M\) into a Lie group \(G\) correspond by integration to certain holomorphic maps (the extended solutions) into the group \(\Omega G\) of based smooth loops in \(G\). If the Fourier series associated to an extended solution has finitely many terms, it is said that this solution and the corresponding harmonic map have ``finite uniton number''. For example, all harmonic maps from the two-sphere \(M=S^2\) have finite uniton number.NEWLINENEWLINEIn this paper all harmonic maps with finite uniton number from \(M\) into the special unitary group \(\mathrm {SU} (n)\) and the corresponding inner symmetric spaces (complex Grassmannians) are classified in terms of certain pieces of the Bruhat decomposition of \(\Omega_{\mathrm{alg}} \mathrm{SU} (n)\) (algebraic loop space).