= Weight decomposition
{title2=$V=\bigoplus_\lambda V_\lambda$}
For a representation on which a commuting torus or Cartan action is semisimple, the weight decomposition is the direct sum of simultaneous <weight spaces>. A finite-dimensional representation of a <compact> <torus> can be made unitary by <Haar measure> averaging; its commuting unitary matrices are simultaneously diagonalizable. Each diagonal character is an integral <weight of a representation>. The decomposition makes tensor and exterior-power weights additive. For nonsemisimple actions, simultaneous generalized eigenspaces give a generalized weight decomposition instead.
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