Isotypic component

The isotypic component of weight of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight .

Definition

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  • Each finite-dimensional irreducible representation of is uniquely identified (up to isomorphism) by its highest weight
, where denotes the highest weight module with highest weight .
  • In the decomposition of , a certain isomorphism class might appear more than once, hence
.

This defines the isotypic component of weight of : where is maximal.

See also

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References

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  • Bürgisser, Peter; Matthias Christandl; Christian Ikenmeyer (2011-02-15). "Even partitions in plethysms". Journal of Algebra. 328 (1): 322–329. arXiv:1003.4474. doi:10.1016/j.jalgebra.2010.10.031. ISSN 0021-8693.