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
- A finite-dimensional module of a reductive Lie algebra (or of the corresponding Lie group) can be decomposed into irreducible submodules
- .
- 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 V: where is maximal.
See also
References
- Bürgisser, Peter; Matthias Christandl; Christian Ikenmeyer (2011-02-15). "Even partitions in plethysms". Journal of Algebra. 328 (1): 322–329. ISSN 0021-8693. doi:10.1016/j.jalgebra.2010.10.031. Retrieved 2011-03-26.
- Heinzner, P.; A. Huckleberry; M. R Zirnbauer (2004-11-10). "Symmetry classes of disordered fermions". arXiv:math-ph/0411040 . doi:10.1007/s00220-005-1330-9.
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