Isotypic component
It has been suggested that this article be merged into Semisimple representation#Isotypic decomposition. (Discuss) Proposed since July 2024. |
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
[edit]- 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 : where is maximal.
See also
[edit]- Lie algebra representation
- Weight (representation theory)
- Semisimple representation#Isotypic decomposition
References
[edit]- 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.
- Heinzner, P.; A. Huckleberry; M. R Zirnbauer (2005). "Symmetry classes of disordered fermions". Communications in Mathematical Physics. 257 (3): 725–771. arXiv:math-ph/0411040. Bibcode:2005CMaPh.257..725H. doi:10.1007/s00220-005-1330-9.