Isotypic component

The isotypic component of weight ${\displaystyle \lambda }$ of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight ${\displaystyle \lambda }$.

Definition

• A finite-dimensional module ${\displaystyle V}$ of a reductive Lie algebra ${\displaystyle {\mathfrak {g}}}$ (or of the corresponding Lie group) can be decomposed into irreducible submodules

${\displaystyle V=\bigoplus _{i=1}^{N}V_{i}}$.

• Each finite-dimensional irreducible representation of ${\displaystyle {\mathfrak {g}}}$ is uniquely identified (up to isomorphism) by its highest weight

${\displaystyle \forall i\in \{1,\ldots ,N\}\,\exists \lambda \in P({\mathfrak {g}}):V_{i}\simeq M_{\lambda }}$, where ${\displaystyle M_{\lambda }}$ denotes the highest weight module with highest weight ${\displaystyle \lambda }$.

• In the decomposition of ${\displaystyle V}$, a certain isomorphism class might appear more than once, hence

${\displaystyle V\simeq \bigoplus _{\lambda \in P({\mathfrak {g}})}(\bigoplus _{i=1}^{d_{\lambda }}M_{\lambda })}$.

This defines the isotypic component of weight ${\displaystyle \lambda }$ of ${\displaystyle V}$: ${\displaystyle \lambda (V):=\bigoplus _{i=1}^{d_{\lambda }}V_{i}\simeq \mathbb {C} ^{d_{\lambda }}\otimes M_{\lambda }}$ where ${\displaystyle d_{\lambda }}$ is maximal.

See also

• Lie algebra representation
• Weight (representation theory)
• Semisimple representation#Isotypic decomposition

References

• 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.