Isotypic component

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

Definition

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

$V=\underset{i=1}{\overset{N}{⨁}}{V}_i$.

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

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

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

$V\simeq \underset{\lambda \in P(\mathfrak{𝔤})}{⨁}(\underset{i=1}{\overset{d_{\lambda }}{⨁}}{M}_{\lambda })$.

This defines the isotypic component of weight $\lambda$ of $V$: $\lambda (V):=\underset{i=1}{\overset{d_{\lambda }}{⨁}}{V}_i\simeq {\mathbb{ℂ}}^{d_{\lambda }}\otimes M_{\lambda }$ where $d_{\lambda }$ is maximal.

See also

• Lie algebra representation
• Weight (representation theory)
• Semisimple representation

References

• Bürgisser, Peter; Matthias Christandl; Christian Ikenmeyer (2011-02-15). "Even partitions in plethysms". Journal of Algebra 328 (1): 322–329. 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. doi:10.1007/s00220-005-1330-9. Bibcode: 2005CMaPh.257..725H.