• 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.

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

• 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.