Isotypic component

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

Definition

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

[math]\displaystyle{ V = \bigoplus_{i=1}^N V_i }[/math].

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

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

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

[math]\displaystyle{ V \simeq \bigoplus_{\lambda \in P(\mathfrak{g})} (\bigoplus_{i=1}^{d_\lambda} M_{\lambda}) }[/math].

This defines the isotypic component of weight [math]\displaystyle{ \lambda }[/math] of [math]\displaystyle{ V }[/math]: [math]\displaystyle{ \lambda(V) := \bigoplus_{i=1}^{d_\lambda} V_i \simeq \mathbb{C}^{d_\lambda} \otimes M_{\lambda} }[/math] where [math]\displaystyle{ d_\lambda }[/math] 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.