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
 * $$V = \bigoplus_{i=1}^N V_i$$.


 * Each finite-dimensional irreducible representation of $$\mathfrak{g}$$ is uniquely identified (up to isomorphism) by its highest weight
 * $$\forall i \in \{1,\ldots,N\} \,\exists \lambda \in P(\mathfrak{g}) : V_i \simeq M_\lambda$$, 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
 * $$V \simeq \bigoplus_{\lambda \in P(\mathfrak{g})} (\bigoplus_{i=1}^{d_\lambda} M_{\lambda})$$.

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