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Isotypic component

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It has been suggested that this article be merged into Semisimple representation#Isotypic decomposition. (Discuss) Proposed since July 2024.

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

Definition

V = ⨁ i = 1 N V i {\displaystyle V=\bigoplus _{i=1}^{N}V_{i}} .
  • Each finite-dimensional irreducible representation of g {\displaystyle {\mathfrak {g}}} is uniquely identified (up to isomorphism) by its highest weight
∀ i ∈ { 1 , … , N } ∃ λ ∈ P ( g ) : V i ≃ M λ {\displaystyle \forall i\in \{1,\ldots ,N\}\,\exists \lambda \in P({\mathfrak {g}}):V_{i}\simeq M_{\lambda }} , where M λ {\displaystyle M_{\lambda }} denotes the highest weight module with highest weight λ {\displaystyle \lambda } .
  • In the decomposition of V {\displaystyle V} , a certain isomorphism class might appear more than once, hence
V ≃ ⨁ λ ∈ P ( g ) ( ⨁ i = 1 d λ M λ ) {\displaystyle V\simeq \bigoplus _{\lambda \in P({\mathfrak {g}})}(\bigoplus _{i=1}^{d_{\lambda }}M_{\lambda })} .

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

See also

References

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