Isotypic component

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

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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

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

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

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

## References

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* 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.