Isotypic component explained

λ of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight λ .

Definition

V of a reductive Lie algebra ak{g} (or of the corresponding Lie group) can be decomposed into irreducible submodulesV = o plus ^{N} _{i =1} V _{i} .

• Each finite-dimensional irreducible representation of

ak{g} is uniquely identified (up to isomorphism) by its highest weight\forall i \in \{ 1 ,\ldots,N\} \exists λ \in P( ak{g}) : V _{i} \simeq M _{λ} , where M _{λ} denotes the highest weight module with highest weight λ .

• In the decomposition of

V , a certain isomorphism class might appear more than once, henceV \simeq o plus _{λ} )} ( o plus ^{d _{λ}} _{i =1} M _{λ} ) .This defines the isotypic component of weight λ of V :λ (V) : = o plus ^{d _{λ}} _{i =1} V _{i} \simeq C ^{d _{λ}} ⊗ M _{λ} where d _{λ} is maximal.

See also

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

References

• 10.1016/j.jalgebra.2010.10.031. 0021-8693. 328. 1. 322–329. Bürgisser. Peter. Matthias Christandl . Christian Ikenmeyer . Even partitions in plethysms. Journal of Algebra. 2011-02-15. 1003.4474.
• 10.1007/s00220-005-1330-9. Heinzner. P.. A. Huckleberry . M. R Zirnbauer . Symmetry classes of disordered fermions. Communications in Mathematical Physics. 257. 3. 725–771. 2005. math-ph/0411040. 2005CMaPh.257..725H.