isotypic component について

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isotypic component ：ウィキペディア英語版

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

\mathfrak

(or of the corresponding Lie group) can be decomposed into irreducible submodules
:

V = \oplus_^N V_i

.
* Each finite-dimensional irreducible representation of

\mathfrak

is uniquely identified (up to isomorphism) by its highest weight
:

\forall i \in \ \exists \lambda \in P(\mathfrak) : 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 \oplus_ ( \oplus_^ M_)

.
This defines the isotypic component of weight

\lambda

of V:

\lambda(V) := \oplus_^ V_i \simeq \mathbb^ \otimes  M_

where

d_\lambda

is maximal.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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