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=\oplus _{{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 \oplus _{{\lambda \in P({\mathfrak {g}})}}(\oplus _{{i=1}}^{{d_{\lambda }}}M_{{\lambda }})

.

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

References

Bürgisser, Peter; Matthias Christandl; Christian Ikenmeyer (2011-02-15). "Even partitions in plethysms". Journal of Algebra. 328 (1): 322–329. doi:10.1016/j.jalgebra.2010.10.031. ISSN 0021-8693. Retrieved 2011-03-26.

Heinzner, P.; A. Huckleberry; M. R Zirnbauer (2004-11-10). "Symmetry classes of disordered fermions". arXiv:math-ph/0411040. Bibcode:2005CMaPh.257..725H. doi:10.1007/s00220-005-1330-9.

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Isotypic component

Definition

See also

References