Symmetry, Representations, and Invariants pp 127-174 | Cite as

# Highest-Weight Theory

## Abstract

In this chapter we study the regular representations of a classical group *G* by the same method used for the adjoint representation. When *G* is a connected classical group, an irreducible regular *G*-module decomposes into a direct sum of weight spaces relative to the action of a maximal torus of *G*. The *theorem of the highest weight* asserts that among the weights that occur in the decomposition, there is a unique *maximal* element, relative to a partial order coming from a choice of positive roots for *G*. We prove that every dominant integral weight of a semisimple Lie algebra g is the highest weight of an irreducible finite-dimensional representation of g. When g is the Lie algebra of a classical group *G*, the corresponding regular representations of *G* are constructed in Chapters 5 and 6 and studied in greater detail in Chapters 9 and 10. A crucial property of a classical group is the *complete reducibility* of its regular representations. We give two (independent) proofs of this: one algebraic using the *Casimir operator*, and one analytic using Weyl’s *unitarian trick*.

## Keywords

Irreducible Representation Classical Group Weyl Group Simple Root Maximal Torus## Preview

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