Editing Compact group (section)

===Representation theory of ''K''===
{{See also|Lie algebra representation#Classifying finite-dimensional representations of Lie algebras}}
[[File:A2example.pdf|thumb|Example of the weights of a representation of the group SU(3)]]
[[Image:meson octet.png|thumb|The "[[Eightfold Way (physics)|eightfold way]]" representation of SU(3), as used in particle physics]]
[[File:Weights_for_A2_root_system.png|thumb|right|Black dots indicate the dominant integral elements for the group SU(3)]]
We now let <math>\Sigma</math> denote a finite-dimensional irreducible representation of ''K'' (over <math>\mathbb{C}</math>). We then consider the restriction of <math>\Sigma</math> to ''T''. This restriction is not irreducible unless <math>\Sigma</math> is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of ''T''. (Note that a given irreducible representation of ''T'' may occur more than once.) Now, each irreducible representation of ''T'' is described by a linear functional <math>\lambda</math> as in the preceding subsection. If a given <math>\lambda</math> occurs at least once in the decomposition of the restriction of <math>\Sigma</math> to ''T'', we call <math>\lambda</math> a '''weight''' of <math>\Sigma</math>. The strategy of the representation theory of ''K'' is to classify the irreducible representations in terms of their weights.

We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on [[Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras|weights in representation theory]]. We need the notion of a '''root system''' for ''K'' (relative to a given maximal torus ''T''). The construction of this root system <math>R\subset \mathfrak{t}</math> is very similar to the [[Semisimple Lie algebra#Cartan subalgebras and root systems|construction for complex semisimple Lie algebras]]. Specifically, the weights are the nonzero weights for the adjoint action of ''T'' on the complexified Lie algebra of ''K''. The root system ''R'' has all the usual properties of a [[root system]], except that the elements of ''R'' may not span <math>\mathfrak{t}</math>.<ref>{{harvnb|Hall|2015}} Section 11.7</ref> We then choose a base <math>\Delta</math> for ''R'' and we say that an integral element <math>\lambda</math> is '''dominant''' if <math>\lambda(\alpha)\ge 0</math> for all <math>\alpha\in\Delta</math>. Finally, we say that one weight is '''higher''' than another if their difference can be expressed as a linear combination of elements of <math>\Delta</math> with non-negative coefficients.

The irreducible finite-dimensional representations of ''K'' are then classified by a '''theorem of the highest [[Weight (representation theory)|weight]]''',<ref>{{harvnb|Hall|2015}} Chapter 12</ref> which is closely related to the analogous theorem classifying [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|representations of a semisimple Lie algebra]]. The result says that: 
# every irreducible representation has highest weight, 
# the highest weight is always a dominant, analytically integral element, 
# two irreducible representations with the same highest weight are isomorphic, and 
# every dominant, analytically integral element arises as the highest weight of an irreducible representation.

The theorem of the highest weight for representations of ''K'' is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an [[Weight (representation theory)#Integral weight|integral element]] is different. The weights <math>\lambda</math> of a representation <math>\Sigma</math> are analytically integral in the sense described in the previous subsection. Every analytically integral element is [[Weight (representation theory)#Integral weight|integral]] in the Lie algebra sense, but not the other way around.<ref>{{harvnb|Hall|2015}} Section 12.2</ref> (This phenomenon reflects that, in general, [[Lie group–Lie algebra correspondence#Lie group representations|not every representation]] of the Lie algebra <math>\mathfrak{k}</math> comes from a representation of the group ''K''.) On the other hand, if ''K'' is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.<ref>{{harvnb|Hall|2015}} Corollary 13.20</ref>