Editing Compact group (section)

==Representation theory of a connected compact Lie group==
Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the [[Representation of a Lie group#An example: The rotation group SO.283.29|rotation group SO(3)]], the [[Representation theory of SU(2)|special unitary group SU(2)]], and the [[Clebsch–Gordan coefficients for SU(3)#Representations of the SU.283.29 group|special unitary group SU(3)]]. We focus here on the general theory. See also the parallel theory of [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|representations of a semisimple Lie algebra]].

Throughout this section, we fix a connected compact Lie group ''K'' and a [[maximal torus]] ''T'' in ''K''.

===Representation theory of ''T''===
Since ''T'' is commutative, [[Schur's lemma]] tells us that each irreducible representation <math>\rho</math> of ''T'' is one-dimensional:
:<math>\rho:T\rightarrow GL(1;\mathbb{C})=\mathbb{C}^* .</math>
Since, also, ''T'' is compact, <math>\rho</math> must actually map into <math>S^1\subset\mathbb{C}</math>.

To describe these representations concretely, we let <math>\mathfrak{t}</math> be the Lie algebra of ''T'' and we write points <math>h\in T</math> as
:<math>h=e^{H},\quad H\in\mathfrak{t} .</math>
In such coordinates, <math>\rho</math> will have the form
:<math>\rho(e^{H})=e^{i \lambda(H)}</math>
for some linear functional <math>\lambda</math> on <math>\mathfrak{t}</math>.

Now, since the exponential map <math>H\mapsto e^{H}</math> is not injective, not every such linear functional <math>\lambda</math> gives rise to a well-defined map of ''T'' into <math>S^1</math>. Rather, let <math>\Gamma</math> denote the kernel of the exponential map:
:<math>\Gamma = \left\{ H\in\mathfrak{t} \mid e^{2\pi H}=\operatorname{Id} \right\},</math>
where <math>\operatorname{Id}</math> is the identity element of ''T''. (We scale the exponential map here by a factor of <math>2\pi</math> in order to avoid such factors elsewhere.) 
Then for <math>\lambda</math> to give a well-defined map <math>\rho</math>, <math>\lambda</math> must satisfy
:<math>\lambda(H)\in\mathbb{Z},\quad H\in\Gamma,</math>
where <math>\mathbb{Z}</math> is the set of integers.<ref>{{harvnb|Hall|2015}} Proposition 12.9</ref> A linear functional <math>\lambda</math> satisfying this condition is called an '''analytically integral element'''. This integrality condition is related to, but not identical to, the notion of [[Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras|integral element]] in the setting of semisimple Lie algebras.<ref>{{harvnb|Hall|2015}} Section 12.2</ref>

Suppose, for example, ''T'' is just the group <math>S^1</math> of complex numbers <math>e^{i\theta}</math> of absolute value 1. The Lie algebra is the set of purely imaginary numbers, <math>H=i\theta,\,\theta\in\mathbb{R},</math> and the kernel of the (scaled) exponential map is the set of numbers of the form <math>in</math> where <math>n</math> is an integer. A linear functional <math>\lambda</math> takes integer values on all such numbers if and only if it is of the form <math>\lambda(i\theta)= k\theta</math> for some integer <math>k</math>. The irreducible representations of ''T'' in this case are one-dimensional and of the form
:<math>\rho(e^{i\theta})=e^{ik\theta},\quad k \in \Z .</math>

===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>

===The Weyl character formula===
{{Main|Weyl character formula}}
If <math>\Pi:K\to\operatorname{GL}(V)</math> is representation of ''K'', we define the '''character''' of <math>\Pi</math> to be the function <math>\Chi : K \to \mathbb{C}</math> given by
:<math>\Chi(x)=\operatorname{trace}(\Pi(x)),\quad x\in K</math>.
This function is easily seen to be a class function, i.e., <math>\Chi(xyx^{-1})=\Chi(y)</math> for all <math>x</math> and <math>y</math> in ''K''. Thus, <math>\Chi</math> is determined by its restriction to ''T''.

The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the [[Peter–Weyl theorem]], is that the characters form an orthonormal basis for the set of square-integrable class functions in ''K''. A second key result is the [[Weyl character formula]], which gives an explicit formula for the character—or, rather, the restriction of the character to ''T''—in terms of the highest weight of the representation.

In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established ''after'' the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of ''K'', the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using [[Verma module]]s. In Weyl's approach, the construction is based on the [[Peter–Weyl theorem]] and an analytic proof of the [[Weyl character formula]].<ref>{{harvnb|Hall|2015}} Sections 12.4 and 12.5</ref> Ultimately, the irreducible representations of ''K'' are realized inside the space of continuous functions on ''K''.

===The SU(2) case===
We now consider the case of the compact group SU(2). The representations are often considered from the [[Representation theory of SU(2)|Lie algebra point of view]], but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form
:<math> \begin{pmatrix}
e^{i\theta} & 0\\
0 & e^{-i\theta}
\end{pmatrix} .
</math>
According to the example discussed above in the section on representations of ''T'', the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers <math>m</math>. The general theory then tells us that for each <math>m</math>, there is a unique irreducible representation of SU(2) with highest weight <math>m</math>.

Much information about the representation corresponding to a given <math>m</math> is encoded in its character. Now, the Weyl character formula says, [[Weyl character formula#The SU.282.29 case|in this case]], that the character is given by
:<math>\Chi\left(\begin{pmatrix}
e^{i\theta} & 0\\
0 & e^{-i\theta}
\end{pmatrix}\right)=\frac{\sin((m+1)\theta)}{\sin(\theta)}.</math>
We can also write the character as sum of exponentials as follows:
:<math>\Chi\left(\begin{pmatrix}
e^{i\theta} & 0\\
0 & e^{-i\theta}
\end{pmatrix}\right)=e^{im\theta}+e^{i(m-2)\theta}+\cdots e^{-i(m-2)\theta}+e^{-im\theta}.</math>
(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)

From this last expression and the standard formula for the [[Weyl character formula#Complex semisimple Lie algebras|character in terms of the weights of the representation]], we can read off that the weights of the representation are 
:<math>m,m-2,\ldots,-(m-2),-m,</math>
each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are <math>m+1</math> weights, each with multiplicity 1, the dimension of the representation is <math>m+1</math>. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.

===An outline of the proof===
We now outline the proof of the theorem of the highest weight, following the original argument of [[Hermann Weyl]]. We continue to let <math>K</math> be a connected compact Lie group and <math>T</math> a fixed maximal torus in <math>K</math>. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.<ref>{{harvnb|Hall|2015}} Sections 12.4 and 12.5</ref>

The tools for the proof are the following:
*The [[Maximal torus#Properties|torus theorem]].
*The [[Maximal torus#Weyl integral formula|Weyl integral formula]].
*The [[Peter–Weyl theorem#Restriction to class functions|Peter–Weyl theorem for class functions]], which states that the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions on <math>K</math>.

With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the [[Weyl character formula]]. The formula states that if <math>\Pi</math> is an irreducible representation with highest weight <math>\lambda</math>, then the character <math>\Chi</math> of <math>\Pi</math> satisfies:
: <math>\Chi(e^H)=\frac{\sum_{w\in W} \det(w) e^{i\langle w\cdot(\lambda+\rho),H\rangle}}{\sum_{w\in W} \det(w) e^{i\langle w\cdot\rho,H\rangle}}</math>
for all <math>H</math> in the Lie algebra of <math>T</math>. Here <math>\rho</math> is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of <math>i</math> in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the <math>L^2</math> norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.

Next, we let <math>\Phi_\lambda</math> denote the function on the right-hand side of the character formula. We show that ''even if <math>\lambda</math> is not known to be the highest weight of a representation'', <math>\Phi_\lambda</math> is a well-defined, Weyl-invariant function on <math>T</math>, which therefore extends to a class function on <math>K</math>. Then using the Weyl integral formula, one can show that as <math>\lambda</math> ranges over the set of dominant, analytically integral elements, the functions <math>\Phi_\lambda</math> form an orthonormal family of class functions. We emphasize that we do not currently know that every such <math>\lambda</math> is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions <math>\Phi_\lambda</math>, and these functions are orthonormal.

Now comes the conclusion. The set of all <math>\Phi_\lambda</math>—with <math>\lambda</math> ranging over the dominant, analytically integral elements—forms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the <math>\Phi_\lambda</math>'s. And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some <math>\lambda</math> that is not the highest weight of a representation, then the corresponding <math>\Phi_\lambda</math> would not be the character of a representation. Thus, the characters would be a ''proper'' subset of the set of <math>\Phi_\lambda</math>'s. But then we have an impossible situation: an orthonormal ''basis'' (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of <math>\Phi_\lambda</math>'s). Thus, every <math>\lambda</math> must actually be the highest weight of a representation.