Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Since about 1990, for the verification of binomial

and hypergeometric series, there are automatic tools

available. The book by Petkovsˇek et al. (1996) is an

excellent introduction into these aspects. The philo-

sophy is as follows. Suppose we are given a binomial

or hypergeometric series S(n) =

F(n, k). The

Gosper–Zeilbergeralgorithm(see‘‘Furtherread-

ing’’)(cf.Petkovsˇek et al. (1996);asimplified

version was presented in the reference Zeilberger in

‘‘Furtherreading’’)willfindalinearrecurrence

ðnÞSðnÞþA

ðnÞSðn þ 1Þþ

þ A

ðnÞSðn þ dÞ¼CðnÞ½70

for some d, where the coefficients A

(n) are

polynomials in n, and where C(n) is a certain

function in n, with proof !

If, for example, we suspected that S(n) = RHS(n),

where RHS(n) is some closed-form expression, then

we just have to verify that RHS(n) satisfies the

recurrence [70] and check S(n) = RHS(n) for suffi-

ciently many initial values of n to have a proof for

the identity S(n) = RHS(n) for all n. On the other

hand, if RHS(n) was a different sum, then we would

apply the algorithm to find a recurrence for RHS(n).

If it turns out to be the same recurrence then, again,

a check of S(n) = RHS(n) for a few initial values will

provide a full proof of S(n) = RHS(n) for all n.

Even in the case that we do not have a conjectured

expression RHS(n), this is not the end of the story.

Given a recurrence of the type [70], the Petkovsˇek

algorithm(see‘‘Furtherreading’’)(cf.Petkovsˇek et al.

(1996)) is able to find a closed-form solution (where

‘‘closed form’’ has a precise meaning), respectively tell

that there is no closed-form solution.

The fascinating point about both algorithms is

that neither do we have to know what the algorithm

does internally nor do we have to check that. For

the Petkovsˇek algorithm, this is obvious anyway

because, once the computer says that a certain

expression is a solution of [70], it is a routine matter

to check that. This is less obvious for the Gosper–

Zeilberger algorithm. However, what the Gosper–

Zeilberger algorithm does is, for a given sum

(n) =

F(n , k), it finds polynomials A

(n),

(n), ..., A

(n) and an expression G(n, k) (which

is, in fact, a rational multiple of F(n, k)), such that

ðnÞFðn; kÞþA

ðnÞFðn þ 1; kÞþ

þ A

ðnÞFðn þ d; kÞ¼Gðn; k þ 1ÞGðn; kÞ½71

for some d. Because of the properties of F(n, k) and

G(n, k), which are part of the theory, this is an

identity which can be directly verified by clearing all

common factors and checking the remaining identity

between rational functions in n and k. However, we

may now sum both sides of [71] over k to obtain a

recurrence of the form [70].

Algorithms for multiple sums are also available

(see‘‘Furtherreading’’).TheyfollowideasbyWilf

and Zeilberger (1992) (of which a simplified

version is presented in a Mohammed and Zeilber-

gerpreprint(see‘‘Furtherreading’’));however,they

run more quickly in capacity problems. Schneider

(2005) is currently developing a very promising

new algorithmic approach to the automatic treat-

ment of multisums. See q-Special Functions and

Statistical Mechanics and Combinatorial Problems.

See also: Classical Groups and Homogeneous Spaces;

Compact Groups and Their Representations; Dimer

Problems; Growth Processes in Random Matrix Theory;

Ordinary Special Functions; q-Special Functions; Saddle

Point Problems; Statistical Mechanics and Combinatorial

Problems.

Further Reading

http://algo.inria.fr This site includes, among its libraries, the

Maple program gdev.

Andrews GE (1976) The Theory of Partitions, Encyclopedia of

MathematicsandItsApplications, vol. 2. (reprinted by Cambridge

University Press, Cambridge, 1998). Reading: Addison–Wesley.

Andrews GE, Askey RA, and Roy R (1999) In: Rota GC (ed.)

Special Functions, Encyclopedia of Mathematics and Its

Applications, vol. 71. Cambridge: Cambridge University Press.

Ayoub R (1963) An Introduction to the Analytic Theory of

Numbers. Mathematical Surveys, vol. 10, Providence, RI:

American Mathematical Society.

Bergeron F, Labelle G, and Leroux P (1998) Combinatorial Species

and Tree-Like Structures. Cambridge: Cambridge University Press.

Bousquet-Me´lou M and Jehanne A (2005), Polynomial equations

with one catalytic variable, algebraic series, and map

enumeration. Preprint,ariv:math.CO/0504018.

Bressoud DM (1999) Proofs and Confirmations – The Story of

the Alternating Sign Matrix Conjecture. Cambridge: Cam-

bridge University Press.

de Bruijn NG (1964) Po´lya’s theory of counting. In: Beckenbach

EF (ed.) Applied Combinatorial Mathematics, New York:

Wiley, (reprinted by Krieger, Malabar, Florida, 1981).

Comtet L (1974) Advanced Combinatorics. Dordrecht: Reidel.

Dolbilin NP, Mishchenko AS, Shtan’ko MA, Shtogrin MI, and

Zinoviev YuM (1996) Homological properties of dimer

configurations for lattices on surfaces. Functional Analysis

and its Application 30: 163–173.

Feller W (1957) An Introduction to Probability Theory and Its

Applications, vol. 1, 2nd edn. New York: Wiley.

Fisher ME (1984) Walks, walls, wetting and melting. Journal of

Statistical Physics 34: 667–729.

Flajolet P and Sedgewick R, Analytic Combinatorics, book

project, available at http://algo.inria.fr.

Di Francesco P, Zinn-Justin P and Zuber J.-B. (2004), Determi-

nant formulae for some tiling problems and application to

fully packed loops, Preprint,ariv:math-ph/0410002.

Di Francesco P and Zinn-Justin P (2005), Quantum Knizhnik–

Zamolodchikov equation, generalized Razumov–Stroganov

sum rules and extended Joseph polynomials. Preprint,

ariv:math-ph/0508059.

Combinatorics: Overview 575

Galluccio A and Loebl M (1999) On the theory of Pfaffian

orientations I. Perfect matchings and permanents. Electronic

Journal of Combinatorics 6: Article #R6, 18 pp.

http://www.fmf.uni-lj.si – website of Faculty of Mathematics of

University of Ljubljana. A Mathematica implementation by

Marko Petkovsˇek is available here.

Gasper G and Rahman M (2004) Basic Hypergeometric Series,

2nd edn. Encyclopedia of Mathematics and Its Applications,

vol. 96. Cambridge: Cambridge University Press.

de Gier J (2005) Loops matchings and alternating-sign matrices.

Discrete Mathematics 365–388.

Humphreys JE (1990) Reflection Groups and Coxeter Groups.

Cambridge: Cambridge University Press.

Johansson K (2002) Non-intersecting paths, random tilings and

random matrices. Probability Theory and Related Fields

123: 225–280.

Kenyon R (2003) An Introduction to the Dimer Model,LectureNotes

for a Short Course at the ICTP, 2002; ariv:math.CO/0310326.

Koekoek R and Swarttouw RF, The Askey–scheme of hypergeo-

metric orthogonal polynomials and its q-analogue, TU Delft,

The Netherlands, 1998; on the www: http://aw.twi.tudelft.nl.

Krattenthaler C (1997) The enumeration of lattice paths with

respect to their number of turns. In: Balakrishnan N (ed.)

Advances in Combinatorial Methods and Applications to

Probability and Statistics, pp. 29–58. Boston: Birkha¨user.

Krattenthaler C (2003), Asymptotics for random walks in alcoves

of affine Weyl groups. Preprint,ariv:math.CO/0301203.

Krattenthaler C (2005a), Watermelon configurations with wall

interaction: exact and asymptotic results. Preprint,

ariv:math.CO/0506323.

Krattenthaler C (2005b) Advanced determinant calculus: a

complement. Linear Algebra Applications 411: 68–166.

Krattenthaler C, Guttmann AJ, and Viennot XG (2000) Vicious

walkers, friendly walkers and Young tableaux II: with a wall.

Journal of Physics A: Mathematical and General 33: 8835–8866.

Kuperberg G (1998) An exploration of the permanent-determi-

nant method. Electronic Journal of Combinatorics 5: Article

#R46, 34 pp.

Labelle G and Lamathe C (2004) A shifted asymmetry index

series. Advances in Applied Mathematics 32: 576–608.

Mohammed M and Zeilberger D (2005) Multi-variable Zeilberger

and Almkvist–Zeilberger algorithms and the sharpening of

Wilf–Zeilberger theory. Advanced Applications in Mathe-

matics (to appear).

Mohanty SG (1979) Lattice Path Counting and Applications.

New York: Academic Press.

Odlyzko AM (1995) Asymptotic enumeration methods. In:

Graham RL, Gro¨tschel M, and Lova´sz L (eds.) Handbook of

Combinatorics, pp. 1063–1229. Amsterdam: Elsevier.

Pemantle R and Wilson MC, Twenty combinatorial examples of

asymptotics derived from multivariate generating functions.

Preprint, available at http://www.cs.auckland.ac.nz.

Petkovsˇek M, Wilf H, and Zeilberger D (1996) A ¼ B Wellesley:

Peters AK.

http://www.mat.univie.ac.at – Website of Faculty of Mathematics,

University of Vienna. It provides the manual of the Mathe-

matica package HYP.

Propp J (1999) Enumeration of matchings: problems and progress.

In: Billera L, Bjo¨rner A, Greene C, Simion R, and Stanley RP

(eds.) New Perspectives in Algebraic Combinatorics,Mathe-

matical Sciences Research Institute Publications, vol. 38,

pp. 255–291. Cambridge: Cambridge University Press.

Razumov AV and Stroganov YG (2005) Enumeration of quarter-

turn symmetric alternating-sign matrices of odd order.

Preprint, ariv:math-ph/0507003.

Robertson N, Seymour PD, and Thomas R (1999) Permanents,

Pfaffian orientations, and even directed circuits. Annals of

Mathematics 150(2): 929–975.

Schneider C (2005) A new Sigma approach to multi-summation.

Advances in Applied Mathematics 34(4): 740–767.

Slater LJ (1966) Generalized Hypergeometric Functions.

Cambridge: Cambridge University Press.

Stanley RP (1986) Enumerative Combinatorics, Pacific Grove,

CA: Wadsworth & Brooks/Cole, (reprinted by Cambridge

University Press, Cambridge, 1998).

Stanley RP (1999) Enumerative Combinatorics, vol. 2. Cambridge:

Cambridge University Press.

Szego

G (1959) Orthogonal Polynomials, American Mathematical

Society Colloquium Publications, vol. 23. New York. Provi-

dence RI: American Mathematical Society.

Tesler G (2000) Matchings in graphs on non-oriented surfaces.

Journal of Combinatorial Theory Series B 78: 198–231.

http://www.risc.uni.linz.ac.at – website of RISC (Research Insti-

tute for Symbolic Computation). Mathematica implementa-

tions written by Peter Paule and Markus Schorn, and Axel

Riese and Kurt Wegschaider are available here.

http://www.math.rutgers.edu – website of Department of Mathe-

matics, Rutgers University. Computer implementations written

by D Zeilberger are available here.

Viennot X and James W Heaps of segments, q-Bessel functions in

square lattice enumeration and applications in quantum

gravity. Preprint.

Wilf HS and Zeilberger D (1992) An algorithmic proof theory for

hypergeometric (ordinary and ‘‘q’’) multisum/integral identi-

ties. Inventiones Mathematicae 108: 575–633.

Zeilberger D (2005) Deconstructing the Zeilberger algorithm.

Journal of Difference Equations and Applications 11: 851–856.

Compact Groups and Their Representations

A Kirillov, University of Pennsylvania,

Philadelphia, PA, USA

A Kirillov, Jr., Stony Brook University,

Stony Brook, NY, USA

In this article, we describe the structure and

representation theory of compact Lie groups.

Throughout the article, G is a compact real Lie

group with Lie algebra g. Unless otherwise stated,

G is assumed to be connected. The word ‘‘group’’

will always mean a ‘‘Lie group’’ and the word

‘‘subgroup’’ will mean a closed Lie subgroup. The

notation Lie(H) stands for the Lie algebra of a Lie

group H. We assume that the reader is familiar

with the basic facts of the theory of Lie groups and

Lie algebras, which can be found in Lie Groups:

General Theory, or in the books listed in the

bibliography.

576 Compact Groups and Their Representations

Examples of Compact Lie Groups

Examples of compact groups include



finite groups,



quotient groups T

= R

, or more generally,

V=L, where V is a finite-dimensional real vector

space and L is a lattice in V, that is, a discrete

subgroup generated by some basis in V – groups

of this type are called ‘‘tori’’; it is known that

every commutative connected compact group is a

torus;



unitary groups U(n) and special unitary groups

SU(n), n  2;



orthogonal groups O(n) and SO(n), n  3; and



the groups U(n, H), n  1, of unitary quaternionic

transformations, which are isomorphic to Sp(n):=

Sp(n, C) \ SU(2n).

The groups O(n) have two connected components,

one of which is SO(n). The groups SU(n) and Sp(n)

are connected and simply connected.

The groups SO(n) are connected but not simply

connected: for n 3, the fundamental group of

SO(n)isZ

. The universal cover of SO(n)isa

simply connected compact Lie group denoted by

Spin(n). For small n, we have isomorphisms:

Spin(3) ’ SU(2), Spin(4) ’ SU(2)  SU(2), Spin(5) ’

Sp(4), and Spin(6) ’ SU(4).

Relation to Semisimple Lie Algebras

and Lie Groups

Reductive Groups

A Lie algebra g is called



‘‘simple’’ if it is nonabelian and has no ideals

different from {0} and g itself;



‘‘semisimple’’ if it is a direct sum of simple ideals;

and



‘‘reductive’’ if it is a direct sum of semisimple and

commutative ideals.

We call a connected Lie group G ‘‘simple’’ or

‘‘semisimple’’ if Lie(G) has this property.

Theorem 1 Let G be a connected compact Lie

group and g = Lie(G). The n

(i) The Lie algebra g = Lie(G) is reductive: g = a 

, where a is abelian and g

= [g, g] is

semisimple.

(ii) The group G can be written in the form G = (A 

K)=Z, where A is a torus, K is a connected, simply

connected compact semisimple Lie group, and Z

is a finite central subgroup in A  K.

(iii) If G is simply connected, it is a product of

simple compact Lie groups.

The proof of these results is based on the fact that

the Killing form of g is negative semidefinite.

Example 1 The group U(n) contains as the center

the subgroup C of scalar matrices. The quotient

group U(n)=C is simple and isomorphic to

SU(n)=Z

. The presentation of Theorem 1 in this

case is

UðnÞ¼ T

 SUðnÞ



¼ C  SUðnÞðÞ= C \ SUðnÞðÞ

For the group SO(4) the presentation is

(SU(2)  SU(2))={(1  1)}.

This theorem effectively reduces the study of the

structure of connected compact groups to the study

of simply connected compact simple Lie groups.

Complexification of a Compact Lie Group

Recall that for a real Lie algebra g, its complex-

ification is g

= g  C with obvious commutator. It

is also well known that g

is semisimple or

reductive iff g is semisimple or reductive, respec-

tively. There is a subtlety in the case of simple

algebras: it is possible that a real Lie algebra is

simple, but its complexification g

is only semi-

simple. However, this problem never arises for Lie

algebras of compact groups: if g is a Lie algebra of a

real compact Lie group, then g is simple if and only if

is simple.

The notion of complexification for Lie groups is

more delicate.

Definition 1 Let G be a connected real Lie group

with Lie algebra g. A complexification of G is a

connected complex Lie group G

(i.e., a complex

manifold with a structure of a Lie group such that

group multiplication is given by a complex analytic

map G

 G

), which contains G as a closed

subgroup, and such that Lie(G

) = g

. In this case,

we will also say that G is a real form of G

It is not obvious why such a complexification

exists at all; in fact, for arbitrary real group it may

not exist. However, for compact groups we do have

the following theorem.

Theorem 2 Let G be a connected compact Lie

group. Then it has a unique complexification G

 G.

Moreover, the following properties hold:

(i) The inclusion G  G

is a homotopy equiva-

lence. In particular, 

(G) = 

) and the

quotient space G

=G is contractible.

(ii) Every complex finite-dimensional representation

of G can be uniquely extended to a complex

analytic representation of G

Compact Groups and Their Representations 577

Since the Lie algebra of a compact Lie group G is

reductive, we see that G

must be reductive; if G is

semisimple or simple, then so is G

. The natural

question is whether every complex reductive group

can be obtained in this way. The following theorem

gives a partial answer.

Theorem 3 Every connected complex semisimple

Lie group H has a compact real form: there is a

compact real subgroup K  H such that H = K

Moreover, such a compact real form is unique up to

conjugation.

Example 2

(i) The unitary group U(n) is a compact real form

of the group GL(n, C).

(ii) The orthogonal group SO(n) is a compact real

form of the group SO(n, C).

(iii) The group Sp(n) is a compact real form of the

group Sp(n, C).

(iv) The universal cover of GL(n, C) has no compact

real form.

These results have a number of important appli-

cations. For example, they show that study of

representations of a semisimple complex group H

can be replaced by the study of representations of its

compact form; in particular, every representation is

completely reducible (this argument is known as

Weyl’s unitary trick).

Classification of Simple Compact Lie Groups

Theorem 1 essentially reduces such classification to

classification of simply connected simple compact

groups, and Theorems 2 and 3 reduce it to the

classification of simple complex Lie algebras. Since

the latter is well known, we get the following result.

Theorem 4 Let G be a connected, simply con-

nected simple compact Lie group. Then g

must be

a simple complex Lie algebra and thus can be

described by a Dynkin diagram of one the following

types: A

, B

, C

, D

, E

, F

, G

Conversely, for each Dynkin diagram in the above

list, there exists a unique, up to isomorphism, simply

connected simple compact Lie group whose Lie

algebra is described by this Dynkin diagram.

For types A

, ..., D

, the corresponding compact

Lie groups are well-known classical groups shown in

the table below:

The restrictions on n in this table are

made to avoid repetitions which appear for

small values of n. Namely, A

= B

= C

, which

gives SU(2) = Spin(3) = Sp(1); D

= A

[ A

, which

gives Spin(4) = SU(2)  SU(2); B

= C

, which gives

SO(5) = Sp(4); and A

= D

, which gives SU(4) =

Spin(6). Other than that, all entries are distinct.

Exceptional groups E

, ..., G

also admit explicit

geometric and algebraic descriptions which are

related to the exceptional nonassociative algebra O

of the so-called octonions (or Cayley numbers). For

example, the compact group of type G

can be

defined as a subgroup of SO(7) which preserves an

almost-complex structure on S

. It can also be

described as the subgroup of GL(7, R) which

preserves one quadratic and one cubic form, or,

finally, as a group of all automorphisms of O.

Maximal Tori

Main Properties

In this section, G is a compact connected Lie group.

Definition 2 A ‘‘maximal torus’’ in G is a maximal

connected commutative subgroup T  G.

The following theorem lists the main properties of

maximal tori.

Theorem 5

(i) For every element g 2 G, there exists a maxi mal

torus T 3 g.

(ii) Any two maximal tori in G are conjugate.

(iii) If g 2 G commutes with all elements of a

maximal torus T, then g 2 T.

(iv) A connected subgroup H  G is a maximal

torus iff the Lie algebra Lie(H) is a maximal

abelian subalgebra in Lie(G).

Example 3 Let G = U(n). Then the set T of

diagonal unitary matrices is a maximal torus in G;

moreover, every maximal torus is of this form after

a suitable unitary change of basis. In particular, this

implies that every element in G is conjugate to a

diagonal matrix.

Example 4 Let G = SO(3). Then the set D of

diagonal matrices is a maximal commutative sub-

group in G, but not a torus. Here D consists of four

elements and is not connected.

Maximal Tori and Cartan Subalgebras

The study of maximal tori in compact Lie groups is

closely related to the study of Cartan subalgebras in

reductive complex Lie algebras. For convenience of

readers, we briefly recall the appropriate definitions

, n  1 B

, n  2 C

, n  3 D

, n  4

SU(n þ 1) Spin(2n þ 1) Sp(n) Spin(2n)

578 Compact Groups and Their Representations

here; details can be found in Serre (2001) or in Lie

Groups: General Theory.

Definition 3 Let a be a complex reductive Lie

algebra. A Cartan subalgebra h  a is a maximal

commutative subalgebra consisting of semisimple

elements.

Note that for general Lie algebras Cartan sub-

algebra is defined in a different way; however, for

reductive algebras the definition given above is

equivalent to the standard one.

A choice of a Cartan subalgebra gives rise to the

so-called root decomposition: if h  a is a Cartan

subalgebra in a complex reductive Lie algebra, then

we can write

a ¼ h 

2R



½1

where



¼fx 2 aj ad h:x ¼h; hix 8h 2 hg

R ¼f 2 h



f0gja



6¼ 0gh



The set R is called the ‘‘root system’’ of a with

respect to Cartan subalgebra h; elements  2 R are

called ‘‘roots.’’ We will also frequently use elements



2 h defined by h

, i= 2(, )=(, ) where ( , )

is a nondegenerate invariant bilinear form on a



and

h, i is the pairing between a and a



. It can be shown

that so defined 

does not depend on the choice of

the form ( , ).

Theorem 6 Let G be a connected compact Lie

group with Lie algebra g, and let T  Gbea

maximal torus in G, t = Lie(T)  g. Let g

, G

the complexification of g, GasinTheorem 2.

Let h = t

 g

. Then h is a Cartan subalgebra in

, and the corresponding root system R  it



Conversely, every Cartan subalgebra in g

can be

obtained as t

for some maximal torus T  G.

Weights and Roots

Let G be semisimple. Recall that the root lattice

Q  it



is the abelian group generated by roots  2

R, and let the coroot lattice Q

 it be the abelian

group generated by coroots 

,  2 R. Define also

the weight and coweight lattices by

P ¼fjh

;i2Z 8 2 Rgit



¼ftjht;i2Z 8 2 Rgit;

where h, i is the pairing between t and the dual

vector space t



It follows from the definition of root system that

we have inclusions

Q  P  it



 P

 it

½2

Both P, Q are lattices in it



; thus, the index (P : Q)

is finite. It can be computed explicitly: if 

is a basis

of the root system, then the fundamental weights !

defined by

h

i¼

form a basis of P. The simple roots 

are related

to fundamental weights !

by the Cartan matrix A:



. Therefore, (P : Q) = (P

: Q

) = jdet Aj.

Definitions of P, Q, P

, Q

also make sense when

g is reductive but not semisimple. However, in this

case they are no longer lattices: rkQ < dim t



, and P

is not discrete.

We can now give more precise information about

the structure of the maximal torus.

Lemma 1 Let T be a compact connected commu-

tative Lie group, and t = Lie(T) its Lie algebra. Then

the exponential map is surjective and preimage

of unit is a lattice L  t. There is an isomorphism

of Lie groups

exp : t=L ! T

In particular, T ’ R

= T

, r = dim T.

Let X(T)  it



be the lattice dual to ð2iÞ

1

XðTÞ¼f 2 it



jh; li22iZ 8l 2 Lg½3

It is called the ‘‘character lattice’’ for T (see the

subsection‘‘Examplesofrepresentations’’).

Theorem 7 Let G be a compact connected Lie

group, and let T  G be a maximal torus in G.

Then Q  X(T)  P. Moreover , the group G is

uniquely determined by the Lie algebra g and the

lattice X(T) 2 it



which can be any lattice between

QandP.

Corollary For a given complex semisimple Lie

algebra a, there are only finitely many (up to

isomorphism) compact connected Lie groups G

with g

= a.

The largest of them is the simply connected group,

for which T = t=2iQ

, X(T) = P; the smallest is the

so-called ‘‘adjoint group,’’ for which T = t=2iP

X(T) = Q.

Example 5 Let G = U(n). Then it = {real diagonal

matrices}. Choosing the standard basis of matrix

Compact Groups and Their Representations 579

units in it, we identify it ’ R

, which also allows us

to identify it



’ R

. Under this identification,

Q ¼ð

; ...;

Þj

2 Z;



¼ 0

P ¼ð

; ...;

Þj

2 R;

 

2 Z



XðTÞ¼Z

Note that Q, P are not lattices: Q ’ Z

n1

P ’ R  Z

n1

Now let G = SU(n). Then it



= R

=R  (1, ..., 1),

and Q, P are the images of Q , P for G = U(n) in this

quotient. In this quotient they are lattices, and

(P : Q) = n. The character lattice in this case is

X(T) = P, since SU(n ) is simply connected. The

adjoint group is PSU(n) = SU(n)=C,whereC =

{  idj

= 1} is the center of SU(n).

Weyl Group

Let us fix a maximal torus T  G.LetN(T)  G be

the normalizer of T in G: N(T) = {g 2 G jgTg

1

= T}.

For any g 2 N(T) the transformation A(g): t 7!gtg

1

an automorphism of T.AccordingtoTheorem 5,this

automorphism is trivial iff g 2 T. So in fact, it is the

quotient group N(T)=T which acts on T.

Definition 4 The group W = N(T)=T is called the

‘‘Weyl group’’ of G.

Since the Weyl group acts faithfully on t and t



,it

is common to consider W as a subgroup in GL(t



). It

is known that W is finite.

The Weyl group can also be defined in terms of

Lie algebra g and its complexification g

Theorem 8 The Weyl group coincides with the

subgroup in GL(it



) generated by reflections



: x 7! x  (2(, x))=(, ),  2 R, where, as

before,(,)is a nondegenerate invariant bilinear

form on g



Theorem 9

(i) Two element s t

, t

2 T are conjugate in G iff

= w(t

) for some w 2 W.

(ii) There exis ts a natural homeomorphism of

quotient spaces G=AdG ’ T=W, where AdG

stands for action of G on itself by conjugation.

(Note, however, that these quotient spaces are

not manifolds: they have singularities.)

(iii) Let us call a function f on G central if

f (hgh

1

) = f (g) for any g, h 2 G. Then the

restriction map gives an isomorphi sm

fcontinuous central functions on Gg

’fW invariant continuous functions on Tg

Example 6 Let G = U(n). The set of diagonal unitary

matrices is a maximal torus, and the Weyl group is the

symmetric group S

acting on diagonal matrices by

permutations of entries. In this case, Theorem 9 shows

that if f (U) is a central function of a unitary matrix,

then f (U) =

f (

, ..., 

), where 

are eigenvalues of

U and

f is a symmetric function in n variables.

Representations of Compact Groups

Basic Notions

By a representation of G we understand a pair

(, V), where V is a complex vector space and  is

a continuous homomorphism G !Aut(V). This

notation is often shortened to  or V. In this article,

we only consider finite-dimensional (f.d.) represen-

tations; in this case, the homomorphism  is

automatically smooth and even real-analytic.

We associate to any f.d. representation (, V)ofG

the representation (



, V)oftheLiealgebrag = Lie(G)

which is just the derivative of the map  : G !AutV at

the unit point e 2 G. In terms of the exponential map,

we have the following commutative diagram:

!



AutV

exp

" "exp

!





EndV

Choosing a basis in V, we can write the operators

(g) and 



(X) in matrix form and consider  and 



as matrix-valued functions on G and g. The diagram

above means that

ðexp XÞ¼e





ðXÞ

½4

Recall that if G is connected, simply connected, then

every representation of g can be uniquely lifted to a

representation of G. Thus, classification of repre-

sentations of connected simply connected Lie groups

is equivalent to the classification of representations

of Lie algebras.

Let (

, V

)and(

, V

) be two representations of

thesamegroupG.AnoperatorA 2 Hom(V

, V

)is

called an ‘‘intertwining operator,’’ or simply an

‘‘intertwiner,’’ if A  

(g) = 

(g)  A for all g 2 G.

Two representations are called ‘‘equivalent’’ if they

admit an invertible intertwiner. In this case, using an

appropriate choice of bases, we can write 

and 

by the same matrix-valued function.

Let (, V) be a representation of G. If all operators

(g), g 2 G, preserve a subspace V

 V, then the

restrictions 

(g) = (g)j

define a ‘‘subrepresenta-

tion’’ (

, V

)of( , V). In this case, the quotient

space V

= V=V

also has a canonical structure of a

representation, called the ‘‘quotient representation.’’

580 Compact Groups and Their Representations

A representation (, V) is called ‘‘reducible’’ if it

has a nontrivial (different from V and {0}) sub-

representation. Otherwise it is called ‘‘irreducible.’’

We call representation (, V) ‘‘unitary’’ if V is a

Hilbert space and all operators (g), g 2 G, are

unitary, that is, given by unitary matrices in any

orthonormal basis. We use a short term ‘‘unirrep’’

for a ‘‘unitary irreducible representation.’’

Main Theorems

The following simple but important result was one

of the first discoveries in representation theory. It

holds for representations of any group, not necessa-

rily compact.

Theorem 10 (Schur lemma). Let (

, V

), i = 1, 2, be

any two irreducible finite-dimensional repre senta-

tions of the same group G. Then any intertwiner

A : V

is either invertible or zero.

Corollary 1 If V is an irreducible f.d. representation,

then any intertwiner A:V !Visscalar: A=c id, c 2C.

Corollary 2 Every irreducible representation of a

commutative group is one dimensional.

The following theorem is one of the fundamental

results of the representation theory of compact

groups. Its proof is based on the technique of

invariant integrals on a compact group, which will

be discussed in the next section.

Theorem 11

(i) Any f.d. representation of a compact group is

equivalent to a unitary representa tion.

(ii) Any f.d. representatio n is completely reducible:

it can be decomposed into direct sum

V ¼

where V

are pairwise nonequivalent unirreps.

Numbers n

2 Z

are called ‘‘multiplicities.’’

Examples of Representations

The representation theory looks rather different for

abelian (i.e., commutative) and nonabelian groups.

Here we consider two simplest examples of both kinds.

Our first example is a one-dimensional compact

connected Lie group. Topologically, it is a circle

which we realize as a set T ’ U(1) of all complex

numbers t with absolute value 1.

Every unirrep of T is one dimensional; thus, it is

just a continuous multiplicative map  of T to itself.

It is well known that every such map has the form



ðtÞ¼t

for some k 2 Z

The collection of all unirreps of T is itself a group,

called ‘‘Pontrjagin dual’’ of T and denoted by

T. This group is isomorphic to Z.

By Theorem 11, any f.d. representation  of T is

equivalent to a direct sum of one-dimensional

unirreps. So, an equivalence class of  is defined by

the multiplicity function  on

T = Z taking non-

negative values:

 ’

k2Z

ðkÞ

The many-dimensional case of compact connected

abelian Lie group can be treated in a similar way.

Let T be a torus, that is, an abelian compact group,

t = Lie(T). Then every irreducible representation

of T is one dimensional and thus is defined by a

group homomorphism  : T !T

= U(1). Such

homomorphisms are called ‘‘characters’’ of T. One

easily sees that such characters themselves form a

group (Pontrjagin dual of T). If we denote by L the

kernel of the exponential map t !T (see Lemma 1),

one easily sees that every character has a form

ðexpðtÞÞ ¼ e

ht;i

; t 2 t;2 XðTÞ

where X(T)  it



is the lattice defined by [3]. Thus,

we can identify the group of characters

T with X(T).

In particular, this shows that

T ’ Z

dim T

The second example is the group G = SU(2), the

simplest connected, simply connected nonabelian

compact Lie group. Topologically, G is a three-

dimensional sphere since the general element of G is

a matrix of the form

g ¼



b a



; a; b 2 C; jaj

þjbj

¼ 1

Let V be two-dimensional complex vector space,

realized by column vectors ð

Þ. The group G acts

naturally on V. This action induces the representa-

tion  of G in the space S(V) of all polynomials in

u, v. It is infinite dimensional, but has many f.d.

subrepresentations. In particular, let S

(V), or

simply S

, be the space of all homogeneous

polynomials of degree k. Clearly, dim S

= k þ 1.

It turns out that the corresponding f.d. representa-

tions (

, S

), k  0, are irreducible, pairwise non-

equivalent, and exhaust the set

G of all unirreps.

Some particular cases are of special interest:

1. k = 0. The space V

consists of constant functions

and 

is the trivial one-dimensional representa-

tion: 

(g)  1.

2. k = 1. The space V

is identical to V and 

just the tautological representation (g)  g.

3. k = 2. The space V

is spanned by monomials

, uv, v

. The remarkable fact is that this

Compact Groups and Their Representations 581

representation is equivalent to a real one. Namely,

in the new basis

x ¼

þ v

; y ¼

 v

; z ¼ iuv

we have





b a

Reða

þb

Þ 2ImðabÞ Imðb

a

2Imða

bÞjaj

jbj

2ReðabÞ

Imða

þb

Þ 2ReðabÞ Reða

b

This formula defines a homomorphism 

:SU(2)!

SO(3). It can be shown that this homomorphism is

surjective, and its kernel is the subgroup

{ 1} SU(2):

1 !f1g,!SUð2Þ!



SOð3Þ!1

The simplest way to see it is to establish the

equivalence of 

with the adjoint representation

of G in g. The corresponding intertwiner is

3ð þiÞu

þ2iuv

þð iÞv

iþi

 þi i



Note that SU(2) and SO(3) are the only compact

groups associated with the Lie algebra sl(2, C).

The group G contains the subgroup H of diagonal

matrices, isomorphic to T

. Consider the restriction

of 

to T

. It splits into the sum of unirreps 

follows:

Res



s¼½n=2

s¼0



n2s

The characters 

which enter this decomposition

are called the weights of 

. The collection of all

weights (together with multiplicities) forms a multi-

set in

T denoted by P(

)orP(S

Note the following features of this multiset:

1. P(

) is invariant under reflection k 7!k.

2. All weights of 

are congruent modulo 2.

3. The nonequivalent unirreps have different multi-

sets of weights.

Below we show how these features are generalized

to all compact connected Lie groups.

Fourier Transform

Haar Measure and Invariant Integral

The important feature of compact groups is the

existence of the so-called ‘‘invariant integral,’’ or

‘‘average.’’

Theorem 12 For every compac t Lie group G, there

exists a unique measure dgonG, called ‘‘Haar

measure,’’ which is invariant under left shifts

: h 7!gh and satisfies

dg = 1.

In addition, this measure is also invariant under

right shifts h 7!hg and under involution h 7!h

1

Invariance of the Haar measure implies that for

every integrable function f (g), we have

f ðgÞdg ¼

f ðhgÞdg ¼

f ðghÞdg ¼

f ðg

1

Þdg

For a finite group G, the integral with respect to

the Haar measure is just averaging over the group:

f ðgÞdg ¼

jGj

g2G

f ðgÞ

For compact connected Lie groups, the Haar

measure is given by a differential form of top degree

which is invariant under right and left translations.

For a torus T

= R

with real coordinates 

R=Z or complex coordinates t

= e

2i

, the Haar

measure is d

 := d

d

d

t :¼

k¼1

2it

In particular, consider a central function f (see

Theorem 9). Since every conjugacy class contains

elements of the maximal torus T (see Theorem 5),

such a function is determined by its values on T, and

the integral of a central function can be reduced to

integration over T. The resulting formula is called

‘‘Weyl integration formula.’’ For G = U(n) it looks

as follows:

UðnÞ

f ðgÞdg ¼

f ðtÞ

i<j

 t

where T is the maximal torus consisting of diagonal

matrices

t ¼ diagðt

; ...; t

Þ; t

¼ e

2i

and d

t is defined above.

Weyl integration formula for arbitrary compact

group G can be found in Simon (1996) or Bump

(2004, section 18).

The main applications of the Haar measure are the

proof of complete reducibility theorem (Theorem 11)

and orthogonality relations (see below).

Orthogonality Relations and Peter–Weyl Theorem

Let V

, V

be unirreps of a compact group G.

Taking any linear operator A : V

and aver-

aging the expression A(g):= 

1

)  A  

(g) over

582 Compact Groups and Their Representations

G, we get an intertwining operator hAi=

A(g)dg.

Comparing this fact with the Schur lemma, one

obtains the following fundamental results.

Let (, V) be any unirrep of a compact group G.

Choose any orthonormal basis {v

,1 k  dim V}

in V and denote by t

,ort



, the function on G

defined by

ðgÞ¼ððgÞv

; v

The functions t

are called ‘‘matrix elements’’ of the

unirrep (, V).

Theorem 13 (Orthogonality relations)

(i) The matrix elements t

are pairwise orthogonal

and have norm ( dim V)

1=2

in L

(G,dg).

(ii) The matrix elements corresponding to equiva-

lent unirreps span the same subspace in

(G,dg).

(iii) The matrix elements of two nonequivalent

unirreps are orthogonal.

(iv) The linear span of all matrix elements of all

unirreps is dense in C(G), C

(G), and in

(G,dg)(generalized Peter–Weyl theorem).

In particular, this theorem implies that the set

G of

equivalence classes of unirreps is countable. For an

f.d. representation (, V) we introduce the character

of  as a function





ðgÞ¼trðgÞ¼

dim V

k¼1



ðgÞ½5

It is obviously a central function on G.

Remark Traditionally, in representation theory

the word ‘‘character’’ has two different meanings:

(1) a multiplicative map from a group to U(1), and

(2) the trace of a representation operator (g). For

one-dimensional representations both notions

coincide.

From the orthogonality relations we get the

following result.

Corollary The characters of unirreps of G form an

orthonormal basis in the subspace of central func-

tions in L

(G,dg).

Noncommutative Fourier Transform

The noncommutative Fourier transform on a com-

pact group G is defined as follows. Let

G denote the

set of equivalence classes of unirreps of G. Choose

for any  2

G a representation (



, V



) of class 

and an orthonormal basis in V



. Denote by d() the

dimension of V



We introduce the Hilbert space L

(

G)asthespace

of matrix-valued functions on

G whose value at a point

 2

G belongs to Mat

d()

(C). The norm is defined as

kFk

GÞ

2

dðÞtrðFðÞFðÞ



For a function f on G define its Fourier transform

as a matrix-valued function on

f ðÞ¼

f ðg

1

Þ



ðgÞdg

Note that in the case G = T

this transform

associates to a function f the set of its Fourier

coefficients. In general this transform keeps some

important features of Fourier coefficients.

Theorem 14

(i) For a function f 2 L

(G,dg) the Fourier transform

f is well defined and bounded (by matrix norm)

function on

(ii) For a function f 2 L

(G,dg) \ L

(G,dg) the

following analog of the Plancherel formula holds:

kf k

ðG;dgÞ

:¼

jf ðgÞj

2

dðÞtrð

f ðÞ



Þ¼: k

f k

GÞ

(iii) The following inversion formula expresses f in

terms of

f :

f ðgÞ¼

2

dðÞtrð

f ðÞ



ðgÞÞ

(iv) The Fourier transform sends the convolution to

the matrix multiplication:

 f



where the convolution product  is defined by

ðf

 f

ÞðhÞ¼

ðhgÞf

ðg

1

Þdg

Note the special case of the inversion formula for

g = e:

f ðeÞ¼

2

dðÞtrð

f ðÞÞ;

ðgÞ¼

2

dðÞ



ðgÞ

where (g) is Dirac’s delta-function:

f (g)

(g)dg = f (e). Thus, we get a presentation of Dirac’s

delta-function as a linear combination of characters.

Compact Groups and Their Representations 583

Classification of Finite-Dimensional

Representations

In this section, we give a classification of unirreps of

a connected compact Lie group G.

Weight Decomposition

Let G be a connected compact group with maximal

torus T, and let (, V) be a f.d. representation of G.

Restricting it to T and using complete reducibility,

we get the following result.

Theorem 15 The vector space V can be written in

the form

V ¼

2XðTÞ



;



¼fv 2 Vj



ðtÞv ¼h; tiv 8t 2 tg

½6

where X(T) is the character group of T defined by [3].

The spaces V



are called ‘‘weight subspaces,’’

vectors v 2 V



– ‘‘weight vectors’’ of weight . The set

PðVÞ¼f 2 XðTÞjV



6¼f0gg ½7

is called the ‘‘set of weights’’ of , or the ‘‘spectrum’’

of Res

, and

mult

ð;VÞ

ðÞ:¼ dim V



is called the ‘‘multiplicity’’ of  in V.

The next theorem easily follows from the defini-

tion of the Weyl group.

Theorem 16 For any f.d. representation V of G,

the set of weights with multiplicities is invariant

under the action of the Weyl group:

wðPðVÞÞ ¼ PðVÞ; mult

ð;VÞ

ðÞ¼mult

ð;VÞ

ðwðÞÞ

for any w 2 W.

Classification of Unirreps

Recall that R is the root system of g

. Assume that

we have chosen a basis of simple roots 

, ..., 



R. Then R = R

[ R



; roots  2 R

can be written

as a linear combination of simple roots with positive

coefficients, and R



= R

A (not necessarily f.d.) representation of g

called a ‘‘highest-weight representation’’ if it is

generated by a single vector v 2 V



(the highest-

weight vector) such that g



v = 0 for all positive

roots  2 R

It can be shown that for every  2 X(T), there is a

unique irreducible highest-weight representation of

with highest weight , which is denoted L().

However, this representation can be infinite dimen-

sional; moreover, it may not be possible to lift it to a

representation of G.

Definition 5 A weight  2 X(T) is called ‘‘domi-

nant’’ if h, 

i2Z

for any simple root 

. The set

of all dominant weights is denoted by X

(T).

Theorem 17

(i) All weights of L() are of the form  =   n



2 Z

(ii) Let  2 X

. Then the irreducible highest-weight

representation L() is f.d. and lifts to a

representation of G.

(iii) Every irreducible f.d. representation of G is of

the form L() for some  2 X

Thus, we have a bijection {unirreps of G} $X

Example 7 Let G = SU(2). There is a unique simple

root  and the unique fundamental weight !, related

by  = 2!. Therefore, X

= Z

 ! and unirreps are

indexed by non-negative integers. The representa-

tion with highest weight k  ! is precisely the

representation 

constructed in the subsection

‘‘Examplesofrepresentations.’’

Example 8 Let G = U(n). Then X = Z

,andX

{(

, ..., 

) 2 Z

j



}. Such objects are

well known in combinatorics: if we additionally

assume that 

 0, then such dominant weights are

in bijection with partitions with n parts. They can

also be described by ‘‘Young diagrams’’ with n rows

(see Fulton and Harris (1991)).

Explicit Construction of Representations

In addition to description of unirreps as highest-

weight representations, they can also be constructed

in other ways. In particular, they can be defined

analytically as follows. Let B = HN

be the

Borel subgroup in G

; here H = exp h,

= exp

2R

)



. For  2 h



, let 



: B !C



be a multiplicative map defined by





ðhnÞ¼e

hh;i

½8

Theorem 18 (Cartan–Borel–Weil). Let  2 X(T).

Denote by V() the space of complex-analytic

functions on G

which satisfy the following trans-

formation property:

f ðgbÞ¼

1



ðbÞf ðgÞ; g 2 G

; b 2 B

The group G

acts on V() by left shifts:

ðgÞfðÞðhÞ¼f ðg

1

hÞ½9

584 Compact Groups and Their Representations