Group
Representations| Group
Representations |
In General > s.a. group.
* Idea: A representation
is the most common way of giving a group, in which one specifies how it acts
on some vector space.
$ Def: A representation of a group G is a homomorphism h: G → GL(V),
for some vector space V.
* History: The theory
originated with a series of papers by Frobenius in 1896–1900, then Schur,
Burnside, Brauer, and others (finite groups), then generalised to compact groups
by Cartan
and Weyl in the 1920s.
* Quote: "Group
representations can be thought of as an abstraction of the Fourier methods
of solving pde's" (meaning?).
@ General references: in Frobenius 68; in Schur 73; Curtis 99 [history].
@ Texts: Kirillov 76; Barut & Raczka 77; Naimark & Stern
82; Chen 86; James 93 [III].
Specific Concepts and Results > s.a. Character;
Fusion Rules [representation ring]; Schur's
Lemma.
* Intertwiner: Given N irreps
{
j}
of G,
an intertwiner is a multilinear map I: 
j=1k
j → j=k+1N j,
for some k and for all g 
G,
such that k+1(g)cp
... n(g)dq I p...qm...n 1(g–1)ma ... k(g–1)nb
= I c...da...b ,
a.k.a. invariant tensor; The intertwiners for SU(2)
are given by Clebsch-Gordan theory.
Regular Representation
* Idea: Define the space C(G)
of complex-valued functions on G;
The left (right) regular representation of G acts on C(G)
by f 
g(f),
with (g(f))(g'):= f(g–1g')
(respectively, f(g'g)).
* Use: Very important
mathematically, because it contains all irr's (n copies of each n-dimensional
one), and physically because it seems that gauge fields
transform like this.
Types of Groups and Representations > s.a. Adjoint
Representation;
sigma models; Special
Functions; supersymmetry
in field theory.
* Conjugate representations:
For example, for SU(2), 2 and 2*
are isomorphic, while for SU(3), 3 and 3*
are not.
* Cyclic: A representation
of a group G on
a vector space V is
cyclic if 
v0 V such
that for all v V,
g G
such that v = g 
v0.
* Ladder: Multiplicity-free wrt the maximal compact subgroup.
* Schwinger: The
multiplicity-free direct sum of all unitary irreducible representations
of the group.
* Projective:
A map h: G → GL(V) such that h(g1) h(g2)
= 
(g1, g2) h(g1 g2),
where is a phase
factor, |(g1, g2)|
= 1; a.k.a. representation up to a phase.
* On function spaces:
From a representation of G on
a vector space V, we get a representation on functions F: V → C by
(gF)(x):= F(g–1x).
@ Unitary representations: Mackey 76, 78.
@ Projective representations: Adler mp/04-in
[and Yang-Mills theory].
@ Other representations: Chaturvedi et al RVMP(06)qp/05 [Schwinger
representation, finite or
compact simple Lie groups].
@ Corepresentations: Kocinski & Wierzbicki
a0905 [continuous groups].
@ Types of groups: Knapp 86 [semisimple]; Manz & Wolf 93 [solvable];
Klink & Ton-That
JMP(96) [compact,
tensor product].
> Types of groups: see finite groups; lie
groups; group theory [categorical groups].
And Physics > s.a. canonical
quantum gravity; diffeomorphisms; group
theory;
knot theory.
@ Intros for physicists: Banino 77; Jones 98 [IIb]; Chen et al 02; Vvedensky
& Evans 09.
@ And quantum mechanics: Mackey 68; Bargmann in(70) [on Hilbert spaces
of
functions].
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send feedback and suggestions to bombelli at olemiss.edu – modified 6
aug 2008