Group Representations| Group Representations |
In General > s.a. group.
* Idea: A representation is
the most common way of specifying a group, in which one defines 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: Also motivated by the development of quantum physics.
* Quote: "Group
representations can be thought of as an abstraction of the Fourier methods
of solving pdes" (meaning?).
@ General references: in Frobenius 68;
in Schur 73;
Curtis 99 [history].
@ Texts: Kirillov 76;
Barut & Rączka 77;
Naimark & Stern 82;
Chen 86;
Huang 99;
James & Liebeck 01 [III].
Specific Concepts and Results > s.a. Character;
Fusion Rules [representation ring]; Schur's Lemma.
* Intertwiner: Given N
irreducible representations {πj}
of G, an intertwiner is a multilinear map I:
⊗j=1k
πj →
⊗j=k+1N
πj,
for some k, such that \(\pi^~_{k+1}(g)^c{}_p\) ... \(\pi^~_N(g)^d{}_q\)
I p...qm...n
\(\pi_1(g^{-1})\)ma
...
\(\pi_k(g^{-1})\)nb
= I c...da...b
for all g ∈ G, i.e., invariant tensor; The intertwiners for SU(2)
are given by Clebsch-Gordan theory.
@ Invariants: Jarvis & Sumner ANZIAM(14)-a1205 [character methods, and case studies].
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 \(\mapsto\) g(f), with (g(f))(g'):=
f(g−1g')
(respectively, f(g'g)).
* Use: Very important
mathematically, because it contains all irreducible representations (n
copies of each n-dimensional one), and physically because it seems
that gauge fields transform like this.
> Online resources:
see Wikipedia page.
Types of Groups and Representations > s.a. Adjoint
Representation; Special Functions.
* 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 \(\circ\) v0.
* Ladder: Multiplicity-free
with respect to 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)
= ω(g1,
g2) h(g1
g2), where ω is a phase
factor, |ω(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 → \(\mathbb C\) 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: Kociński & 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
@ 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].
> Related topics: see Character;
diffeomorphisms; group theory; knot theory.
> Specific areas: see canonical quantum
gravity; σ-models;
supersymmetry in field theory.
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send feedback and suggestions to bombelli at olemiss.edu – modified 22 jan 2016