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: 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 {*π*_{j}}
of *G*,
an intertwiner is a multilinear map *I*: ⊗_{j=1}^{k}
*π*_{j} → ⊗_{j=k+1}^{N} *π*_{j},
for some *k* and for all *g* ∈ *G*,
such that *π*_{k+1}(*g*)^{c}_{p}
... *π*_{n}(*g*)^{d}_{q} *I*^{ p...q}_{m...n} *π*_{1}(*g*^{–1})^{m}_{a} ... *π*_{k}(*g*^{–1})^{n}_{b}
= *I*^{ c...d}_{a...b} ,
a.k.a. 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*^{–1}*g*')
(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 ∃ *v*_{0} ∈ *V* such
that for all *v* ∈ *V*, ∃ *g* ∈ *G*
such that *v* = *g* \(\circ\) *v*_{0}.

* __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*(*g*_{1}) *h*(*g*_{2})
= *ω*(*g*_{1}, *g*_{2}) *h*(*g*_{1}* g*_{2}),
where *ω* is a phase
factor, |*ω*(*g*_{1}, *g*_{2})|
= 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*^{–1}*x*).

@ __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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jan 2016