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 {*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,* such that \(\pi^~_{k+1}(g)^c{}_p\) ... \(\pi^~_N(g)^d{}_q\)
*I*^{ p...q}_{m...n}
\(\pi_1(g^{-1})\)^{m}_{a}
...
\(\pi_k(g^{-1})\)^{n}_{b}
= *I*^{ c...d}_{a...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*^{−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.

main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 22 jan 2016