 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 gG, 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.

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 ∃ v0V such that for all vV, ∃ gG 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.