= Representation theory
{wiki}
= Representation
{disambiguate=group theory}
{synonym}
Basically, a "representation" means associating each group element as an invertible <matrices>, i.e. a matrix in (possibly some subset of) <GL(n)>, that has the same properties as the group.
Or in other words, associating to the more abstract notion of a <group (mathematics)> more concrete objects with which we are familiar (e.g. a matrix).
Each such matrix then represents one specific element of the group.
This is basically what everyone does (or should do!) when starting to study <Lie groups>: we start looking at <matrix Lie groups>, which are very concrete.
Or more precisely, mapping each group element to a <linear map> over some <vector field> $V$ (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
$$
R(g) : G \to GL(V)
$$
As shown at <Physics from Symmetry by Jakob Schwichtenberg (2015)>
* page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
* 3.6 classifies the <representations of SU(2)>. There is only one possibility per dimension!
* 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the <Lorentz group>, not all representations can be described in terms of matrices, and that we can construct such representations with the help of <Lie group> theory, and that they have fundamental physical application
Motivation:
* https://math.stackexchange.com/questions/1628464/what-is-representation-theory
Bibliography:
* https://www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by <MathDoctorBob> (2011). Too much theory, give me the motivation!
* https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by <Quanta Magazine> (2020). Maybe there is something in there amidst the "the reader might not know what a <matrix> is" stuff.