# What is Representation: Definition and 764 Discussions

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.Representation theory is pervasive across fields of mathematics for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:

illuminates and generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.Second, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

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1. ### POTW Does the Taylor series for arctan converge at x = 1?

Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$
2. ### I How can you represent a point by "z = x + iy" as shown here?

Snapshot of Mary L. Boas' Mathematical Physics book So, the marked lines say If we think of P as the point z = x +iy in the complex plane, we could replace (2.3) by a single equation to describe the motion of P But, until now I have only learned of representing points in the form (x,y), now...
3. ### I Infinite product representation of Bessel's function of the 2nd kind

An infinite product representation of Bessel's function of the first kind is: $$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$ Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this...
4. ### I Uncovering the Mystery of Kallen-Lehmann Spectral Representation

The Kallen-Lehmann representation is a (non perturbative) result in QFT that is proved with what seems to me like very minimal assumptions: https://en.m.wikipedia.org/wiki/Källén–Lehmann_spectral_representation According to this wiki page, in gauge theories something goes wrong and you can no...
5. ### B Array Representation Of A General Tensor Question

So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything. At about 5:50, he states that "The array for Q is...
6. ### B Graphical representation of the weak mixing angle

The graphical pattern of particles in the weak hypercharge and weak isospin plane, visible in this wiki page, shows the mixing angle between the Yw and Q axes. Actually , from the weak hypercharge (-2) of a right-handed electron and its electric charge (-1), one obtains an angle Pi/3, not the...
7. ### Mathematical representation of two-entangled q-bits?

Dear all,I have four questions. Hopefully, someone can answer. Thank you :) 1. A qubit is described as a two-orthogonal basis state. How about two entangled qubits? 2. What is the actual reason for a qubit cannot be cloned/copied? Is it because without knowing the value of the complex...
8. ### A Spin networks with different intertwiners

Hi Pfs Spin networks are defined by the way their links and their nodes are equipped with SU(2) representations and intertwiners. Could you give an example of two different spin networks with the same number of nodes, links between them, the same coloring of the links (and their orientations)...
9. ### Functional representation of the oscillating graph

Hi; This is in fact not a homework question, but it rather comes out of personal curiosity. If you look at the graph of the two functions in the image attached, what is the simplest functional representation for such a symmetrical pattern?
10. ### Matrix representation in QM Assignment -- Need some help please

This screenshot contains the original assignment statement and I need help to solve it. I have also attached my attempt below. I need to know if my matrices were correct and my method and algebra to solve the problem was correct...

40. ### Help with the matrix representation of <-|+|->. Does "+"=|+>?

Trying to use <+|+>=1=<-|-> and <-|+>=0 to prove each iteration of the equation, so I have 6 different versions to prove. But the part I'm currently stuck on is understanding how to simplify any given version. I've written out [S_x,S_y]=S_xS_y\psi-S_yS_x\psi and expanded it in terms of the...
41. ### A Crystallographic representation of a material, two sources seem very d

While searching for a software to plot a crystallographic representation of a particular material, I have come across two sources that seem to give two very different views of a same material. In this case, it is CoSb3. On the one hand there is from ASE (Atomic Simulation Environment), a...
42. ### Is an interpreter without an intermediate representation even possible?

I was reading the page about interpreters on wikipedia and one particular section caught my eye: "An interpreter generally uses one of the following strategies for program execution: Parse the source code and perform its behavior directly; Translate source code into some efficient...
43. ### A Adjoint representation and spinor field valued in the Lie algebra

I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf. On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM. The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...
44. ### I How is this a representation of a 3 dimensional torus?

In a differential geometry text, a torus is defined by the pair of equations: I initially thought this was somehow a torus embedded in 4 dimensions, but I do not see how we can visualize two orthogonal 2-dimensional Euclidian spaces. How is this a representation of a 2 dimensional torus...

49. ### I Commutator's Matrix representation

Hello! I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and $$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$ but what is the second term matrix representation...
50. ### A Questions about representation theory of Lie algebra

I have confusions about representation theory. In the following questions, I will try to express it as best as possible. For this thread say representation is given as ρ: L → GL(V) where L is the Lie group(or symmetry group for a physicist) GL(V) is the general linear...