Representation Definition and 766 Threads

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. H

    A Representation in second quantization

    For setting the fermionic state ## |n_1 n_2\rangle## where ##n_i## is the number of particles in the orbital ##i## we can use the following both representations for the state ##|1 1\rangle##: $$ \hat {\mathbf c_1} ^{\dagger} \hat {\mathbf c_2} ^{\dagger} |vac\rangle $$ or $$ \hat {\mathbf c_2}...
  2. redtree

    I Definition of the special unitary group

    In matrix representation, the special unitary group is distinguished from the more general unitary group by the sign of the matrix determinant. However, this presupposes that the special unitary group is formulated in matrix representation. For a unitary group action NOT formulated in matrix...
  3. Euge

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

    Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$
  4. Slimy0233

    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...
  5. T

    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...
  6. AndreasC

    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...
  7. Vanilla Gorilla

    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...
  8. S

    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...
  9. Emrissa

    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...
  10. H

    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)...
  11. Y

    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?
  12. Ashish Somwanshi

    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...
  13. Euge

    POTW Contour Integral Representation of a Function

    Suppose ##f## is holomorphic in an open neighborhood of the closed unit disk ##\overline{\mathbb{D}} = \{z\in \mathbb{C}\mid |z| \le 1\}##. Derive the integral representation $$f(z) = \frac{1}{2\pi i}\oint_{|w| = 1} \frac{\operatorname{Re}(f(w))}{w}\,\frac{w + z}{w - z}\, dw +...
  14. V

    Comp Sci Converting Float to Char in C: Exploring the Use of sprintf

    I am very new to C programming so am struggling with this question and how exactly to begin it, when we are doing this are we to use something like sprintf? Thank you.
  15. M

    I Are There Other Ways To Represent Vectors Besides Arrows?

    Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented? As far as I know two: Geometrically and Algebraically. There is only one way to represent vectors in geometrically: arrows, however there are several or more methods to...
  16. V

    What Are the Smallest and Largest Numbers in an 8-bit Floating Point System?

    Homework Statement:: We have an 8-bit floating point representation, one bit for the sign, 3-bit biased exponent, and 4-bit for the normalized mantissa. What is the smallest and largest number you can represent? Relevant Equations:: - Would the smallest number just be 0 0 0 0 0 0 0 0 and...
  17. K

    A Matrix representation of a unitary operator, change of basis

    If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|## ##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|## And we've a general vector ##|\alpha\rangle## such that...
  18. A

    Engineering Representation of a threaded section

    Greetings! I´m trying to solve the following exercice I have done the following drawing for the cross section and I want to know if I need to add a top view or any other additional view? (is the cross section enough) thank you!
  19. A

    I Arithmetic representation of symbols according to certain rules

    Hi, Suppose I am given a sequence of special symbols and I want to produce the next sequence of symbols according to certain rules that transform one symbol into one or more symbols. They are 10 symbols in total, say; A, B, C, D, E, F, G, H, I, J The rules are: A transformed into B and it is...
  20. N

    Metamerism (vector-matrix representation of color perception)

    For the illuminant A for the illuminant B Did I got them correct? Thank you
  21. M

    B Representation of infinitesimals in different ways

    Hello. There are 4 types of infinitesimals: 1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers) 2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number. 3) Surreal numbers: { 0, 1...
  22. murshid_islam

    I Matrix representation for closed-form expression for Fibonacci numbers

    From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is: \begin{pmatrix} 1 & 1 \\ 1 & 0\\ \end{pmatrix} ^ n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1}\\ \end{pmatrix} That only works...
  23. P

    Spin matrix representation in any arbitrary direction

    I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction ##\hat{n}##, but do I need to find the eigenstates of ##\sigma \cdot \hat{n}## first and check if they form a diagonal matrix...
  24. K

    I A doubt regarding position representation of product of operators

    We've two operators ##\hat{a}##,##\hat{b}##. I know their position representation ##\langle r|\hat{b} \mid \psi\rangle=b## ##\langle r|\hat{a}| \psi\rangle=a ## Is it generally true that the position representation of the combined operator ##\hat{a}\hat{b}## is ##a b## where ##a, b## are the...
  25. K

    I Position representation of angular momentum operator

    One of the component of angular momentum operator is ##\hat{L}_{x}=\hat{y} \hat{P}_{z}-\hat{z} \hat{P}_{y}## I want it's position representation. My attempt : I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms. The...
  26. shivajikobardan

    Comp Sci Every student loves to party in semantic network representation

    1 example to represent the question in semantic net. Second question-: every student loves to party. Represent this in semantic net. The answer is below-:
  27. G

    I Exploring the Relationship Between Spinors and Clifford Algebras

    I've used a particular clifford algebra expression for a quantum mechanics wave vector to see if the Born rule can become a simple linear inner product in a clifford algebra formulation. The expression $$\Psi=\sum_i (e_i+Jf_i)\psi_i$$ turned out to be successful, where ##J## is the imaginary...
  28. P

    A Lehmann Kallen and spectral representation

    Hello, My question pertains to the formula below: In particular, I would like to ask about the spectral density function shown below: What does the spectral function physically represent? Is there any interpretation of its meaning, whether it has a relation to the physical spectrum of the...
  29. F

    A Majorana representation of higher spin states

    In the article by E. Majorana "Oriented atoms in a variable magnetic field", in particular, it's considered (and solved) the problem of describing a state with spin J using 2J points on the Bloch sphere. That is, if the general state of the spin system , (1) then, according...
  30. fluidistic

    I How to "think" of a polarizer in matrix representation?

    From what I remember of my optics course, any element such as a lens (be it thick or thin), can be represented by a matrix. So they are sort of operators, and it is then easy to see how they transform an incident ray, since we can apply the matrix to the electric field vector and see how it gets...
  31. M

    A Array variable of envelope function (parameter representation)

    Hi, I have a question regarding the envelope function in parameter representation. Let an array of curves in cartesian coordinates be given in parameter representation, with curve parameter 𝑡 and array variable 𝑐 𝑥=𝑥(𝑡,𝑐) 𝑦=𝑦(𝑡,𝑐) Condition for envelope is: 𝜕/𝜕𝑡 𝑥(𝑡,𝑐) 𝜕/𝜕𝑐 𝑦(𝑡,𝑐)=𝜕/𝜕𝑐...
  32. M

    MHB Representation of signed integers of base B

    Hey! :giggle: Consider a representation of signed integers of base $B$, in which the digits are listed in descending order of importance, with the least significant digit corresponding to a positive, and the next digits to an alternate negative and positive value. Thus, a number of this...
  33. M

    I Graph Representation Learning: Question about eigenvector of Laplacian

    Hi, I was reading the following book about applying deep learning to graph networks: link. In chapter 2 (page 22), they introduce the graph Laplacian matrix ##L##: L = D - A where ##D## is the degree matrix (it is diagonal) and ##A## is the adjacency matrix. Question: What does an...
  34. joneall

    A Why does D(1,1) representation of SU(3) give baryon octet?

    The question may be ambiguous but it's really simple. One says that the baryon octet is the D(1,1) representation of SU(3), but then uses the same one for mesons. D(1,1) means one quark and one antiquark, which corresponds perfectly to mesons. But how can it explain baryons? My information and...
  35. patric44

    I Integral representation of incomplete gamma function

    hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...
  36. LCSphysicist

    I Unit Norm Axis Rotation in R3: Exploring Representation & Algebra

    Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?
  37. N

    Draw a rectangle that gives a visual representation of the problem

    A regulation NFL playing field of length x and width y has a perimeter of 346_2/3 or 1040/3 yards. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is y = (520/3)...
  38. cianfa72

    Two-port linear network general representation AV + BI = 0

    Hi, as follow-up to this thread I've a question about general representation of a two-port network. Basically it is ad hoc built four-terminal linear network (using controlled sources + nullator-norator pair): for it I found a general representation ##AV + BI = 0## as in the picture above. If...
  39. A

    I Unitary Representation of Poincaré Group: Classical Relativistic Mechanics

    This thread is a shameless self-promotion of a recent work of mine: https://arxiv.org/abs/2105.13882 In the paper an operational version of classical relativistic dynamics (for massive particles) is obtained from an irreducible representation of the Poincaré group. The formalism has kets...
  40. Haorong Wu

    The representation matrix for alpha and beta in Dirac equation

    In the 4-dimensional representation of ##\beta##, ## \beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,## and we can suppose ## \alpha_i=\begin{pmatrix}\mathbf A_i & \mathbf B_i \\ \mathbf C_i & \mathbf D_i\end{pmatrix} ##. From the anti-commutation relation...
  41. LCSphysicist

    How is the Matrix in Momentum Representation Derived?

    $$\langle p | W | p' \rangle = \int \langle p | x \rangle \langle x W | x' \rangle \langle x' p' \rangle dx dx'$$ $$\langle p | W | p' \rangle = \int \langle p | x \rangle \delta(x-x') W(x) \langle x' | p' \rangle dx dx'$$ $$\langle p | W | p' \rangle = \int \langle p | x' \rangle W(x') \langle...
  42. Y

    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...
  43. fluidistic

    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...
  44. T

    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...
  45. V

    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...
  46. D

    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...
  47. Q

    A Representation of Z2 acting on wavefunctions

    If I have a wavefunction ##\Psi(X)## that is invariant under the group ##Z_2##, what specifically does that mean? There can be several operators that are representations of the group ##Z_2##, for example the operators $$\mathbb{Z}_2=\{ \mathbb{I}, -\mathbb{I} \},$$ or $$\mathbb{Z}_2=\{...
  48. M

    Is this a more accurate representation of a molecule?

    I see the top version used everywhere. But isn't the bottom version better? Because the top version makes it seem like H2 bonds with the C to its right and the other H2 bonds with the N. Whereas the bottom version shows the bonds correctly. Should I draw it like the top version and stop annoying...
  49. R

    B Gravitational Field Representation: Why 1-Plane?

    How did you find PF?: Surfing web Can someone advise on this? In most diagrams showing how mass effects the gravitation field (earth for instance), bending fabric of space, it is demonstrated on one plane. Why is it shown this way and is there any other way of illustrating this?
  50. G

    How to Derive Matrix Representations for Spin Operators?

    $$\hat{S_+} = \hbar \begin{bmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{bmatrix}$$ $$\hat{S_-} = \hbar \begin{bmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{bmatrix}$$ $$\hat{S_x} = \hbar/\sqrt{2} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0...
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