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.
does anyone know what representation in group Oh = O x i , is the polar vector E a basis? E has components Ex, Ey, and Ez. How would I go about showing this? Thanks.
Hi,
I have a question regarding group theory. For the cyclic group C2 with elements e and a, what are the matrices of the regular representation? How do you find this? How would I reduce this representation into irreducible representation? Lastly, how do I find a matrix which brings the...
I'm in India, I don't know about other places. I can't seem to find a channel broadcasting programmes and commercials without a woman. It's not that I don't want women on TV, but its the way they potray women that I don't like. Is this what Men and women who fought for equal rights for women...
Hey, jus an after thought:
Which greek god would you associate with biology
and
Which person(s) would you associate with biology
and
which work do u tink best represent biology
lastly
what breakthrought(s) do u tink represent biology??
Louis de Broglie hypothesized every particle moving with momentum p has a wavelength of
\lambda=\frac{h}{p}
If I understand it correctly, is the de Broglie wavelength directly related to the wavelength of \psi(x)? But because according to quantum physics, the particle coexists with the...
I, in fact, know the correct Fourier representation
for the following (it was given to me):
f(t)=0 \text { if } -\pi \leq \omega t \leq 0
and
f(t)=sin(\omega t) \text { if } 0 \leq \omega t \leq \pi
\hrule
I'm curious about the derivation that led to it -- specifically...
here your are my last contribution to number theory, i tried to send it to several journals but i had no luck and i was rejected, i think journals only want famous people works and don,t want to give an oportunity to anybody.
the work is attached to this message in .doc format only use Mellin...
Let's say I wanted to prove that, given n points, it takes a maximum of a (n-1)th degree polynomial to represent them all. How would I do it? My instinct is to just say because you need a max of (n-1) max/mins ...
For the Schrodinger equation for a hydrogen atom, we need to write out:
p^2/2m for the electron.
If we define our basis states to be a linear discrete array of points, let's say 4 points. 0,d,2d, and 3d, where is some distance, on the order of a Bohr radius. How do I write p as an...
I'm rather new to physics in general, so bear with me in my potential ignorance.
Considering we have no idea of the absolute properties of higher dimensions, how is it that they're identified in equations? This especially perplexes me when thinking about the Kaluza-Klein theory, or even...
find the series for sin(x)/x. I believe this would just mean dividing the series representation of sin(x) by x, therefore sin(x)/x=1-x^2/3!+x^4/5!-x^6/7!...=sigma(x^2n/(2n+1)!)
how then would we find the radius of convergence and interval of convergence.
is the series n/sigma(1/k(k+2))...
Is there an accurate way to write the value of an Irrational number?
If there is no an accurate way to write the value of an Irrational number, then can we conclude that no irrational number has an exact place on the real line?
And if there is an exact place to an irrational number on the...