group representations Definition and Topics - 10 Discussions
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations (i.e. automorphisms) of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...
Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail):
1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...
Consider the pseudoscalar and vector meson family, as well as the baryon
J = 1/2 family and baryon J = 3/2 family.
Within each multiplet, for each particle state write down its complete set
of quantum numbers, its mass, and its quark state content. Furthermore, for
each multiplet draw the (Y...
Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like:
$$\Phi^\dagger...
I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this:
My understanding of this question is that, given a...
In quantum field theory, we use the universal cover of the Lorentz group SL(2,C) instead of SO(3,1). (The reason for this is, of course, that representations of SO(3,1) aren't able to describe spin 1/2 particles.)
How is the invariant speed of light enocded in SL(2,C)?
This curious fact of...
Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...
An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as
U(x) = e^{ixT_1} = \left(
\begin{array}{cc}
\cos\frac{x}{2} & i\sin\frac{x}{2} \\
i\sin\frac{x}{2} & \cos\frac{x}{2} \\
\end{array}
\right)
=
\left(...
Problem
This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions.
Relevant Example
Take SO(3) for example; it's the...
Hi to all the readers of the forum.
I cannot figure out the following thing.
I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V).
I know that a scalar (in Galileian Physics) is something that is invariant under rotation.
How can I reconcile this...