- #1
jimmy.neutron
- 17
- 0
Hey guys, I'm pretty new to group theory at the moment, what's the best way of understanding a 'representation' of a group?
Thanks
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Group representations are mathematical objects used to describe the symmetries of a group. They are used to study the behavior of a group under transformations and can help us understand the structure and properties of the group.
Group representations have many applications in mathematics, physics, and other sciences. They can be used to study symmetry in geometric objects, analyze the behavior of particles in quantum mechanics, and understand the properties of molecules and crystals.
The most common types of group representations are matrix representations, where elements of the group are represented by matrices, and permutation representations, where elements are represented by permutations of a set. Other types include character representations and projective representations.
Group representations are an important tool in group theory, as they can help us understand the structure and behavior of a group. In fact, group representations are often used to define and classify different types of groups, such as finite groups, Lie groups, and topological groups.
To construct a group representation, we first need to choose a vector space on which the group will act. Then, we define a homomorphism from the group to the group of invertible linear transformations on the vector space. This homomorphism will be the group representation, and it can be represented by matrices or other mathematical objects.