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Matrix representations | Representation theory episode 2
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#grouptheory #linearalgebra #matrices #representationtheory
Whether it's classical mechanics or quantum particles, physics makes heavy use of vectors. When a physical system satisfies a set of symmetries, we have to implement those symmetries as matrices. That way, they can operate directly on all those vectors. Welcome to the world of matrix representations.
Many thanks to professor Karel Dekimpe from the university of Leuven for helping me understand representations and characters.
Don't forget to check out these links:
Wikipedia article about the regular representation.
Wikipedia article about the permutation representations in general.
A good introduction to representation theory for beginners.
A free PDF book full of great examples. It builds up the theory from the ground up, and contains all the proofs in a logical order. Quite a challenging read.
Another free PDF about representation theory.
Tobias Osborne: How representations are used in physics. He explains the theory and all the proofs in detail. By giving many specific examples, he manages to convince you of just how important representation theory is. Highly recommended.
0:00 Introduction
0:35 The group of rotations of a square
1:13 A matrix representation must be a homomorphism
2:45 A group has many different matrix representations
5:13 The regular representation
6:26 Matrix representations are essential for physics
8:57 Creating new representations from existing ones
11:13 The homomorphism restricts the matrices
12:55 Decomposing representations into smaller pieces
This video is published under a CC Attribution license
Whether it's classical mechanics or quantum particles, physics makes heavy use of vectors. When a physical system satisfies a set of symmetries, we have to implement those symmetries as matrices. That way, they can operate directly on all those vectors. Welcome to the world of matrix representations.
Many thanks to professor Karel Dekimpe from the university of Leuven for helping me understand representations and characters.
Don't forget to check out these links:
Wikipedia article about the regular representation.
Wikipedia article about the permutation representations in general.
A good introduction to representation theory for beginners.
A free PDF book full of great examples. It builds up the theory from the ground up, and contains all the proofs in a logical order. Quite a challenging read.
Another free PDF about representation theory.
Tobias Osborne: How representations are used in physics. He explains the theory and all the proofs in detail. By giving many specific examples, he manages to convince you of just how important representation theory is. Highly recommended.
0:00 Introduction
0:35 The group of rotations of a square
1:13 A matrix representation must be a homomorphism
2:45 A group has many different matrix representations
5:13 The regular representation
6:26 Matrix representations are essential for physics
8:57 Creating new representations from existing ones
11:13 The homomorphism restricts the matrices
12:55 Decomposing representations into smaller pieces
This video is published under a CC Attribution license
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