Group Representations and the Group Algebra
Realise an abstract group as matrices acting on a vector space, and recognise a representation as a module over the group algebra.
What a learner can do afterwards
- Write down a faithful matrix representation of a small group and check the homomorphism property
- Translate between a representation, a linear action and a module over the group algebra
- Identify subrepresentations and say what irreducible means in this setting
1 · Read
A representation turns an abstract group into matrices you can compute with. Each group element acts as an invertible linear map on a vector space, and the assignment must preserve multiplication: the matrix of a product is the product of the matrices. The same object can be read as a linear action of the group, or as a module over the group algebra, where the group acts linearly.
The cyclic group of order 2 has two elements: doing nothing, and a flip done twice is doing nothing. Send them to the 1 by 1 matrices [1] and [negative 1]. Multiplication is preserved because negative 1 times negative 1 is 1, mirroring the flip done twice. Distinct elements give distinct matrices, so this representation is faithful. For contrast, the regular representation of the symmetric group on 3 letters has dimension 6, the order of the group.
A subrepresentation is a smaller space inside that the whole group preserves. A nonzero representation with no proper nonzero subrepresentation is called irreducible. Permutations give a concrete picture: permuting basis vectors fixes the all ones vector, so its span is a trivial subrepresentation, and the orthogonal complement forms a smaller piece beside it.
To verify a candidate, check that multiplication is preserved and hunt for two elements sharing one matrix, which breaks faithfulness. Rotations give a quick sanity check: a 120 degree rotation composed twice is a 240 degree rotation, and your matrices must compose the same way.
Send each element to an invertible matrix, keep multiplication intact, and read the stable subspaces.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.