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.
More to read
7.2 Permutations
Permutations give matrix representations by permuting basis vectors. The all ones vector spans a trivial subrepresentation. Its orthogonal complement gives a smaller piece. Checking homomorphism means checking multiplication is preserved. Faithfulness means distinct group elements give distinct matrices.
5.9 Systems of Linear Equations in Two Variables
Systems of linear equations describe invariant subspaces. Solving for stable lines tests subrepresentations. Rotation matrices show how cyclic groups act faithfully. Traces and dimensions summarize pieces. Linear algebra over the group algebra unifies the viewpoints.
3.6 Zeros of Polynomial Functions
Zeros of polynomials appear as eigenvalues in examples. Characteristic polynomials encode rotation angles. Minimal polynomials distinguish diagonalizable from Jordan cases. These polynomial tools support representation checks. They also bridge to module structure.
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Where it sits
17 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.