--- title: "Group Representations and the Group Algebra" description: "Realise an abstract group as matrices acting on a vector space, and recognise a representation as a module over the group algebra." canonical: https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW source: https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Abstract Algebra · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW ## Ready when they can - 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 ## Lesson: Groups acting as matrices 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. **Example.** 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. **Tip.** 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. **Recap.** Send each element to an invertible matrix, keep multiplication intact, and read the stable subspaces. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Modules: Linear Algebra Over a Ring](https://lightmysky.com/learn/mathematics/modules-linear-algebra-over-a-ring-mt_0GMVp-s8wD) - [Linear Maps and Their Matrices](https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto) - [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR) ## Opens up - [Maschke's Theorem and Complete Reducibility](https://lightmysky.com/learn/mathematics/maschkes-theorem-and-complete-reducibility-mt_FVNCwUqeTx)