--- title: Group Representations description: Group representation theory studies abstract groups by acting them as linear transformations on vector spaces. Learners will understand how to translate group-theoretic problems into linear algebra, analyze group actions, and construct matrix representations of finite and continuous groups. category: mathematics subcategory: representation-theory difficulty: beginner, intermediate, advanced url: /subject/group-representations --- # Group Representations Group representation theory studies abstract groups by acting them as linear transformations on vector spaces. Learners will understand how to translate group-theoretic problems into linear algebra, analyze group actions, and construct matrix representations of finite and continuous groups. ## Where to start Start with Representations and Characters of Groups by Gordon James and Martin Liebeck, an undergraduate textbook that needs only basic group theory and linear algebra. If you only use one resource, make it Representation Theory: A First Course by William Fulton and Joe Harris, which runs from finite groups through Lie groups and Lie algebras. Pavel Etingof's free Introduction to Representation Theory (MIT 18.712) is the natural next step. ## Available Resources 2 Books • 2 Courses • 4 Websites ## Websites ### 1. nLab: Representation Theory The nLab wiki entry on representation theory, written from a category-theoretic viewpoint. It frames representations as functors and modules, links to related notions such as characters, Tannaka duality, and geometric representation theory, and lists key references for further study. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/representation+theory **Tags:** representation-theory, category-theory, algebra, reference-wiki, mathematics ### 2. Etingof's Introduction to Representation Theory Free lecture notes by Pavel Etingof and co-authors from MIT, introducing representations of algebras, finite groups, quivers, and Lie algebras, with many exercises. Readers learn character theory, Schur-Weyl duality, and Gabriel's theorem, assuming a solid linear algebra and abstract algebra background. **Difficulty:** Beginner | **Price:** Free **Link:** https://math.mit.edu/~etingof/replect.pdf **Tags:** representation-theory, group-theory, lie-algebras, quivers, lecture-notes ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 4. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki ## Courses ### 1. Applied Geometric Algebra (MIT RES.8-001) **Author:** László Tisza Typeset 1976 lecture notes (11 chapters, about 110 pages) with problems, applying group theory and geometric algebra to physics: rotations, Lorentz transformations, spinors, and the unifying role of symmetry in classical, relativistic and quantum mechanics. Builds geometric intuition for advanced mathematical tools. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-8-001-applied-geometric-algebra-spring-2009/ **Tags:** geometric-algebra, group-theory, spinors, lorentz-group, mathematical-physics ### 2. Introduction to Representation Theory (MIT 18.712) **Author:** Pavel Etingof Representations of finite groups, Lie algebras, associative algebras and quivers, with character theory and basic category theory. The notes form a full open textbook with exercises, alongside problem sets, building the algebraic tools for studying symmetry in linear spaces. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/ **Tags:** representation-theory, group-representations, character-theory, lie-algebras, quiver-representations ## Books ### 1. Representation Theory: A First Course **Author:** William Fulton, Joe Harris Graduate textbook (Springer GTM 129) that builds representation theory from finite groups through Lie groups and Lie algebras, emphasizing worked examples. Readers learn character theory, Weyl's construction, and the classification of complex semisimple Lie algebras and their representations. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387974954?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, representation-theory ### 2. Representations and Characters of Groups **Author:** Gordon James, Martin Liebeck Undergraduate textbook on representations of finite groups over the complex numbers, requiring only basic group theory and linear algebra. Readers learn Maschke's theorem, character tables, orthogonality relations, induced characters, and applications such as Burnside's paqb theorem. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521445906?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, representation-theory --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/group-representations