2023
Groups and Representations
Name: Groups and Representations
Code: MAT14352M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: Presencial
Presentation
In this CU we study the representations of groups. Namely finite groups, Lie groups as well as the associated Lie algebras.
Sustainable Development Goals
Learning Goals
- Acquiring experience abut finite groups, and the ability of recognizing several representations as well as their reducibility .
- Compute characters
- Understanding and applying Young diagrams and de Frobenius characters formula.
- Knowing Lie groups and álgebras as well as its representations and introductory results.
- Obtaining a general overview of na introduction to group and algebra theory.
- Compute characters
- Understanding and applying Young diagrams and de Frobenius characters formula.
- Knowing Lie groups and álgebras as well as its representations and introductory results.
- Obtaining a general overview of na introduction to group and algebra theory.
Contents
- Finite groups representations. Complete reducebility. Schurs lemma.
- Characters. Projection formulas.
- Examples. External power and standard representation of a symmetrical group. Induced representations.
- Young diagrams and Frobenius charecters formula. Irreducible representations of symmetric groups.
- Lie groups and algebras. Exponencial map. Engel´s theorem and Lies theorem.
- Classical Lie Algebras and their representations.
- Characters. Projection formulas.
- Examples. External power and standard representation of a symmetrical group. Induced representations.
- Young diagrams and Frobenius charecters formula. Irreducible representations of symmetric groups.
- Lie groups and algebras. Exponencial map. Engel´s theorem and Lies theorem.
- Classical Lie Algebras and their representations.
Teaching Methods
Systematic introduction to groups and their representations. Illustration of concepts and arguments of the theory by means of concrete examples.
Processing by exercises and applications in other areas of mathematics. Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of solving individually two lists of exercises in which themes or problems related to the syllabus are addressed. The final classification is the average of the classifications on the lists.
2) Evaluation by exam.
Formative evaluation is done in to improve the learning process; the elements of formative evaluation will have no weight on the final classification.
Processing by exercises and applications in other areas of mathematics. Verification of the level of acquisition of the subject-matter during the period of lecturing by means of classroom questions.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of solving individually two lists of exercises in which themes or problems related to the syllabus are addressed. The final classification is the average of the classifications on the lists.
2) Evaluation by exam.
Formative evaluation is done in to improve the learning process; the elements of formative evaluation will have no weight on the final classification.
Teaching Staff
- Pedro Correia Gonçalves Macias Marques [responsible]
Certificações
