CourseMATHEMATICS
Code057AA
Credits6
PeriodSemester 2
LanguageItalian
Knowledge of the basics of representation theory of finite groups, with a focus on the representation theory of the symmetric group: young diagrams and tableaux, characters, symmetric functions, and so on.
During the exam the student must be able to demonstrate their knowledge of the course material and be able to discuss the reading matter thoughtfully and with propriety of expression, and to solve simple exercises.
The student will be able to work with representations of finite groups and to apply their knowledge also in several other settings (action on varieties, cohomology, study of finite groups generated by reflections).
Many excercises will be solved together with the students during the lessons. Some exercises will be left for the students to solve at home, and they can send their solutions as part of the exam.
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Linear algebra, basic group theory.
- Delivery: in person.
- Attendance: advised.
- Learning activities: lectures, exercises, individual study
Representation theory of finite groups: definition, subrepresentation, irreducible representation, quotients complete reducibility, Schur's lemma characters, and orthonormality of the characters of irreducible representations. Representation of associative algebras, semisimple algebras representation of the symmetric group: construction of representations and characters, Schurs' duality. Symmetric functions: bases, structure constants, algebraic properties.
Course Syllabus
Bruce E. Sagan: The Symmetric Group
The exam will consist of either written exercises and a final seminar, or an oral examination that includes both theory and exercises.
Written exercises will be assigned throughout the course, and the students who deliver them consistently will access to a final oral examination consisting of a seminar on a topic related to the course but not included in it.
The students who do not deliver the exercises will undergo a final oral examination consisting of both exercises and questions on the content of the course.