Groups and representations
Objectives
At the end of the course, the student must
-know the theory of finite group representations (TR) on the field C with emphasis on the representations of symmetric groups
-master techniques that will be used in the tests of the main results of TR
-know the orthogonality relations of characters, know how to obtain characters from group constructions to deduce new characters
-should be able to use the results taught on Character Theory to construct the character tables of some groups; should be able to recognize the importance of Burnside''s Theorem p^a q^b in obtaining intrinsic results from Group Theory based on Character Theory
-know the irreducible representations of the symmetric group, namely the construction of representations for their characters
-know Algebraic Combinatorics derived from the study of combinatorial objects arising in the context of TR of symmetric groups
-have skills in critical analysis, synthesis and organization of theoretical knowledge; written communication skills
General characterization
Code
11588
Credits
6.0
Responsible teacher
António José Mesquita da Cunha Machado Malheiro
Hours
Weekly - 4
Total - 52
Teaching language
Inglês
Prerequisites
Available soon
Bibliography
-
W. Fulton. Young Tableaux: With Applications to Representation Theory and Geometry (London Mathematical Society Student Texts). Cambridge: Cambridge University Press, 1996, pp. x+260.
G. James e M. Liebeck. Representations and Characters of Groups. Cambridge University Press, 2001, pp. xviii+321 pp.
W. Fulton e J. Harris. Representation Theory: A First Course. Graduate texts in mathematics. Vol. 129. New York, NY: Springer, 1991..
B. Sagan. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer-Verlag New York, 2001, pp. xvi+240 pp.
J. P. Serre. Linear Representations of Finite Groups. Graduate texts in mathematics. Vol. 42. New York, NY: Springer-Verlag, 1977.
Teaching method
There are classes in which theory is lectured and illustrated by examples. There are also problem-solving
sessions. Some exercises are left to the students to be solved on their own as part of their learning process.
Students can ask questions during the classes, in weekly scheduled sessions or in special sessions accorded directly with the professor.
Evaluation method
The evaluation of this course will have two parameters:
1. Evaluation of the solutions of the exercises proposed periodically by the teacher. The teacher, when correcting the exercises, will make comments that will allow the student to improve their resolution. This parameter will correspond to 50% of the final evaluation;
2. A written exam that will correspond to 50% of the final evaluation.
Subject matter
I Finite group representations
(a) algebras, modules, group representations and homomorphisms of representations;
b) Reducibility and Maschke''s Theorem;
c) Schur''s lemma and some of its applications: representation of abelian finite groups; diagonalization;
d) The space of G-homomorphisms.
II Character Theory
a) Characters, class functions and examples;
b) Inner product, orthogonal relations and decomposition of G-modules;
c) Character and group constructions: normal subgroups; tensor product;
d) Restricted characters, induced characters and Frobenius''s reciprocity theorem.
III The Burnside Theorem p^a q^b
(a) algebraic numbers and integers;
b) Burnside''s Theorem p^a q^b.
IV Representations of symmetric groups
a) Young tableaux and partitions;
b) Symmetric polynomials and symmetric functions;
c) The action of the symmetric group in Young tableaux;
d) Specht modules and the Young symmetrizer;
e) The ring of representations and symmetric functions.