|INTRODUCTION TO REPRESENTATION THEORY
Modules over rings and algebras, simple modules, Schur's lemma. Actions of
groups on vector spaces, representations. Group algebras, modules, complete
reducibility, Wedderburn's theorem. Characters, orthogonality relations.
Tensor product of representations. Restriction and induction.
INTRODUCTION TO GROUP THEORY|
Actions of groups on sets, Symmetric group and alternating group. Cayley's
theorem. Direct products of
groups, Sylow's theorem. Applications: classification
of groups of small order, The alternating group is simple. Classification
of finite abelian groups, finitely-generated abelian groups.
TOPICS IN REPRESENTATION THEORY|
Chwas Abbas Ahmed
Conjugacy classes of symmetric groups and parameterizations of simple
modules. Counting standard tableaux of fixed shape: Young diagrams and
tableaux, standard tableaux, Young--Frobenius formula, hook formula.
Construction of fundamental modules for symmetric groups: Action of
symmetric groups on tableaux, tabloids and polytabloids; permutation
modules on Young subgroups.
INTRODUCTION TO GALOIS THEORY|
Field extensions. Degree of extension. Algebraic numbers. Geometric
constructions with ruler and compasses.The Galois group of an extension.
The Galois correspondence between subgroups and intermediate
fields.Splitting field for a polynomial. Transitivity of the Galois group
on the zeros of an irreducible polynomial in a normal extension. Properties
equivalent to normality. Galois groups of normal separable extensions.
Properties of Galois correspondence for normal separable extensions. Normal
subgroups and normal intermediate extensions. The Fundamental Theorem of
REPRESENTATION OF LIE GROUPS AND LIE ALGEBRAS|
Introduction to Lie groups and Lie algebras (over R or C)
Generalisation: Lie groups and algebras over finite fields (the so called finite groups of Lie type)
Representations of finite groups of Lie type (Harish-chandra approach)
Representations of finite groups of Lie type (Deligne-Lusztig approach using l-adic cohomology)
Computing character tables using perverse sheaves (the so called character sheaves)