## UE Algebra 2

Degrees incorporating this pedagocial element :

### Description

Description (subject to change)

1. Motivation and preliminary notions
• Definition of a representation of a group; motivations (crystallography, polyhedra, virology, group classification); morphisms, isomorphisms.
• Focus: classification of representations
2. Examples
• Representations of dimension 1, representation associated with an action, regular representation. Groups of isometrics retaining a finite part.
• Automorphisms of structures. Direct amounts.
3. Decomposition of representations
• Sub-representations, quotient, supplementary.
• Maschke's theorem. Irreducible representations, a sufficient condition for the existence of a decomposition in sum of irreducible representations. Schur's Lemma, applications.
• Isotypic components, description of morphisms.
4. Character theory, classification of complex representations.
• Definition of characters, examples, character orthogonality relation, Schur orthonormality relation. Number of irreducible representations.
5. Character tables
• Definition, the case of finite abelian groups, dihedral groups, the alternating group on 4 elements.
6. Linear algebra complements
• Module on a ring (not necessarily commutative). Link between representations and modules on group algebra. Simple submodules and irreducible representations, Schur's lemma. Matrix computation in the non-commutative case, group algebra decomposition theorem. Tensor product for modules. Universal property, compatibility with direct sum, base. Tensor product of representations, extension of scalars. Subgroups, restriction, induction, character of induced representation.
7. Character integrity and applications
• The notion of an integer element in algebra, the centre of the algebra of a group, the dimension of an irreducible complex representation divides the cardinal of the group. Second Schur orthogonality relation, Burnside theorem.
8. Discrete Fourier Transform
• Group of characters, definition of transform, analogy with other Fourier transforms, utility

### Targeted skills

To master the notions related to representations, to find the table of the characters of small cardinal groups, to know how to use the tensor product in simple situations

### Bibliography

Jean-Pierre Serre, Représentations linéaires des groupes finis, Hermann, 1998