Representation theory of finite groups and algebras

The course offered an introduction to the representation theory of finite groups and algebras.

News

The course took place during the winter semester 2022-2023.

Lectures

The course consisted of approximately 15 lectures.

• Lecture 1. Linear representations. Examples: one-dimensional representations, the regular representation, permutation representations. The category of representations: subrepresentations, sums, products and homomorphisms. Irreducibility and indecomposablility. Schur's Lemma.
• Lecture 2. Maschke's theorem. Characters. Properties of characters. Class functions. Inner products. Orthogonality relations. Irreducibles and conjugacy classes: first look. Examples of character tables.
• Lecture 3. Semisimple algebras and modules. Jacobson radical and the Jacobson density theorem. Double centralizers. Artin--Wedderburn decomposition theorem. The group algebra. The converse to Maschke's Theorem in the modular case.
• Lecture 4. Arithmetical considerations and algebraic integers. Frobenius divisibility, Schur divisibility and splitting fields for finite groups. Solvable and simple groups. Burnside's theorem on the solvability of groups of order $p^aq^b$.
• Lecture 5. Induction and restriction of groups and scalars. Frobenius reciprocity. The character of an induced representation and the Frobenius formula. Mackey's decomposition theorem and Mackey's irreducibility criterion.
• Lecture 6. Representation theory of symmetric groups I: Partitions and their statistics. Young diagrams. Young symmetrisers. Specht modules through idempotents and full classification of irreducibles. The rationals are a splitting field for the symmetric group.
• Lecture 7. Representation theory of symmetric groups II: the representations $U_\lambda$ induced from row stabilizer subgroups. Symmetric functions, the ring of characters and the character formula for Specht modules${}^{1}$.
• Lecture 8. Quivers and their path algebras. Canonical primitive idempotents. Connected and acyclic quivers. Universal property. The arrow ideal and the radical. Finite connected acyclic quivers produce finite dimensinonal connected basic algebras. Some examples.
• Lecture 9. Details on basic algebras and equivalence of module categories. Admissible ideals and bound quiver algebras. Relations. The ordinary quiver of a connected basic finite dimensional algebra. Presentations as bound quiver algebras.
• Lecture 10. Every finite dimensional connected basic algebra can be presented as a bound quiver algebras. Projective, injective and simple modules of finite dimensional basic algebras. The standard duality functor.
• Lecture 11. The category of representations of bound quiver algebras. The simple projective indecomposables and injective indecomposable modules. Computation of the socle, radical and top of a representation. The Nakayama functor.
• Lecture 12. The radical of the category of modules. Short exact sequences and extensions. Minimal morphisms and projective presentations. The transpose of a module. The Auslander--Reiten translations.
• Lecture 13. Irreducible morphisms. Left and right minimal morphisms. Left and right almost split morphisms. Irreducible morphisms between indecomposable modules and their relation to the radical. Almost split sequences.
• Lecture 14. Further properties of the Auslander--Reiten translation. Projectively and injectively stable categories and the Auslander--Reiten formulas. The existence of almost split sequences.
• Lecture 15. The Auslander--Reiten quiver of an algebra. Further characterisations of almost split sequences. Computing irreducible morphisms and the construction of the Auslander--Reiten quiver.

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Exercises

Consult the corresponding Moodle page of the course.

Literature

• Introduction to representation theory. P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, Student Mathematical Library (Volume 59), American Mathematical Society.
• Representation theory: a first course. W. Fulton, J. Harris. Graduate Texts in Mathematics (Volume 129), Springer New York, NY.
• Linear Representations of Finite Groups. J.-P. Serre. Graduate Texts in Mathematics (Volume 42), Springer New York, NY.
• Associative Algebras. Richard S. Pierce. Graduate Texts in Mathematics (Volume 88), Springer New York, NY.
• Elements of the Representation Theory of Associative Algebras 1. I. Assem, A. Skowroński and D. Simson. LMS Student Texts (Volume 65), Cambridge University Press.

Further material

There are slides and video recordings available at N. Mascot's webpage from a past course covering similar topics for group representations.