Representation theory of rings and groups for master's students (2019 spring)

Course Outline

Problem sets

Problem Set 1 an its solution

Problem Set 2 an its solution

Problem Set 3 an its solution

Problem Set 4 an its solution

Problem Set 5 an its solution

Problem Set 6 an its solution

Problem Set 7 and its solution

Problem Set 8 and its solution

Problem Set 9 and its solution

Problem Set 10 and its solution

Tests

Test 1, April 4, Thursday, 12.15, T604

Consultation before the test, April 1, Monday, 12.15, H45a

Test 2, May 9, Thursday, 12.15, T604

Consultation before the test, May 7, Monday, 12.15, H46

Review for Test 1

Proofs for Test 1:

1. Characterization of projective and injective modules (the three and two equivalent conditions)
2. Every Abelian group can be embedded into an injective Abelian group.
3. Equivalent conditions for the semisimplicity of a module
4. If A is a finite dimensional algebra, and then every indecomposable projective module over A is local.
5. Irreducible morphisms going to a projective module
6. Harada-Sai-lemma

Review for Test 2

Proofs for Test 2:

1. Maschke's Theorem
2. Degrees of the irreducible representations (together with the two lemmas)
3. Number of irreducible characters
4. First orthogonality relation
5. Frobenius reciprocity
6. χ(g)|K|/χ(1) is an algebraic integer (where K is the conjugacy class of g)

Recommended literature

Assem, Simson, Skowronski: Elements of the Representation Theory of Associative Algebras, vol. 1, Chapters I–IV

Isaacs: Character Theory, Chapters 1–6

Lecture notes in Hungarian

Gyűrűk reprezentációi

Csoportreprezentációk

Tests from 2013

Test 1 in Hungarian and in English

Test 2 in Hungarian and in English