Practical information: Winter semester 2026/2027, Mondays at 12-14 in HS 4, building E2.4, Saarland University. (No lecture 21st of December)
Tutorials: Thurdays at 12-14 in HSIII, building E2.5. First tutorial on Thursday 29th of October, then every second week.
Sketch of topics: Representations of (finite) groups, Maschke’s Theorem, Schur’s Lemma, characters, induced and restricted representations, Young tableaux, tabloids, Specht modules.
References: J.-P. Serre: Linear representations of finite groups, W. Fulton: Young tableaux: Applications to Representation Theory and Geometry, B. E. Sagan: The symmetric group, W. Fulton; J. Harris: Representation Theory: A First Course.
Registration: Send an e-mail to billi[at]math.uni-sb.de.
Requisites to pass: Hand-in exercise sheet every second week (correct at least at 50%), oral exam.
Exercise sheets and tutorials:
Tutorial_1, Sheet_1 (sheet due by 05/11/2026)
Tutorial_2, Sheet_2 (sheet due by 19/11/2026)
You can hand in the exercises by sending an email to billi[at]math.uni-sb.de.
(if you cannot download the PDFs or you notice anything off, please contact me!)
Lecture record:
Definition of linear representation, examples. Subrepresentations, morphisms of representations. Kernels and images are subrepresentations. Bijection between complements of a subspace and projectors. Theorem: every subrepresentation admits a complement that is a subrepresentation.
Maschke's theorem: every representation is direct sum of irreducible representations. Recollections about the trace of an endomorphism. Characters, basic properties of charcters with values in the complex numbers. Hom representation, dual representation, tensor representation. The invariant space of a representation, morphisms of representations are invariant elements of the representation Hom.
Isomorphic representations have the same character. Characters of direct sums, tensor products, dual representations, and Hom representations. Schur's Lemma. The Hermitian form on the space of class functions. Theorem: computation of the Hermitian form on charaters of irreducible representations. Corollary: representations with the same character are isomorphic. Other consequences: the Hermitian form computed on characters counts the number of isomorphic irreducible subrepresentations, it is an integer in particular, and the irreducible subrepresentation don't depend on the chosen decomposition in irreducibles.
A representation is irreducible if and only if the norm of its character is 1. Every irreducible representation appears in the regular representation with multiplicity its dimension. Corollary: the irreducible representations are in finite number and the squares of their dimensions sum to the cardinality of the group. Examples: found all the irreducible representation of the cyclic group of order n and of the symmetric group acting on 3 elements. Theorem: the characters of irreducible representations form an orthonormal basis of the space of class functions. Corollary: the number of irreducible representations equals the number of conjugacy classes. A finite group is abelian if and only if every irreducible representation has dimension 1.
PhD courses:
November 2024, Geometric Invariant Theory (16h).
Teaching assistant:
September 2022 – February 2023, Algebra 2 (40h).
September 2022 – February 2023, Geometria 2 (15h)
February 2022 – September 2022, Geometria 2 (30h).
September 2021 – February 2022, Analisi Matematica 2 (20h).
February 2021 – September 2021, Geometria 2 (30h).
October 2020 – February 2021, Geometria 1A (30h).
September 2018 – February 2019, Algebra 2 (50h).