(in Polish) Reprezentacje grup i geometria

General data

Course ID:	1000-1M22RGG
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	(unknown)
Name in Polish:	Reprezentacje grup i geometria
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Main fields of studies for MISMaP:	computer science mathematics physics
Type of course:	elective monographs
Requirements:	Algebra I 1000-113bAG1a Topology II 1000-134TP2
Prerequisites:	Algebra II 1000-134AG2 Algebraic topology 1000-135TA Lie groups and Lie algebras 1000-135AGL
Prerequisites (description):	Groups, rings, topological spaces and cohomology.
Mode:	Classroom
Short description:	1. Classical character theory, 2. Representations of permutation groups, 3. Relations with geometry, Schubert's calculus 4. Hecke algebras and Kazhdan-Lusztig polynomials
Full description:	Generalities on linear representations. Basic examples, tensor product, symmetric, exterior powers, Irreducible representations. Character theory, Schur's lemma. Orthogonality relations, Decomposition of the regular representation, Permutation group as an example of a Coxeter group. Representations of permutation groups. Young diagrams, Construction of irreducible representations, Characters of irreducible representations and symmetric polynomials, Decomposing the tensor product of irreducible representations, Pieri and Littlewood-Richardsone rule Schubert varieties in flag manifolds, Schubert polynomials and divided difference operators. Hecke algebra as a deformation of the group ring. Geometric interpretation of the Hecke algebra. Kazhdan-Lusztig polynomials.
Bibliography:	Fulton - Young Tableaux Gruson, Serganova - A Journey Through Representation Theory Humphreys - Reflection Groups and Coxeter Groups Serre - Linear Representations of Finite Groups
Learning outcomes:	Knows general facts about linear representations of finite groups, character theory. Knows connection of permutation group representation with Young diagrams and symmetric functions. Understands connection between permutation groups and geometry of flag varieties.
Assessment methods and assessment criteria:	Evaluation based on activity during problem sessions, essay and oral exam

This course is not currently offered.

Course descriptions are protected by copyright.
Copyright by University of Warsaw.