(in Polish) Reprezentacje grup i geometria 1000-1M22RGG
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.
Main fields of studies for MISMaP
physics
computer science
Type of course
Mode
Requirements
Prerequisites
Prerequisites (description)
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 criteria
Evaluation based on activity during problem sessions, essay and oral exam
Bibliography
Fulton - Young Tableaux
Gruson, Serganova - A Journey Through Representation Theory
Humphreys - Reflection Groups and Coxeter Groups
Serre - Linear Representations of Finite Groups
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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