Representation Theory
(Elective course Aug--Nov 2009)
The course meets on Tuesdays and Thursdays during 1645--1800 hours
in Matscience room 123.
The mid-term test was held on 24th September.
Click here for the pdf file of the test paper.
If you want to cheat :-) and look at solutions, click here.
The notes below are alpha version. Use at your own discretion.
Suggestions for improvement are welcome.
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G-sets: G-sets; orbits and stabilisers; class equation; applications to finite groups: Sylow theory.
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semi-direct products, extensions: semi-direct products; example: dihedral groups; extensions and their splitting.
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solvable and nilpotent groups: solvable, super-solvable, and nilpotent groups; elementary properties; structure of finite nilpotent groups.
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linear representations as modules for the group ring: associative algebras and their modules; the group ring; linear representations of groups; modules for the group ring.
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Commutation (après Bourbaki Algebra Chapter 8 §1): projections; commutants and bicommutants; direct summands and bicommutants.
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Artinian and Noetherian modules (après Bourbaki Algebra Chapter 8 §2): Artinian and Noetherian modules; Fitting lemma; decomposition into indecomposables of a module of finite
length;
Artinian and Noetherian rings.
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simple and semisimple modules (après Bourbaki Algebra Chapter 8 §3): simple modules; semisimple modules; isotpyic
components; length of a semisimple module.
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the commutant and bicommutant of a semisimple module (après Bourbaki Algebra Chapter 8 §4): the bicommutant of a semisimple module; the density theorem; the commutant of a simple module; the commutant of a semisimple module; application: stable submodules of tensor products.
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simple and semisimple rings (après Bourbaki Algebra Chapter 8 §5): semisimple rings; simple rings; simple components of a semisimple ring; structure of simple rings; semisimple
subalgebras of semisimple algebras; degrees, heights, and indices.
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The radical (après Bourbaki Algebra Chapter 8 §6): nil and nilpotent ideals; the radical of a module;
the radical of a ring; radicals in the presence of the Artinian condition; modules over Artinian rings.
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Local representation theory (après Alperin's book):
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Chapter 1 ``Semisimple modules'':
radical and socle series; when is the group algebra semisimple?; Brauer's
theorem: number of simple kG-modules equals the number of p-regular conjugacy classes; restrictions to normal subgroups of semisimple modules are semisimple;
Examples: cyclic groups, p-groups, SL(2,p).
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Chapter II ``Projective modules'':
indecomposable modules and local algebras;
uniserial modules;
free, projective, and injective modules;
projective indecomposable modules (PIMs);
bijective correspondence between PIMs and simples;