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International School on Algebra and Algebraic Geometry
August 18–20, 2011, Ekaterinburg
 


Reflection groups

E. Yu. Smirnov

National Research University "Higher School of Economics"
Video records:
Flash Video 600.9 Mb
Flash Video 552.7 Mb
Flash Video 559.1 Mb
MP4 600.9 Mb
MP4 559.1 Mb
MP4 552.7 Mb
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Adobe PDF 55.9 Kb
Adobe PDF 398.7 Kb

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E. Yu. Smirnov


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Abstract: Reflection group is a discrete group of motions of a space of constant curvature (a sphere, Euclidean or hyperbolic space) which is generated by a set of reflections. Reflection groups appear remarkably often in various algebraic problems.
A very preliminary (and rather optimistic) plan of my lectures is as follows.
Lecture 1. Finite reflection groups. Examples: dihedral, permutation and hyperoctahedral groups. Regular tilings (kaleidoscopes) of a sphere, Euclidean space, Lobachevskii space. Coxeter cones and polytopes. Root systems, positive and simple roots.
Lecture 2. Classification of finite reflection groups by Coxeter graphs. An explicit construction of exceptional root systems. Crystallographic groups, Dynkin diagrams, relation to Lie algebras. Affine reflection groups. If time permits: hyperbolic reflection groups (an overview).
Lecture 3. A non-geometric example: representations of quivers, following Bernstein-Gelfand-Ponomarev. Indecomposable representations. Gabriel's theorem: the underlying graph of a finite type quiver is a simply-laced Dynkin diagram (ADE classification). Reflection functors. Bijection between the indecomposable representations and the positive roots. Independence on the orientation of a quiver. Kac's theorem.

Materials: smirnov_1.pdf (55.9 Kb), smirnov_slides.pdf (398.7 Kb)

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