In representation theory, one of the most important problems is to give a (good) basis for each irreducible representation, which enables us to obtain an explicit formula for its character.
In this course, we explain a combinatorial model of finite-dimensional, irreducible (highest weight) reprersentations of complex finite-dimensional semi-simple Lie algebras; this model is called Littelmann's path model.
The aim of this course is to give an explicit combinatorial parametrization of a certain good basis of each finite-dimensional, irreducible representation of a complex finite-dimensional semi-simple Lie algebra.
There exists a one-to-one correspondence between the set of equivalence classes of finite-dimensional irreducible highest weight representations of a complex finite-dimensional semi-simple Lie algebra and the set of dominant integral weights.
The goal of this course is become able to write down explicitly all the Lakshmibai-Seshadri (LS) paths of an arbitrary fixed shape (or, dominant integral weight), which indexes a certain good basis of the finite-dimensional irreducible representation with the given highest weight; here an LS path is a certain combinatorial object, which is described in terms of root systems and Weyl groups of semi-simple Lie algebras.
complex semi-simple Lie algebra, irreducible highest weight representation, crystal basis, Lakshmibai-Seshadri path, Littelmann's path model
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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Standard lecture course.
Course schedule | Required learning | |
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Class 1 | Complex semi-simple Lie algebras and their root systems | Details will be provided during each class session. |
Class 2 | Weyl groups and the Bruhat order | Details will be provided during each class session. |
Class 3 | Action of root operators (Kashiwara operators) on paths | Details will be provided during each class session. |
Class 4 | Properties of root (Kashiwara operators) operators | Details will be provided during each class session. |
Class 5 | Lakshmibai-Seshadri (LS) paths | Details will be provided during each class session. |
Class 6 | Properties of LS paths | Details will be provided during each class session. |
Class 7 | Action of root operators on LS paths | Details will be provided during each class session. |
Class 8 | Littelmann's path model | Details will be provided during each class session. |
None.
M. Kashiwara, Bases cristallines des groupes quantiques, Cours Specialises, Vol. 9, SMF.
Based on evaluation of assignments. Details will be announced during each class.
None.
Do not hesitate to ask any questions.