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|
Standard lecture course.
|Course schedule||Required learning|
|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.|
M. Kashiwara, Bases cristallines des groupes quantiques, Cours Specialises, Vol. 9, SMF.
Based on evaluation of assignments. Details will be announced during each class.
Do not hesitate to ask any questions.