20-22/09/2023
Content
Kazhdan-Lusztig polynomials form a family of polynomials associated with any Coxeter group via a recursive definition. Introduced in the 1970s, over the course of the last decades Kazhdan-Lusztig (KL) polynomials have played a central role in many different areas of representation theory: they determine the characters formulas in several settings, from semisimple Lie algebras in characteristic zero to algebraic groups in positive characteristic or to quantum groups at roots of unity. Moreover, they provide a profound link between representation theory and the geometry of Schubert varieties.Although their definition is elementary, KL polynomials remain still today somehow obscure objects. We know that their coefficients are positive integers, yet we are unable to give them a combinatorial meaning except in some special cases. There are still several significant open questions concerning their combinatorics, with the most pivotal being the "combinatorial invariance conjecture," which precisely pinpoints the data upon which KL polynomials depend.
The minicourse will be divided into three parts. We will first introduce Coxeter groups in great generality, then define Kazhdan-Lusztig polynomials and explain their role in representation theory and finally explore some of their combinatorial properties.
Some exercises on the content of the lectures are available here
Literature
- B. Elias, S. Makisumi, U. Thiel, G. Williamson, Introduction to Soergel bimodules. Vol. 5. RSME (Chapter 1-3)
- Bjorner, Brenti: Combinatorics of Coxeter Groups.
Useful links
- Supplementary material for the book Introduction to Soergel bimodules Further exercises, pictures of Coxeter complexes, examples of computations of KL polynomials
- Joel Gibson: Visualization software of Affine Weyl Groups and Kazhdan-Lusztig polynomials. (Joel's page has a lot of amazing stuff to visualize Lie theory. Have a look!)