NatWi - Leonardo Patimo

Kazhdan-Lusztig polynomials: recent results and open questions

The proof of the Kazhdan-Lusztig conjecture in the 80s was a cornerstone in modern mathematics. This conjecture asserts that the most important structures in representation theory are governed by the Kazhdan-Lusztig polynomials, a family of polynomials that are defined via a recursive algorithm. Although their definition is elementary, KL polynomials remain still today somehow obscure objects. For example, we know that their coefficients are positive integers, yet we are unable to give them a combinatorial meaning except in some special cases. In the talk I will present KL polynomials in some examples and discuss some related open questions.