KazhdanLusztig

The Kazhdan-Lusztig polynomials are a family of polynomials that arose from the study of representations of Coxeter groups. They were introduced by David Kazhdan and George Lusztig in the late 1970s. These polynomials have a deep connection to algebraic geometry, particularly to the cohomology of flag varieties.

The definition of Kazhdan-Lusztig polynomials relies on a combinatorial object called the Bruhat order on the elements of a Coxeter group. For any two elements x and y in a Coxeter group such that x is less than or equal to y in the Bruhat order, there is an associated Kazhdan-Lusztig polynomial. These polynomials have coefficients that are integers.

A significant property of Kazhdan-Lusztig polynomials is their relation to the characters of irreducible representations of certain Lie groups and algebraic groups. Specifically, they appear in the calculation of the dimensions of cohomology groups of line bundles over flag varieties. This connection has made them a powerful tool in representation theory and algebraic combinatorics.

The initial motivation for their development was to understand the structure of Verma modules for semisimple Lie algebras. The coefficients of the Kazhdan-Lusztig polynomials encode information about the subquotients of these modules. Over time, their importance has expanded to various areas of mathematics, including the representation theory of finite groups of Lie type and the study of Schubert calculus. While their combinatorial definition is elegant, their deeper meaning and applications continue to be an active area of research.