Abstract:

Kazhdan-Lusztig polynomials were first defined by Kazhdan and Lusztig in [Invent. Math., 53 (1979), 165-184]. Since then, many applications have been found, especially to representation theory and to the geometry of Schubert varieties. In 1987 Deodhar introduced parabolic analogues of these polynomials. These are related to their ordinary counterparts in several ways, and also play a direct role in other areas, including geometry of partial flag manifolds and the theory of Macdonald polynomials.

In this talk I consider the parabolic Kazhdan-Lusztig polynomials of the quasi-maximal quotients of the symmetric group. I first show how the elements of these quotients are encoded by ``rooted partitions'' and then I give explicit, closed combinatorial formulas for the polynomials. These are based on a special class of rooted partitions, the ``quasi-Dyck'' ones, and imply that they are always (either zero or) a power of $q$.

I conclude with some recent enumerative results on Dyck and quasi-Dyck partitions, showing a connection with random walks on regular trees.

This is partly based on a joint work with Francesco Brenti and Mario Marietti.