This talk will introduce the notion of a -kernel, due to R.P. Stanley. I will explain how -kernels can regarded as an abstraction of Kazhdan--Lusztig theory in which the collection of all Kazhdan--Lusztig polynomials is treated as a single combinatorial object in terms of an incidence algebra. The relationship between -kernels and IC ("Intersection Cohomology") bases will also be discussed.
P-kernels and IC bases