An update on Haiman's conjectures

TL;DR

This paper investigates Haiman's conjecture on the reducibility of Kazhdan-Lusztig basis elements in the Hecke algebra, providing a geometric proof for smooth cases and counterexamples for singular cases.

Contribution

It offers a geometric proof for smooth permutations and demonstrates counterexamples for singular permutations, clarifying the conjecture's scope.

Findings

Proved the conjecture for smooth permutations using geometric methods.

Counterexamples show the conjecture does not hold for singular permutations.

Clarified the conditions under which Kazhdan-Lusztig basis elements are reducible.

Abstract

We revisit Haiman's conjecture on the relations between characters of Kazdhan-Lusztig basis elements of the Hecke algebra over the symmetric group. The conjecture asserts that, for purposes of character evaluation, any Kazhdan-Lusztig basis element is reducible to a sum of the simplest possible ones (those associated to so-called codominant permutations). When the basis element is associated to a smooth permutation, we are able to give a geometric proof of this conjecture. On the other hand, if the permutation is singular, we provide a counterexample.