Equivalence classes for the mu-coefficient of Kazhdan-Lusztig polynomials in S_n

TL;DR

This paper investigates the structure of Kazhdan-Lusztig mu-coefficients in symmetric groups, proposing equivalence classes to explain value scarcity and providing computed data for S_10 and S_11.

Contribution

It introduces the concept of equivalence classes for mu-coefficients and conjectures a 'crosshatch' pair within each class, supported by computational results.

Findings

Equivalence classes help explain the limited distinct mu-values.

Conjecture that each class contains a 'crosshatch' pair.

Computed mu-values for S_10 and S_11 groups.

Abstract

We study equivalence classes relating to the Kazhdan-Lusztig mu(x,w) coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a "crosshatch" pair. We also compute the values attained by mu(x,w) for the permutation groups S_10 and S_11.