Adjustment matrices for the principal block of the Iwahori-Hecke algebra $\mathcal{H}_{5e}$

TL;DR

This paper investigates the adjustment matrix for the principal block of the Iwahori-Hecke algebra, confirming James's Conjecture in most cases and providing partial calculations for the exceptional case when e=4.

Contribution

It proves that the adjustment matrix is the identity for the principal block of $ ext{H}_{5e}$ when the characteristic is at least 5 and e≠4, and computes most entries when e=4.

Findings

Adjustment matrix is identity for e≠4 when characteristic ≥ 5.

Partial calculation of adjustment matrix entries for e=4.

Supports James's Conjecture in specific cases.

Abstract

James's Conjecture predicts that the adjustment matrix for blocks of the Iwahori-Hecke algebra of the symmetric group is the identity matrix when the weight of the block is strictly less than the characteristic of the field. In this paper, we consider the case when the characteristic of the field is greater than or equal to 5, and prove that the adjustment matrix for the principal block of $\mathcal{H}_{5e}$ is the identity matrix whenever $e\neq4$. When $e=4$, we are able to calculate all but two entries of the adjustment matrix.