The minimal counterexample to James's conjecture

TL;DR

This paper presents the explicit construction of the smallest known counterexample to James's conjecture in the representation theory of symmetric groups, specifically in the Hecke algebra at a primitive fourth root of unity.

Contribution

It provides the first explicit minimal counterexample in the case where the quantum characteristic is not 2, along with detailed graded decomposition numbers.

Findings

Explicit minimal counterexample in Hecke algebra of rank 24

Graded decomposition numbers computed for the new counterexample

Counterexample occurs at a primitive fourth root of unity

Abstract

In 2017, Geordie Williamson proved the existence of counterexamples to James's conjecture on the decomposition matrices of symmetric groups and their Hecke algebras. The smallest counterexample detectable by Williamson's method occurs in the symmetric group $\mathfrak{S}_n$ for $n=1 \thinspace 744 \thinspace 860$, in characteristic $p=2237$. Those detected by Williamson remain the only known counterexamples to James's conjecture. In this work, we calculate an explicit new counterexample, occurring in the principal block of the Hecke algebra $\mathscr{H}_{24}$ when $q$ is a primitive fourth root of unity, and give explicit graded decomposition numbers in this case. This is the minimal rank counterexample for $e\neq 2$.