Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The minimal counterexample to James's conjecture

Liron Speyer

TL;DR

The paper addresses James's conjecture on the characteristic-independence of decomposition matrices for Hecke algebras. It uses cyclotomic KLR algebras and graded decomposition numbers, together with GAP-based Gram-matrix computations, to exhibit an explicit minimal counterexample in the principal block of $\mathscr{H}_{24}$ at $e=4$, $p=7$. It shows that $d_{\lambda,\mu}^0(v)=3v^2$ in characteristic $0$ but $d_{\lambda,\mu}^7(v)=3v^2+1$ for the chosen partitions, and that the corresponding weight-space Gram matrix has a rank deficiency, violating James's conjecture. The result highlights that the abelian-defect restriction does not suffice for consistency of simple-module dimensions and points to a richer structure of decomposition numbers in small-rank blocks, with a broader landscape of counterexamples in higher ranks.

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 $e\neq 2$.

The minimal counterexample to James's conjecture

TL;DR

The paper addresses James's conjecture on the characteristic-independence of decomposition matrices for Hecke algebras. It uses cyclotomic KLR algebras and graded decomposition numbers, together with GAP-based Gram-matrix computations, to exhibit an explicit minimal counterexample in the principal block of at , . It shows that in characteristic but for the chosen partitions, and that the corresponding weight-space Gram matrix has a rank deficiency, violating James's conjecture. The result highlights that the abelian-defect restriction does not suffice for consistency of simple-module dimensions and points to a richer structure of decomposition numbers in small-rank blocks, with a broader landscape of counterexamples in higher ranks.

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 for , in characteristic . 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 when is a primitive fourth root of unity, and give explicit graded decomposition numbers in this case. This is the minimal rank counterexample for .
Paper Structure (6 sections, 1 theorem, 8 equations)

This paper contains 6 sections, 1 theorem, 8 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Hecke algebras and their decomposition matrices
  4. Cyclotomic KLR algebras and their graded decomposition matrices
  5. Specht modules over KLR algebras
  6. The minimal counterexample to James's conjecture

Key Result

Theorem 1.1

Let $e=4$, $\lambda=(5,4,3^2,2,1^7)$, and $\mu=(1^{24})$. Then in characteristic $0$, the graded decomposition number $d_{\lambda,\mu} (v)$ is equal to $3v^2$, while in characteristic $7$ it is equal to $3v^2+1$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Conjecture 2.1
  • Conjecture 2.2
  • Conjecture 3.1