Wild blocks of type $A$ Hecke algebras are strictly wild

TL;DR

The paper establishes that, for quantum characteristic $e\ge 3$, every wild block of type $A$ Hecke algebras with weight at least $2$ is strictly wild, with a possible exception for the weight-$2$ Rouquier block when $e=3$. It achieves this by a reduction to submatrices of graded decomposition numbers, using runner removal, Scopes equivalences, and analysis of abacus partitions to produce characteristic-free submatrices that embed strictly wild quivers like $A_3^{(1)\wedge}$ or $D_4^{(1)\wedge}$. The methods extend to $q$-Schur algebras, giving a stronger result that all wild blocks are strictly wild for $e\ge 3$. The work combines a detailed case analysis for weights $2$ and $3$ with a general weight $\ge 4$ argument via $e$-quotients and row-removal, providing a comprehensive classification of strict wildness in this setting and strengthening Erdmann–Nakano’s wildness results.

Abstract

We prove that all wild blocks of type $A$ Hecke algebras with quantum characteristic $e \geqslant 3$ -- i.e. blocks of weight at least $2$ -- are strictly wild, with the possible exception of the weight $2$ Rouquier block for $e = 3$. As a corollary, we show that for $e \geqslant 3$, all wild blocks of the $q$-Schur algebras are strictly wild, without exception.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Remark
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• Corollary 3.2