The projective cover of the trivial module in characteristic $11$ for the sporadic simple Janko group $J_4$ revisited

TL;DR

This work fixes the remaining ambiguity in the $11$-modular decomposition of the permutation module $1_H^G$ for the largest sporadic Janko group $J_4$ by leveraging a condensation framework for induced modules together with long-orbit enumeration and table-automorphism analysis. By comparing with a well-understood maximal subgroup $U\cong U_3(11)\colon 2$ and using explicit endomorphism and idempotent techniques on condensed modules, the authors determine that the parameter $a$ must be $0$ and derive the four projective indecomposable summands for $1_H^G$ up to admissible automorphisms. A detailed analysis of class fusions, subgroup embeddings, and the action of a specific $11$-subgroup yields eight altered fusion entries relative to the ModularAtlas data, and the condensation steps identify the multiplicities needed to realize the four summands. The results advance the goal of the full $11$-modular decomposition matrix of $J_4$ and demonstrate a robust, scalable method for deep modular representation-theoretic investigations of large sporadic groups via condensation and long-orbit enumeration.

Abstract

This is a sequel to arXiv:2509.05805 [math.RT], where we have determined the $11$-modular projective indecomposable summands of the permutation character of $J_4$ on the cosets of an $11'$-subgroup of maximal order, amongst them the projective cover of the trivial module, up to a certain parameter. Here, we fix this parameter, by applying a new condensation method for induced modules which uses enumeration techniques for long orbits.