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

TL;DR

This paper refines the understanding of the projective cover of the trivial module in characteristic 11 for the sporadic simple group J_4 by applying a new condensation method involving enumeration of long orbits.

Contribution

It introduces a novel condensation technique for induced modules that enables precise determination of the projective cover of the trivial module in characteristic 11 for J_4.

Findings

Fixed the previously undetermined parameter in the projective cover.

Applied a new condensation method using enumeration of long orbits.

Enhanced the understanding of modular representations of J_4.

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.