Is the projective cover of the trivial module in characteristic $11$ for the sporadic simple Janko group $J_4$ a permutation module?

TL;DR

This work settles whether the projective cover of the trivial module for the sporadic simple group $J_4$ in characteristic $11$ is a permutation module. By computing the endomorphism algebra of the permutation module arising from the action on cosets of a maximal $11'$-subgroup and analyzing its ordinary and modular representations via orbit enumeration (ORB) and the MeatAxe, the author determines the ordinary projective indecomposable character $\Psi_{\mathbb F_G}$ and related projectives within the principal $11$-block. The $11$-modular decomposition of the endomorphism algebra is explicitly derived, revealing a four-column basic set and showing that the projective indecomposable corresponding to the trivial constituent is non-rational. Consequently, $P_{\mathbb F_{J_4}}$ is not a permutation module in characteristic $11$, marking progress toward the full $11$-modular decomposition matrix of $J_4$ and illustrating substantial computational methods for large sporadic groups.

Abstract

We determine the ordinary character of the projective cover of the trivial module in characteristic $11$ for the sporadic simple Janko group $J_4$, and answer the question posed in the title.