The $5$-modular representations of the Tits simple group in the principal block

Abstract

In this paper we show how to construct the 5-modular absolutely irreducible representations of the Tits simple group in the principal block, which is the only block of positive defect. Starting with the smallest nontrivial ones, all the others except one pair are obtained as constituents of tensor products of dimension at most 729. The last two we get from a permutation representation of degree 1600. We give an exact description of the construction of the first one of degree 26 by extending its restrictions to several subgroups, a method first used in the existence proof of the Janko group J4 . Using the explicit matrices obtained from the above constructions, we work out the Green correspondents and sources of all the representations and state their socle series.