Cuspidal Unipotent Brauer Characters

On the decomposition numbers of the Ree groups F42(q2) in non-defining characteristic

“…The Levi subgroup L a has a unique cuspidal unipotent Brauer character (the modular Steinberg character, see [19]) and we denote the corresponding modular Harish-Chandra series of G by A 1 . The Levi subgroup L b has two cuspidal unipotent Brauer characters: the restrictions of the ordinary irreducible characters of degree Table D.3.…”

“…The principal series is abbreviated by ps. The Levi subgroup L a has a unique cuspidal unipotent Brauer character, the modular Steinberg character; see [19]. We denote the corresponding modular Harish-Chandra series of G by A 1 .…”

“…By [19,Theorem 4.2], the modular Steinberg character of G is cuspidal. So, the construction of Ψ 5 in Table B.4 implies that the projective cover of φ 21 is not a summand of the projective module corresponding to Ψ 5 .…”

“…By the decomposition matrix (1), φ 19 (1), φ 20 (1) into the right-hand side of (4) leads to a lower bound for deg(φ 21 ). Unfortunately, this bound turns out to be negative and does not give any information at all.…”

“…In this case, Theorem 3.3 determines all decomposition numbers of G, except for several decomposition numbers in the ordinary characters χ 18 , χ 19 , χ 20 , χ 21 . Thanks to the unitriangular shape of the decomposition matrix, this gives us the degrees of all irreducible Brauer characters of G except for φ 18 (1), φ 19 (1), φ 20 (1), φ 21 (1). We see…”