On the decomposition numbers ofSO8+(2f)

Decomposition matrices for groups of Lie type in non-defining characteristic

“…Assuming (T ℓ ), the result in Table 1 remains true for even q. In fact, it was shown by Paolini [45] that (T ℓ ) holds for D 4 (q) in characteristic 2 and thus the unipotent decomposition matrix of D 4 (2 f ) agrees with the one given in Table 1, up to the knowledge of a 1 and c which will require (T ℓ ) for the types D 5 and D 6 .…”

On the characters of Sylow $p$-subgroups of finite Chevalley groups $G(p^f)$ for arbitrary primes

“…One of the approaches to this problem is to relate the modular representations of G with the irreducible characters of a Sylow p-subgroup U of G. Namely by inducing elements of Irr(U ) to G one gets -projective characters, which yield approximations to the -decomposition matrix of G. This is particularly important when p is a bad prime for G, in that a definition of generalized Gelfand-Graev characters is yet to be formulated. Such an approach has proved to be successful in the cases of SO 7 (q), Sp 6 (q) [12] and SO 8 (q) [22]. In order to achieve this, obtaining a suitable parametrization of the set Irr(U ) is an unavoidable step.…”

“…In particular, if ρ ∈ Irr(G) is one of the cuspidal characters F I 4 [1] or F II 4 [1] in the notation of [1, §13.9], then ρ(1) 2 = q 4 /8, and we do find irreducible characters of U of degree q 4 /8 in the family F 8 7,2 in Table 3. We lay the groundwork for a package in GAP4 [6], whose code is available at [16], in order to build a database for the generic character table of UF 4 (2 f ), in particular to find suitable replacements of generalized Gelfand-Graev characters as in [22]. Furthermore, we verify the generalization of Higman's conjecture in [9] for the group UF 4 (2 f ), namely the number of its irreducible characters is a polynomial in q = 2 f with integral coefficients.…”