Decomposition numbers ofSO7(q)andSp

Abstract

We complete the ℓ-modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups SO 7 (q) and the symplectic groups Sp 6 (q) for all prime powers q and all odd primes ℓ different from the defining characteristic.

Decomposition matrices for low-rank unitary groups

“…The modular counterpart of this property is proved in [1,Proposition 8.10]. It gives some control on the multiplicity of projective covers of cuspidal modules in the virtual module R w , and has already been shown to be a powerful tool to determine decomposition numbers in [6,20]. In this paper, we combine this new ingredient with standard methods involving Harish-Chandra induction from proper Levi subgroups, but also Harish-Chandra restriction from suitable groups of larger rank, tensor products of characters and the known fact that the decomposition matrix is unitriangular to determine, up to only few unknown entries when n = 10, the decomposition matrices of unipotent -blocks of SU n (q) for n 10 and a unitary prime, as well as the repartition of the simple modules into -modular Harish-Chandra series.…”

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

“…In contrast with [18,Theorem 4], we have that two cores with the same form [z, m, c] are not necessarily isomorphic. Namely we see from Table 3 that in UF 4 (2 f ) the cores of the form [4,8,4] split into at least two isomorphism classes, since the sets of the form Irr(X S ) Z are evidently different, and so do cores of the form [4,12,9] and [5, 9,4].…”

“…Two examples of graphs as described above in the case G = F 4 (2 f ). On the left, S and Z correspond to the [4,12,9]-core associated to F 7,1 . On the right, S and Z are taken with respect to the only [5, 11, 6]-core; here |T | = 1, and ∆ ∈ T is the graph with edges {α 4 , α 6 } and {α 6 , α 1 }.…”

“…• the family F 4,2 corresponding to a [4,8,4]-core, which provides the smallest example where the expression of the cardinality of a family F i is PORC, but not polynomial, • the family F 7,2 corresponding to a [4,12,9]-core, where X is not a subgroup, which presents a more intricate branching and contains characters of degree q 4 /8, and • the family F 11 corresponding to a [6, 10, 4]-core, whose study requires the determination of solutions of complete cubic equations over F q . The difficulty of the computations related to all other families in Table 3 is bounded by that of the three families described above.…”

“…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.…”