Some simple modules for classical groups and p-ranks of orthogonal and Hermitian geometries

Abstract

2016-12-24T18:13:07

Oppositeness in Buildings and Simple Modules for Finite Groups of Lie Type

“…In such cases, it is possible to deduce the character of L((p − 1)ω). Numerous examples of this method were worked out in detail in [2].…”

“…The usefulness of the sum formula is based on the fact that the characters of the Weyl modules themselves are given by Weyl's Character Formula, so that the right hand side can be computed from p, R and λ. In [2] there is a detailed description of a procedure for performing this computation. We shall refer to the quantity in (13) as the Jantzen sum for V (λ).…”

“…Example 6.2. For the examples of types B ℓ , C ℓ and D ℓ , with J = I \ {1} concerning singular points in classical spaces, the p-ranks have been computed in [2] by analysis of the Weyl modules, as outlined above. The Weyl modules in question are V ((p − 1)ω 1 ) and for type C ℓ they are simple, so work is only needed for types B ℓ and D ℓ .…”

On p-Brauer characters of p′-degree and self-normalizing Sylow p-subgroups

“…If [1]. In both of the latter two cases, the Weyl module with highest weight $ has a composition series with two factors, Lð$Þ and Lð0Þ, cf.…”

New bounds for partial spreads of H(2d−1,q2) and partial ovoids of the Ree–Tits octagon

“…Crucially, the calculation of the dimension of this module can be reduced to the prime field case (see Proposition 3.1). In the prime case computations have been made for many simple modules, including those needed for this paper ( [2], [16]), as part of the representation theory of semisimple algebraic groups. In this way, we obtain the p-ranks of the oppositeness graphs for O(2 t ) and H(3, q 2 ).…”