The Projective Indecomposable Modules of SL(2, p ⁿ )

The maximal subgroups of positive dimension in exceptional algebraic groups

“…Now recall that y = 1. Also, (1 ) shows that e + y + s ≤ 4. Since we have already observed that s ≤ 2, this is a contradiction, completing the proof of the Sublemma.…”

“…Proof First observe that from [1] we see that the relevant Weyl modules have the following structure, where in each case the module is uniserial: W (2) = 2|0, W (4) = 4|0|2, and W (6) = 6|4|0.…”

“…Consider first case (1). Here [31, p.65] shows that there is a 1-dimensional torus T < A 1 A 5 , projecting to a torus in a regular A 1 in each factor and having the same weights on L as T .…”

“…In cases (18), (20) and (21) 1 . In neither case can we obtain a compatible labelling of the E 6 factor.…”

“…From a consideration of the weights on long roots we see that if V = V G (λ 1 ) then V ↓ X = 8/6 4 /4/0 6 . The Weyl module W X (8) is uniserial with the following structure (see [1]): 8|0|4|6. Consider a weight vector v ∈ V of weight 8 and the corresponding cyclic module Xv , which is an image of this Weyl module.…”