Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Abstract

Let G G be a classical group with natural module V V over an algebraically closed field of good characteristic. For every unipotent element u u of G G , we describe the Jordan block sizes of u u on the irreducible G G -modules which occur as composition factors of V ⊗ V ∗ V \otimes V^* , ∧ 2 ( V ) \wedge ^2(V) , and S 2 ( V ) S^2(V) . Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of u u , for which recursive formulae are known.