On the behaviour of unipotent elements from subsystem subgroups of small ranks in irreducible representations of the classical algebraic groups in positive characteristic

T. S. Busel ; I. D. Suprunenko

Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 117-129

Résumé

In the paper we considered some results on determining the Jordan block sizes (disregarding their multiplicities) for the images of unipotent elements from subsystem subgroups of small ranks in modular irreducible representations of the classical algebraic groups. The principal attention is given to regular unipotent elements from subsystem subgroups of type $A_3$ and $A_5$ or $C_2$ and $C_3$ in representations of groups of types $A_n$ or $C_n$, respectively. For $p$-restricted irreducible representations, it is proved that the images of such elements have Jordan blocks of all a priori possible sizes if some sequences of consecutive coefficients of the highest weight satisfy certain special conditions.