Regular unipotent elements from subsystem subgroups of type $A_2$ in representations of the special linear groups

For $p>2$ odd, Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type $A_2$ in irreducible $p$-restricted representations for groups of type $A_r$ over the field of characteristic $p$, the weights of which are locally small with respect to $p$, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than $p$. This result is a part of a more common programme investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.