Estimates for the number of large composition factors in the restrictions of representations of special linear groups on subsystem subgroups of type $A_{2}$

One of the main problems of representation theory is the description of the restrictions of irreducible representations of algebraic groups to subgroups, i.e., of the branching rules for representations. H. Weyl and I. Schur obtained the classical branching rules that describe the restrictions of representations of classical algebraic groups of rank $r$ to a naturally embedded classical subgroup of rank $r$ or $r-1$ in the characteristic $0$. In a positive characteristic, obtaining such rules in explicit form in the nearest future is unlikely. Therefore it is reasonable to develop methods for studying the modular representations, which do not require the knowledge of the characters. One of the directions of such studies is finding asymptotic analogs of branching rules to subgroups of small rank. Earlier we described the restrictions of modular representations of algebraic groups to subgroups of type $A_1$. In the present paper we study the restrictions of irreducible representations of the special linear group over an algebraically closed field of positive characteristic $p$ to subsystem subgroups of type $A_2$. An estimate is obtained for the number of factors that are large with respect to the subgroup for representations of groups of rank 3 and 4.