On the behaviour of small quadratic elements in representations of the special linear group with large highest weights

For almost all $p$-restricted irreducible representations of the groups $A_n(K)$ in characteristic $p>0$ with highest weights large with respect to $p$ the Jordan block structure of images of small quadratic unipotent elements in these representations is determined. It is proved that if $\varphi$ is an irreducible $p$-restricted representation of $A_n(K)$ in characteristic $p>0$ with highest weight
$$ m_1\omega_1+\ldots+m_n\omega_n, \quad \sum_{i=1}^n m_i\ge p-1, $$
not too few of the coefficients $m_i$ are less than $p-1$ and $n$ is large enough with respect to the codimension of the fixed subspace of an element $z$ under consideration, then $\varphi(z)$ has blocks of all sizes from 1 to $p$.