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For irreducible rational representations of the classical algebraic groups in characteristic p > 2, which are not equivalent to the composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order p > p has at least two Jordan blocks of size > p; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size > 1 are classified. [ABSTRACT FROM AUTHOR]

The behavior of regular unipotent elements from a subsystem subgroup of type A × A in p-restricted irreducible representations of a special linear group of rank greater than 5 over a field of characteristics p > 2 is investigated. For a certain class of such representations with locally small highest weights, it is shown that the images of these elements have Jordan blocks of all a priori possible sizes. In particular, the following is proved. Let K be an algebraically closed field of characteristics p, G = A( K), r ≥ 9, x ∈ G be a regular unipotent element from a subsystem subgroup of type A × A, and let φ be a p-restricted representation of G with highest weight $$ \sum\limits_{j = 1}^r {{a_j}{\omega_j}} $$. Set Assume that more than 6 coefficients a are note equal to p − 1 and that fo some i < r, a + a < p − 2 for p > 3 and a = a = 0 or 1 p = 3. Then the element φ( x) has Jordan blocks of all sizes from 1 to l. Bibliography: 30 titles. [ABSTRACT FROM AUTHOR]