Search

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]

In this paper branching rules for the polynomial irreducible representations of the general linear groups in positive characteristic with highest weights labeled by partitions of the form (2a, 1b, 0c) and their restrictions to the special linear groups are found. The submodule structure of the restrictions of the corresponding irreducible modules for the group GLn(F) (or SLn(F)) to the naturally embedded subgroup GLn-1(F) (or SLn-1(F)) is determined. As a corollary, inductive systems of irreducible representations for GL∞(F) and SL∞(F) that consist of representations indicated above, are classified. The submodule structure of the relevant Weyl modules is refined. [ABSTRACT FROM AUTHOR]