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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]

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [ n],where [ n] is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2. For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

For almost all p-restricted irreducible representations of the group An(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of An(K) with highest weight , not too few of the coefficients mi 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 φ(z) has blocks of all sizes from 1 to p. Bibliography: 15 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]

In this paper, branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure of the fundamental Weyl modules is refined. 2000 Mathematics Subject Classification 20G05. [ABSTRACT FROM AUTHOR]

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of types B n, C n, and D n in positive characteristic p are found under some minor restrictions on p. If G = B n( K), C n( K), or D n( K), n ≥ 8, p > 2 for types B n and D n and p > 7 for type C n, then either the maximal weight multiplicity for an irreducible representation of G is at least n − 7 or all its weight multiplicities are equal to 1. [ABSTRACT FROM AUTHOR]