RESTRICTIONS OF MODULAR IRREDUCIBLE REPRESENTATIONS OF THE SPECIAL LINEAR GROUP WITH HIGHEST WEIGHTS LARGE ENOUGH WITH RESPECT TO THE CHARACTERISTIC TO SMALL NATURAL SUBGROUPS ARE NOT COMPLETELY REDUCIBLE

Restrictions of irreducible representations of the special linear grou p SLr+1(K) over an algebraically closed field K of characteristic p > 0 to naturally embedded subgroups of small ranks are considered. It is proved that if phi is an irreducible representation of this group wit h highest weight Sigma(i=1)(r) a(i)omega(i), all a(i) < p and Sigma(i= 1)(r) a(i) greater than or equal to 2p-1, then the restriction of phi to a naturally embedded subgroup of type A and rank n cannot be comple tely reducible provided n less than or equal to r/2. A conjecture that a similar result holds if Sigma(i=1)(r) a(i) greater than or equal to p is formulated. Recent results of J. Brundan, A. S. Kleshchev and th e author imply that for r large enough there exist irreducible represe ntations cp with all a(i) < p and Sigma(i=1)(r) arbitrarily large with respect to p wheose restrictions to SLr(K) are completely reducible.