%0 Journal Article
%T Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety
%J Mathematics | An Open Access Journal from MDPI
%D 2019
%R https://doi.org/10.3390/math7030295
%X Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G. In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when £¿ ¦Ë , ¦Á ¡Å £¿ = £¿ p n a £¿ 1 , ( 1 ¡Ü a ¡Ü p , n > 0 ) or £¿ ¦Ë , ¦Á ¡Å £¿ = £¿ p n £¿ r , ( r ¡Ý 2 , n ¡Ý 0 ) . We also give the corresponding results for the two dimensional modules N ¦Á ( ¦Ë ) . These results will help us understand the representations of S L 3 ( k ) in the given cases. View Full-Tex
%U https://www.mdpi.com/2227-7390/7/3/295