Some questions arising from the study of cohomology on flag varieties

Abstract

A fundamental problem at the confluence of algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties over a field of characteristic  p p . When p = 0 p=0 , the solution is given by the celebrated Borel–Weil–Bott Theorem, while for p > 0 p>0 the problem is widely open. In this note we describe a collection of open questions that arise from the study of particular cases of the general theory, focusing on their combinatorial and commutative algebra aspects.