The $F$-pure threshold of a Schubert cycle

The $F$-pure threshold is the characteristic $p$ counter part of the log canonical threshold in characteristic zero. It is a numerical invariant associated to the singularities of a variety, hence computing its value is important. We give a closed formula for the $F$-pure threshold of the irrelevant maximal ideal of Schubert cycles, which are the homogeneous coordinate rings of Schubert subvarieties of a Grassmannian. The main point of the computation is to give an explicit formula for the $a$-invariant of a Schubert cycle. The derivation of both formulas is made possible through the combinatorics of the underlying poset of these rings.