When is a Schubert variety Gorenstein? (Q855773)

In the variety of complete flags Flag\((\mathbb C^n)\), Schubert varieties are defined via the intersection properties with a reference flag \(\mathcal E\). A Schubert variety \(X_w\) is always locally Cohen-Macaulay, although often singular. The authors study conditions under which \(X_w\) is locally Gorenstein. In this setting, \(X_w\) is locally Gorenstein exactly when its dualizing sheaf \(\omega_{X_w}\) is locally free. Using the characterization of the dualizing sheaf of \(X_w\) given by Ramanathan, the authors find an equation, in the group of Cartier divisors of \(X_w\), which is satisfied exactly when \(X_w\) is locally Gorenstein. The existence of solutions for the equation is then translated in necessary and sufficient combinatorial conditions on the permutation \(w\) associated with \(X_w\). These condition are expressed in terms of pattern avoidance and alignment of inner corners. It turns out that \(X_w\) is locally Gorenstein if and only if it is such along its maximal singular locus. As a consequence, the authors are able to give a precise description of the dualizing line bundle of locally Gorenstein Schubert varieties.