Pattern avoidance and rational smoothness of Schubert varieties
Let \(X\) be the full flag variety of a connected simple complex algebraic group \(G\) and let \(X_w\subset X\) be a Schubert variety (associated to a Weyl group element \(w\)). If \(G\) is of type \(A_n\), \textit{V. Lakshmibai} and \textit{G. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] have shown that \(X_w\) is rationally smooth iff \(w\) avoids certain length-4 patterns. Further, \textit{V. Gasharov} has shown that (again in the \(A_n\)-case) \(X_w\) is rationally smooth iff the Poincaré polynomial \(p_w(t):= \sum_{v\leq w}t^{\ell(v)}\) (where \((\ell(v)\) is the length of \(v\)) factors into polynomials of the form \((1+t+ t^2+\cdots+ t^r)\). Now, Billey, in the paper under review, generalizes both of these results for classical groups \(G\) of type \(B\) and \(C\). As a corollary, she obtains that \(X_w\) (for \(G\) of type \(A\) and \(B\)) is rationally smooth iff the poset \([e,w]:= \{v\leq w\}\) has the Peck property (i.e., is rank symmetric, rank unimodal, and \(k\)-Sperner for all \(k\)). She also obtains a characterization for smooth \(X_w\) (for \(G\) of type \(B,C\) and \(D\)) in terms of pattern avoidance.
scientific article; zbMATH DE number 1236157
scientific article