Smooth Schubert varieties in \(G/B\) and \(B\)-submodules of \(\mathfrak{g}/\mathfrak{b}\) (Q649078)

Let \(G\) be a complex semisimple algebraic group without \(G_2\)-factors and let \(B\subset G\) and \(T\subset B\) be a Borel subgroup and maximal torus respectively. Let \(W\) be the Weyl group of \(G\). For any \(w\in W\), let \(X(w)\subset G/B\) be the associated Schubert variety, which is the closure of the Bruhat cell \(Bwb/B\) inside \(G/B\). The main result of the paper asserts that the Schubert variety \(X(w)\) is smooth if and only if the following two conditions are satisfied: (a) its Poincaré polynomial is palindromic, and (b) the span of the tangent spaces at the identity \(e\), inside the Zariski tangent space of \(X(w)\) at \(e\), of all the \(T\)-stable curves through \(e\) in \(X(w)\) is a \(B\) submodule of \({\mathfrak g}/{\mathfrak b}\), where \({\mathfrak g}\) (resp. \({\mathfrak b})\) is the Lie algebra of \(G\) (resp. \(B\)). The proof uses the notion of Stellar root systems due to Billey-Postnikov. The condition (a) is equivalent to (by earlier results of Carrell and Peterson) the condition that \(X(w)\) is rationally smooth. The condition (b) can be interpreted as a `pattern avoidance' condtion. As a corollary, Carrell obtains that \(X(w)\) is smooth if and only if \(X(w^{-1})\) is smooth.