B-sub-modules of Lie(G)/Lie(B) and Smooth Schubert Varieties in G/B

TL;DR

This paper establishes combinatorial criteria based on Weyl group properties to determine the smoothness of Schubert varieties in G/B, with specific conditions involving palindromic Poincaré polynomials and tangent space modules.

Contribution

It provides necessary and sufficient combinatorial conditions for smoothness of Schubert varieties in G/B, extending understanding beyond classical cases.

Findings

Schubert variety X(w) is smooth iff its Poincaré polynomial is palindromic and tangent space conditions hold.

The criteria are expressed in terms of Weyl group combinatorics and root systems.

Conditions do not fully characterize smoothness in type G_2 cases.

Abstract

Let G be a complex semi-simple linear algebraic group without G_2 factors, B a Borel subgroup of G and T a maximal torus in B. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is isomorphic, as a B-module, to Lie(G)/Lie(B). Recall that if w is an element of the Weyl group W of the pair (G,T), the Schubert variety X(w) in G/B is by definition the closure of the Bruhat cell BwB. In this note we prove that X(w) is non-singular iff the following two conditions hold: 1) its Poincar\'e polynomial is palindromic and 2) the tangent space TE(X(w)) to the set T-stable curves in X(w) through the identity is a $B$-submodule of Lie(G)/Lie(B). This gives two criteria in terms of the combinatorics of W which are necessary and sufficient for X(w) to be smooth: \sum_{x\le w} t^{\ell(x)} is palindromic, and every root of (G,T) in the convex hull of the set of negative roots whose reflection is less than w (in the Bruhat order on W) has the property that its T-weight space (in Lie(G)/Lie(B)) is contained in TE(X(w)). However, as we show by example, these conditions don't characterize the smooth Schubert varieties when G has type G_2.