Let \(G\) be a connected complex reductive algebraic group, \(B\) a Borel subgroup of \(G\) and \(T\) a maximal torus in \(B\). By the Bruhat decomposition, the generalized flag variety \(G/B\) is the disjoint union of the subsets \(BwB/B\) where \(w\) runs over all elements of the Weyl group \(W\) of \(G\). The subset \(BwB/B\) is called a Schubert cell, and its closure \(X_w\) is called a Schubert variety. The action of \(B\) on \(G/B\) leaves stable Schubert varieties, and in general there are multiple intermediate parabolic subgroups \(B\subset P\subset G\) that leave a given Schubert variety stable.
The article under review focuses on the question whether, given such a Schubert variety \(X_w\) and parabolic subgroup \(P\), \(X_w\) is spherical under the action of a Levi subgroup \(H\) of \(P\). The latter means that a Borel subgroup of \(H\) acts with an open orbit in \(X_w\). An obvious example is given by the full generalized flag variety \(G/B\) which, by the Bruhat decomposition again, is a spherical Schubert variety under the action of \(H=P=G\). The article studies, more precisely, a combinatorial criterion which conjecturally characterizes when such a property holds. This conjecture is proved for rank two simple groups in the paper, as well as some other special cases, and has since been proved for type A groups [Y. Gao et al., “Classification of Levi-spherical Schubert varieties”, Preprint, arXiv:2104.10101]. Without stating the technical criterion in detail here, it should be noted that the criterion depends only on the Weyl group. In particular, the definition of the combinatorial counterpart of the Schubert variety being spherical is extended in the paper to finite Coxeter systems.
A significant part of the article focuses on the type A case, that is, when \(G=GL_n\) and the Weyl group is \(\mathfrak{S}_n\). In this setting, the combinatorial counterpart of sphericality is given a conjectural reformulation in terms of a pattern avoidance property for permutations, which has since been proved by C. Gaetz [“Spherical Schubert varieties and pattern avoidance”, Preprint, arXiv:2104.03264]. The sphericality property is further studied from a polynomial ring point of view by introducing the notion of split-symmetric polynomials (polynomials in \(n\)-variables which are symmetric in subsets of variables, according to a partition of \(n\)). This point of view allows to prove the aforementioned partial results towards the conjecture, and initial steps towards an algorithmic characterization of sphericality. This approach is also used in a companion paper [R. Hodges and A. Yong, “Multiplicity-free key polynomials”, Preprint, arXiv:2007.09229].