Galois Groups of Schubert Problems of Lines Are at Least Alternating

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. The remaining cases use an integral formula for Kostka numbers of two-rowed tableaux which comes from their realization as differences of certain polynomial coefficients that generalize binomial coefficients. This rewrites the inequality as an integral, which we estimate to establish the inequality. Introduction Galois (monodromy) groups of problems from enumerative geometry were first treated by Jordan in 1870 [8], who studied several classical problems with intrinsic structure, showing that their Galois group was not the full symmetric group on the set of solutions to the enumerative problem. Others [14, 21] refined this work, which focused on the equations for the enumerative problem. This line of inquiry remained dormant untill a 1977 letter of Serre to Kleiman [11, p. 325]. The modern, geometric, theory began with Harris [7], who showed that the algebraic Galois group is equal to a geometric monodromy group and determined the Galois groups of several classical problems, including many whose Galois group is equal to the full symmetric group. In general, we expect that the Galois group of an enumerative problem is the full symmetric group and when it is not the geometric problem possesses some intrinsic structure. Despite this, there are relatively few enumerative problems whose Galois group is known. For a discussion, see Harris [7] and Kleiman [11, pp. 356-7]. The Schubert calculus of enumerative geometry [10] is a method to compute the number of solutions to Schubert problems, which are a class of geometric problems involving linear subspaces. The algorithms of Schubert calculus reduce the enumeration to combinatorics. For example, the number of solutions to a Schubert problem involving lines is a Kostka number for a rectangular partition with two parts. This well-understood class of problems provides a laboratory with which to study Galois groups of enumerative problems. 2010 Mathematics Subject Classification. 14N15, 05E15.