On Alternating and Symmetric Groups as Galois Groups

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 tableau...

Double Transitivity of Galois Groups in Schubert Calculus of Grassmannians

We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We use these results to give a new proof that Schubert problems on Grassmannians of 2-planes have Galois groups t...

Alternating group coverings of the affine line for characteristic two

Unramified coverings of the affine line in characteristic two are constructed having alternating groups as Galois groups. The proof uses Jacobson’s criterion for the Galois group of an equation to be contained in the alternating group. Alternative proofs use the Berlekamp discriminant or the Revoy discriminant. These are related to the Arf invariant.

Finite Groups with p-Sylow Coverings

Labelling the character tables of symmetric and alternating groups

Let X be a character table of the symmetric group Sn. It is shown that unless n = 4 or n = 6, there is a unique way to assign partitions of n to the rows and columns of X so that for all λ and ν, Xλν is equal to χ(ν), the value of the irreducible character of Sn labelled by λ on elements of cycle type ν. Analogous results are proved for alternating groups, and for the Brauer character tables of...