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In order to gain better understanding of the multiplication in the integral cohomology of the complex Grassmann manifold G k , n ( C ) (in the Borel picture) a minimal strong Gröbner basis for the ideal I k , n determining this cohomology is obtained. These results are applied to obtain recurrence relations among Kostka numbers which completely determine these numbers. Corresponding results for real Grassmann manifolds are also presented. [ABSTRACT FROM AUTHOR]

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. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality. [ABSTRACT FROM AUTHOR]