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 2- and 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 that contain the alternating group. We also investigate the Galois group of every Schubert problem on Gr(4,8), finding that each Galois group either contains the alternating group or is an imprimitive permutation group and therefore fails to be doubly transitive. These imprimitive examples show that our results are the best possible general results on double transitivity of Schubert problems.Comment: 25 page

Lieder von Vergangenheit, November 1, 1986

This is the concert program of the Lieder von Vergangenheit performance on Saturday, November 1, 1986 at 8:00 p.m., at the Concert Hall, 855 Commonwealth Avenue. Works performed were An Chloe, KV 524 by Wolfgang Amadeus Mozart, Abendempfindung, KV 523 by W. A. Mozart, Kantate, KV 619 by W. A. Mozart, Arianna a Naxos by Franz Joseph Haydn, Lebensmut, D. 937 by Franz Schubert, Nachtviolen, D. 752 by F. Schubert, Das Zügenglöcklein, D. 871 by F. Schubert, Delphine, D. 857 by F. Schubert, Du bist die Ruh, D. 776 by F. Schubert, Gretchen am Spinnrade, D. 118 by F. Schubert, and Rastlose Liebe, D. 138 by F. Schubert Digitization for Boston University Concert Programs was supported by the Boston University Humanities Library Endowed Fund

Skew Schubert polynomials

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.Comment: 10 pages, 7 figure

Schubert Calculus according to Schubert

We try to understand and justify Schubert Calculus the way Schubert did it.Comment: 17 pages in english, 7 figures. This is the english, extended version of a previously posted preprint math.AG/040928

Schubert decompositions for ind-varieties of generalized flags

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G}/\mathbf{P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces. We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian in "Direct limits of Schubert varieties and global sections of line bundles" (J. Algebra 320 (2008), 3187--3198).Comment: Keywords: Classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety. [26 pages