293,902 research outputs found
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 be one of the ind-groups , ,
and be a splitting parabolic
ind-subgroup. The ind-variety 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 as a -orbit on
, where is any Borel ind-subgroup of
which intersects in a maximal ind-torus. A
significant difference with the finite-dimensional case is that in general
is not conjugate to an ind-subgroup of , whence
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
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