65 research outputs found
Torified varieties and their geometries over F_1
This paper invents the notion of torified varieties: A torification of a
scheme is a decomposition of the scheme into split tori. A torified variety is
a reduced scheme of finite type over that admits a torification. Toric
varieties, split Chevalley schemes and flag varieties are examples of this type
of scheme. Given a torified variety whose torification is compatible with an
affine open covering, we construct a gadget in the sense of Connes-Consani and
an object in the sense of Soul\'e and show that both are varieties over \F_1
in the corresponding notion. Since toric varieties and split Chevalley schemes
satisfy the compatibility condition, we shed new light on all examples of
varieties over \F_1 in the literature so far. Furthermore, we compare
Connes-Consani's geometry, Soul\'e's geometry and Deitmar's geometry, and we
discuss to what extent Chevalley groups can be realized as group objects over
\F_1 in the given categories.Comment: 34 pages; includes some clarifications of the definitions from the
previous versio
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
Quantum Pieri rules for isotropic Grassmannians
We study the three point genus zero Gromov-Witten invariants on the
Grassmannians which parametrize non-maximal isotropic subspaces in a vector
space equipped with a nondegenerate symmetric or skew-symmetric form. We
establish Pieri rules for the classical cohomology and the small quantum
cohomology ring of these varieties, which give a combinatorial formula for the
product of any Schubert class with certain special Schubert classes. We also
give presentations of these rings, with integer coefficients, in terms of
special Schubert class generators and relations.Comment: 59 pages, LaTeX, 6 figure
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