94 research outputs found
The Prism tableau model for Schubert polynomials
The Schubert polynomials lift the Schur basis of symmetric polynomials into a
basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these
polynomials. A novel aspect of this alternative to earlier results is that it
directly invokes semistandard tableaux; it does so as part of a colored tableau
amalgam. In the Grassmannian case, a prism tableau with colors ignored is a
semistandard Young tableau. Our arguments are developed from the Groebner
geometry of matrix Schubert varieties.Comment: 23 page
The geometry and combinatorics of Springer fibers
This survey paper describes Springer fibers, which are used in one of the
earliest examples of a geometric representation. We will compare and contrast
them with Schubert varieties, another family of subvarieties of the flag
variety that play an important role in representation theory and combinatorics,
but whose geometry is in many respects simpler. The end of the paper describes
a way that Springer fibers and Schubert varieties are related, as well as open
questions.Comment: 18 page
A geometric Littlewood-Richardson rule
We describe an explicit geometric Littlewood-Richardson rule, interpreted as
deforming the intersection of two Schubert varieties so that they break into
Schubert varieties. There are no restrictions on the base field, and all
multiplicities arising are 1; this is important for applications. This rule
should be seen as a generalization of Pieri's rule to arbitrary Schubert
classes, by way of explicit homotopies. It has a straightforward bijection to
other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's
puzzles.
This gives the first geometric proof and interpretation of the
Littlewood-Richardson rule. It has a host of geometric consequences, described
in the companion paper "Schubert induction". The rule also has an
interpretation in K-theory, suggested by Buch, which gives an extension of
puzzles to K-theory. The rule suggests a natural approach to the open question
of finding a Littlewood-Richardson rule for the flag variety, leading to a
conjecture, shown to be true up to dimension 5. Finally, the rule suggests
approaches to similar open problems, such as Littlewood-Richardson rules for
the symplectic Grassmannian and two-flag varieties.Comment: 46 pages, 43 figure
Gysin maps, duality and Schubert classes
We establish a Gysin formula for Schubert bundles and a strong version of the
duality theorem in Schubert calculus on Grassmann bundles. We then combine them
to compute the fundamental classes of Schubert bundles in Grassmann bundles,
which yields a new proof of the Giambelli formula for vector bundles.Comment: Version 3: published versio
The integral cohomology ring of E_8/T
We give a complete description of the integral cohomology ring of the flag
manifold E_8/T, where E_8 denotes the compact exceptional Lie group of rank 8
and T its maximal torus, by the method due to Borel and Toda. This completes
the computation of the integral cohomology rings of the flag manifolds for all
compact connected simple Lie groups.Comment: 5 pages, AMS-LaTeX, Minor errors corrected
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