Corpus ID: 220646738

Coxeter combinatorics and spherical Schubert geometry

@article{Hodges2020CoxeterCA,
  title={Coxeter combinatorics and spherical Schubert geometry},
  author={Reuven Hodges and Alexander Yong},
  journal={arXiv: Representation Theory},
  year={2020}
}
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N… Expand

Tables from this paper

Classification of Levi-spherical Schubert varieties
A Schubert variety in the complete flag manifold GLn/B is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has a dense orbit. We give a combinatorialExpand
Spherical Schubert varieties and pattern avoidance
A normal variety X is called H-spherical for the action of the complex reductive group H if it contains a dense orbit of some Borel subgroup of H . We resolve a conjecture of Hodges–Yong by showingExpand
Multiplicity-free key polynomials.
The key polynomials, defined by A. Lascoux-M.-P. Sch\"utzenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorialExpand
Slide multiplicity free key polynomials
Schubert polynomials are refined by the key polynomials of Lascoux-Schutzenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine whichExpand
Proper permutations, Schubert geometry, and randomness
We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation isExpand

References

SHOWING 1-10 OF 59 REFERENCES
Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences
Let Lw be the Levi part of the stabilizer Qw in GLN (for left multiplication) of a Schubert variety X(w) in the Grassmannian Gd,N. For the natural action of Lw on ℂ[X(w)]$\mathbb {C}[X(w)]$, theExpand
Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry
In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was usedExpand
ARITHMETIC COHEN-MACAULAYNESS AND ARITHMETIC NORMALITY FOR SCHUBERT VARIETIES
Let G be a semisimple, simply connected Chevalley group over a field k. Let T be a maximal torus, B, a Borel subgroup D T and W, the Weyl group of G (relative to T). Let P be a maximal parabolicExpand
Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties
A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showingExpand
Schubert polynomials and quiver formulas
Fulton's universal Schubert polynomials [F3] represent degeneracy loci for morphisms of vector bundles with rank conditions coming from a permutation. The quiver formula of Buch and Fulton [BF]Expand
Rook placements and cellular decomposition of partition varieties
  • K. Ding
  • Computer Science, Mathematics
  • Discret. Math.
  • 1997
TLDR
It is proved that the cell structure of a partition variety is in one-to-one correspondence with rook placements on a Ferrers board defined by a corresponding partition, enabling one to characterize the geometric attachment between a cell and the closure of another cell combinatorially. Expand
On a Class of Richardson Varieties
We investigate Schubert varieties that are spherical under a Levi subgroup action in a partial flag variety. We show that if the Schubert variety is nonsingular, then it is a spherical $L$-variety,Expand
Multiplicity-free key polynomials.
The key polynomials, defined by A. Lascoux-M.-P. Sch\"utzenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorialExpand
Multiplicity-Free Schubert Calculus
Abstract Multiplicity-free algebraic geometry is the study of subvarieties $Y\,\subseteq \,X$ with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of $\left[ YExpand
Grothendieck polynomials and quiver formulas
Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structureExpand
...
1
2
3
4
5
...