# 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…

## Tables from this paper

## 6 Citations

Spherical Schubert varieties and pattern avoidance

- Mathematics
- 2021

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 showing…

Classification of Levi-spherical Schubert varieties

- Mathematics
- 2021

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 combinatorial…

Proper elements of Coxeter Groups

- Mathematics
- 2021

We extend the notion of proper elements to all Coxeter groups. For all infinite families of Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This…

Multiplicity-free key polynomials.

- Mathematics
- 2020

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 combinatorial…

Proper permutations, Schubert geometry, and randomness

- Mathematics
- 2020

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 is…

Slide multiplicity free key polynomials

- Mathematics
- 2020

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 which…

## References

SHOWING 1-10 OF 67 REFERENCES

Multiplicity-free key polynomials.

- Mathematics
- 2020

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 combinatorial…

Classification of Levi-spherical Schubert varieties

- Mathematics
- 2021

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 combinatorial…

Spherical Schubert varieties and pattern avoidance

- Mathematics
- 2021

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 showing…

and randomness

- J. Comb., to appear,
- 2021

Dizier

- Zero-one Schubert polynomials, Math. Z.
- 2020

Multiplicity-free key polynomials, preprint

- 2020

Proper permutations, Schubert geometry, and randomness

- Mathematics
- 2020

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 is…

Zero-one Schubert polynomials

- MathematicsMathematische Zeitschrift
- 2020

We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S…