TY - JOUR

T1 - Coxeter Combinatorics and Spherical Schubert Geometry

AU - Hodges, Reuven

AU - Yong, Alexander

N1 - Funding Information:
ward Richmond, and John Shareshian for helpful discussions. We thank David Brewster, Jiasheng Hu, and Husnain Raza for writing useful computer code (in the NSF RTG funded ICLUE program). We are grateful to the anonymous referee for their detailed comments which significantly improved the final presentation. We used the Maple packages ACE and Coxeter/Weyl in our investigations. AY was partially supported by a Simons Collaboration Grant, and an NSF RTG grant. RH was partially supported by an AMS-Simons Travel Grant.
Publisher Copyright:
© 2022 Heldermann Verlag.

PY - 2022

Y1 - 2022

N2 - 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 and A. Petukhov, M. Can and R. Hodges, R. Hodges and V. Lakshmibai, P. Karuppuchamy, P. Magyar and J. Weyman and A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. Lascoux and M.P. Schützenberger, multiplicity-freeness, and split-symmetry in algebraic combinatorics. Thereby, we invoke theorems of A. Kohnert, V. Reiner and M. Shimozono, and C. Ross and A. Yong.

AB - 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 and A. Petukhov, M. Can and R. Hodges, R. Hodges and V. Lakshmibai, P. Karuppuchamy, P. Magyar and J. Weyman and A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. Lascoux and M.P. Schützenberger, multiplicity-freeness, and split-symmetry in algebraic combinatorics. Thereby, we invoke theorems of A. Kohnert, V. Reiner and M. Shimozono, and C. Ross and A. Yong.

KW - key polynomials

KW - Schubert varieties

KW - spherical varieties

KW - split symmetry

UR - http://www.scopus.com/inward/record.url?scp=85136879270&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85136879270&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85136879270

SN - 0949-5932

VL - 32

SP - 447

EP - 474

JO - Journal of Lie Theory

JF - Journal of Lie Theory

IS - 2

ER -