Coxeter combinatorics and spherical Schubert geometry
Abstract
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. AvdeevA. Petukhov, M. CanR. Hodges, R. HodgesV. Lakshmibai, P. Karuppuchamy, P. MagyarJ. WeymanA. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. Lascoux M.P. Schützenberger, multiplicityfreeness, and splitsymmetry in algebraic combinatorics. Thereby, we invoke theorems of A. Kohnert, V. ReinerM. Shimozono, and C. RossA. Yong.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.09238
 Bibcode:
 2020arXiv200709238H
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 26 pages, corrections made to examples 1.4, 1.5, 2.16