Geodesic
Coxeter Combinatorics and Spherical Schubert Geometry
Journal of Lie theory, Tome 32 (2022) no. 2, pp. 447-474
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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.
Classification : 14M15, 05E05, 05E10, 14L30
Mots-clés : Schubert varieties, spherical varieties, key polynomials, split symmetry 
@article{JLT_2022_32_2_JLT_2022_32_2_a6,
     author = {R. Hodges and A. Yong},
     title = {Coxeter {Combinatorics} and {Spherical} {Schubert} {Geometry}},
     journal = {Journal of Lie theory},
     pages = {447--474},
     year = {2022},
     volume = {32},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2022_32_2_JLT_2022_32_2_a6/}
}
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R. Hodges; A. Yong. Coxeter Combinatorics and Spherical Schubert Geometry. Journal of Lie theory, Tome 32 (2022) no. 2, pp. 447-474. http://geodesic.mathdoc.fr/item/JLT_2022_32_2_JLT_2022_32_2_a6/