Coxeter combinatorics and spherical Schubert geometry

Reuven Hodges; Alexander Yong

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arXiv · 2007.09238

Coxeter combinatorics and spherical Schubert geometry

Reuven Hodges·Alexander Yong

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

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Reuven Hodges, Alexander Yong. 2020-07-17. Coxeter combinatorics and spherical Schubert geometry. https://arxiv.org/abs/2007.09238

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