Journal of Lie Theory 32 (2022), No. 2, 447--474
Copyright Heldermann Verlag 2022
Coxeter Combinatorics and Spherical Schubert Geometry
Reuven Hodges
Dept. of Mathematics, University of California at San Diego, La Jolla, U.S.A.
rhodges@ucsd.edu
Alexander Yong
Dept. of Mathematics, University of Illinois, Urbana-Champaign, U.S.A.
ayong@illinois.edu
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.
Keywords: Schubert varieties, spherical varieties, key polynomials, split symmetry.
MSC: 14M15; 05E05, 05E10, 14L30.
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