TY - JOUR
T1 - Coxeter Combinatorics and Spherical Schubert Geometry
AU - Hodges, Reuven
AU - Yong, Alexander
N1 - 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.
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 - Schubert varieties
KW - key polynomials
KW - spherical varieties
KW - split symmetry
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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 -