Exponents for $B$-stable ideals
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- by Eric Sommers and Julianna Tymoczko PDF
- Trans. Amer. Math. Soc. 358 (2006), 3493-3509 Request permission
Abstract:
Let $G$ be a simple algebraic group over the complex numbers containing a Borel subgroup $B$. Given a $B$-stable ideal $I$ in the nilradical of the Lie algebra of $B$, we define natural numbers $m_1, m_2, \dots , m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types $A_n, B_n, C_n$ and some other types. When $I = 0$, we recover the usual exponents of $G$ by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.References
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Additional Information
- Eric Sommers
- Affiliation: Department of Mathematics and Statistics, University of Massachusetts–Amherst, Amherst, Massachusetts 01003
- Email: esommers@math.umass.edu
- Julianna Tymoczko
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
- MR Author ID: 746237
- Email: tymoczko@umich.edu
- Received by editor(s): May 27, 2004
- Published electronically: March 24, 2006
- Additional Notes: The first author was supported in part by NSF grants DMS-0201826 and DMS-9729992. The authors thank Vic Reiner for a helpful discussion regarding hyperplane arrangements
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 3493-3509
- MSC (2000): Primary 20G05; Secondary 14M15, 05E15
- DOI: https://doi.org/10.1090/S0002-9947-06-04080-3
- MathSciNet review: 2218986