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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Exponents for $ B$-stable ideals


Authors: Eric Sommers and Julianna Tymoczko
Journal: Trans. Amer. Math. Soc. 358 (2006), 3493-3509
MSC (2000): Primary 20G05; Secondary 14M15, 05E15
Published electronically: March 24, 2006
MathSciNet review: 2218986
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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.


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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
Email: tymoczko@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04080-3
PII: S 0002-9947(06)04080-3
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
Article copyright: © Copyright 2006 American Mathematical Society