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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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


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:
  • Julianna Tymoczko
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
  • MR Author ID: 746237
  • Email:
  • 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:
  • MathSciNet review: 2218986