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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-06-04080-3
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.


References [Enhancements On Off] (What's this?)

  • [AC] E. Akyldiz and J. Carrell, A generalization of the Kostant-Macdonald identity, Proc. Natl. Acad. Sci. U.S.A., 86, 1989, 3934-3937. MR 0998938 (90h:32060)
  • [BEZ] A. Björner, P. H. Edelman, and G. M. Ziegler, Hyperplane arrangements with a Lattice of Regions, Discrete and Computational Geometry, 5, 1990, 263-288. MR 1036875 (90k:51036)
  • [BC] M. Brion and J. Carrell, The equivariant cohomology ring of regular varieties, Michigan Math. J. 52 (2004), 189-203. MR 2043405 (2005h:14112)
  • [ER] P. H. Edelman and V. Reiner, Free hyperplane arrangements between $ A_{n-1}$ and $ B_n$, Math. Zeit., 215, 1994, 347-366.MR 1262522 (95b:52021)
  • [GKM] M. Goresky, R. Kottwitz, and R. MacPherson, Purity of equivalued affine Springer fibers, arXiv:math.RT/0305141.
  • [Jo] A. Joseph, A characteristic variety for the primitive spectrum of a semisimple Lie algebra, in Non-Commutative Harmonic Analysis (Lecture Notes, No. 587, Springer-Verlag, Berlin), 1977, 102-118. MR 0450350 (56:8645)
  • [Ko1] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., 81, 1959, 973-1032. MR 0114875 (22:5693)
  • [Ko2] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem., Ann. Math. (2), 74, 1961, 329-387.MR 0142696 (26:265)
  • [Ko3] B. Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight $ \rho $, Selecta Math. (N. S.) 2, 1996, 43-91. MR 1403352 (97e:17029)
  • [Ma] I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann., 199, 1972, 161-174. MR 0322069 (48:433)
  • [MPS] F. de Mari, C. Procesi, and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332, 1992, 529-534. MR 1043857 (92j:14060)
  • [OST] P. Orlik, L. Solomon, and H. Terao, On Coxeter arrangements and the Coxeter number. Complex analytic singularities, 461-477, Adv. Stud. Pure Math., 8, North-Holland, Amsterdam, 1987. MR 0894305 (88g:32020)
  • [OT] P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften, 300, Springer-Verlag, 1982.MR 1217488 (94e:52014)
  • [R] K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc., 16, 2003, 363-392.MR 1949164 (2004d:14081)
  • [So] E. Sommers, $ B$-stable ideals in the nilradical of a Borel subalgebra, Canadian Bull. Math. 48 (2005), 460-472.MR 2154088
  • [St] R. Stanley, An Introduction to Hyperplane Arrangements, www-math.mit.edu/ rstan/arrangements/
  • [Ty] J. Tymoczko, Decomposing Hessenberg varieties over classical groups, Ph.D. thesis, Princeton University, 2003.

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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: https://doi.org/10.1090/S0002-9947-06-04080-3
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

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