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B-Stable Ideals in the Nilradical of a Borel Subalgebra

Published online by Cambridge University Press:  20 November 2018

Eric N. Sommers
Eric N. Sommers*
Affiliation:
University of Massachusetts—Amherst, Amherst, MA 01003, Institute for Advanced Study, Princeton, NJ 08540 e-mail: esommers@math.umass.edu
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Abstract

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We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Keywords

20F5517B2005E99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005 , pp. 460 - 472
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Athanasiadis, C., Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. London Math. Soc. 36(2004), 294–302.Google Scholar
[2] Broer, A., Lectures on decomposition classes. In: Representation Theories and Algebraic Geometry, Nato Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Lkuwer, Dordrecht, 1998, pp. 39–83.Google Scholar
[3] Cellini, P. and Papi, P., ad-nilpotent ideals of a Borel subalgebra. J. Algebra 225(2000), 130–141.Google Scholar
[4] Cellini, P. and Papi, P., ad-nilpotent ideals of a Borel subalgebra. II. J. Algebra 258(2002), 112–121.Google Scholar
[5] Douglass, J. M., The adjoint representation of a reductive group and hyperplane arrangements. Represen. Theory 3(1999), 444–456.Google Scholar
[6] Fan, C. K., Euler characteristic of certain affine flag varieties. Transform. Groups 1(1996), 35–39.Google Scholar
[7] Fomin, S. and Zelevinsky, A., Y -systems and generalized associahedra. Ann.of Math. 158(2003), 977–1018.Google Scholar
[8] Haiman, M., Conjectures on the quotient ring by diagonal invariants. J. Algebraic Combin. 3(1994), 17–76.Google Scholar
[9] Headley, P., On a family of hyperplane arrangements related to the affine Weyl groups. J. Algebraic Combin. 6(1997), 331–338.Google Scholar
[10] Orlik, P. and Solomon, L., Coxeter arrangements. In: Singularities, Proc. Sympos Pure Math. 40, American Mathematical Society, Providence, RI, 1983, pp. 269–291.Google Scholar
[11] Panyushev, D., Ad-nilpotent ideals of a Borel subalgebra: generators and duality, math.RT/0303107, 2003.Google Scholar
[12] Shi, J.-Y., Sign types corresponding to an affine Weyl group. J. LondonMath. Soc.(2) 35(1987), 56–74.Google Scholar
[13] Shi, J.-Y., The number of ⊕-sign types. Quart. J. Math. Oxford Ser. (2) 48(1997), 93–105.Google Scholar
[14] Sommers, E., A family of affine Weyl group representations. Transform. Groups 2(1997), 375–390.Google Scholar
[15] Sommers, E., A generalization of the Bala-Carter theorem for nilpotent orbits. Internat. Math. Res. Notices 1998, 539–562.Google Scholar
[16] Terao, H., Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula. Invent.Math. 63(1981), 159–179.Google Scholar