Exponents for B-stable ideals

By E. Sommers, Julianna Tymoczko. 2004

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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 ,..., 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 / = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincare 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.

Subfields: Algebra and Number Theory

View Paper: https://www.semanticscholar.org/paper/41202c7bfe537cacdf9609cbc1801c5271022b45

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In Algebra and Number Theory

Citations in the corpus

A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, background): “Harada/ Julianna Tymoczko Department of Mathematics University of Iowa 14 MacLean Hall Iowa City, IA 52242-1419 USA tymoczko@uiowa.edu http://www.math.uiowa.edu/ ∼tymoczko/”
The freeness of ideal subarrangements of Weyl arrangements (2013, unknown): “1 was conjectured by Sommers and Tymoczko in [11] w here they defined and studied the ideal exponents, which is essentially the sa me as ourDP(I).”
The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, methodology): “rd basis of Cn, where µ1,µ2,...,µn are mutually distinct complex numbers. We also note that the dimensions of Hess(N,h) and Hess(S,h) have been computed explicitly in terms of the Hessenberg function [14, 45] and they coincide: dimCHess(N,h) = dimCHess(S,h) = Xn j=1 (2.8) (h(j) −j). Note that this n”
Schubert polynomials and classes of Hessenberg varieties (2007, methodology): “he corresponding Hessenberg varieties are called regular nilpotent; these have been used to compute the quantum cohomology of the ﬂag variety [Ko], [R] and they are related to hyperplane arrangements [ST]. We also consider regular nilpotent Hessenberg schemes, which may have additional nonreduced st”
Hessenberg varieties and hyperplane arrangements (2016, methodology): “3 was posed as a conjecture, and verified by Sommers and Tymoczko in [38] for types A, B, C and G2 by inductive arguments, and F4 and E6 by using computer calculation (see [38, Theorem 4.”