#### Document Type

Article

#### Publication Date

8-1-2006

#### Publication Title

Transactions of the American Mathematical Society

#### 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,. . .,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.

#### Volume

358

#### Issue

8

#### First Page

3493

#### Last Page

3509

#### DOI

10.1090/S0002-9947-06-04080-3

#### ISSN

00029947

#### Rights

Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.

#### Recommended Citation

Sommers, Eric and Tymoczko, Julianna, "Exponents for *B*-Stable Ideals" (2006). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.

https://scholarworks.smith.edu/mth_facpubs/111

## Comments

Peer reviewed accepted manuscript.