The University of Michigan Combinatorics Seminar
Fall 2004
September 17, 4:10-5:00, 3866 East Hall



Exponents for B-stable ideals

Julianna Tymoczko

University of Michigan


Abstract

The Poincare polynomial of the flag variety $G/B$ can be written as the sum $\sum_{w \in W} t^{\ell(w)}$, where the length of an element $w$ in the Weyl group $W$ is denoted $\ell(w)$. A beautiful classical result factors this polynomial as the product $\prod_{i=1}^{\textup{rk} G}(1 + t
+ t^2 + \cdots + t^{m_i})$ for integers $m_i$ called the exponents of $G$. These exponents can also be realized as the dual partition to the height partition, whose $i^{th}$ part $\lambda_i$ is the number of positive roots of height $i$.

In fact, each $B$-stable ideal ${I}$ in the nilradical of the Lie algebra naturally defines a set of equivalence classes $W^I$ of the Weyl group which generate a function $\sum_{w \in W^I}
t^{\ell(w)}$. This function also factors as a product $\prod_{i=1}^{\textup{rk} G}(1 + t + t^2 + \cdots +
t^{m_i^{{I}}})$ for integers $m_i^{{I}}$ called the exponents of the ideal. Ideal exponents can alternately be expressed as the dual partition to the height partition on a certain subset of positive roots. There is even an algebraic variety with this polynomial as its Poincare polynomial, namely the regular nilpotent Hessenberg variety associated to the $B$-stable ideal. We prove these claims in types $A_n$, $B_n$, and $C_n$, and discuss other cases.

This is joint work with Eric Sommers.