The University of Michigan Combinatorics Seminar


Abstract 

The Poincare polynomial of the flag variety can be written as the sum , where the length of an element in the Weyl group is denoted . A beautiful classical result factors this polynomial as the product for integers called the exponents of . These exponents can also be realized as the dual partition to the height partition, whose part is the number of positive roots of height . In fact, each stable ideal in the nilradical of the Lie algebra naturally defines a set of equivalence classes of the Weyl group which generate a function . This function also factors as a product for integers 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 stable ideal. We prove these claims in types , , and , and discuss other cases. This is joint work with Eric Sommers. 