The University of Michigan Combinatorics Seminar
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.