Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract

A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q=0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula

The combinatorics of Steenrod operations on the cohomology of Grassmannians

The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians

Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued

Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.Comment: 11 page

The K-theory of the Flag Variety and the Fomin-Kirillov Quadratic Algebra

We propose a new approach to the multiplication of Schubert classes in the K-theory of the flag variety. This extends the work of Fomin and Kirillov in the cohomology case, and is based on the quadratic algebra defined by them. More precisely, we define K-theoretic versions of the Dunkl elements considered by Fomin and Kirillov, show that they commute, and use them to describe the structure constants of the K-theory of the flag variety with respect to its basis of Schubert classes

The Albany Movement and the Origin of Freedom Songs

“We became visible.” This is how Bernice Johnson Reagon, a Civil Rights Movement worker, a member of the Freedom Singers, and the founder of Sweet Honey In The Rock explained how songs uplifted and inspired those blacks and whites who worked tirelessly for freedom throughout the 1950’s and 1960’s. Indeed, freedom songs in the movement gave participants the ability to stand up against their fears, express their hopes and desires, and unite the diverse range of people who participated in the movement. Reagon, now a history professor and music legend, grew up right outside of Albany, Georgia, where freedom songs first became an integral part of the Civil Rights Movement. Nestled in a land entrenched with racial segregation, the Albany campaign was notable because almost every single black member of the community became visible through his or her work. Whether it was by going to jail, marching, or attending a mass meeting, most citizens actively participated. Albany was and is still considered today to be one of the birthplaces of the mass movement for racial equality. Albany, therefore, came to symbolize for the larger struggle for black freedom not only the birthplace of a true grassroots campaign, but also the birthplace of freedom songs that would spread throughout the country and become a familiar and helpful tool for many freedom fighters