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

Ladder operators and endomorphisms in combinatorial Physics

Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we first show how the ordering of the non-commuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but row-finite, matrices, which may also be considered as endomorphisms of C[x]. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladder-operators familiar in physics

Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention