101 research outputs found
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 combinatorics of Steenrod operations on the cohomology of Grassmannians
The study of the action of the Steenrod algebra on the mod 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
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
Skew Schubert polynomials
We define skew Schubert polynomials to be normal form (polynomial)
representatives of certain classes in the cohomology of a flag manifold. We
show that this definition extends a recent construction of Schubert polynomials
due to Bergeron and Sottile in terms of certain increasing labeled chains in
Bruhat order of the symmetric group. These skew Schubert polynomials expand in
the basis of Schubert polynomials with nonnegative integer coefficients that
are precisely the structure constants of the cohomology of the complex flag
variety with respect to its basis of Schubert classes. We rederive the
construction of Bergeron and Sottile in a purely combinatorial way, relating it
to the construction of Schubert polynomials in terms of rc-graphs.Comment: 10 pages, 7 figure
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