Symmetric functions and Springer representations

Springer Representations on the Khovanov Springer Varieties

Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S). We show that if Xn is embedded antipodally in (S) then the natural Sn-action on (S) induces an Sn-rep...

Alcove walks, buildings, symmetric functions and representations

For a complex simple Lie algebra g, the dimension Kλμ of the μ weight space of a finite dimensional representation of highest weight λ is the same as the number of Littelmann paths of type λ and weight μ. In this paper we give an explicit construction of a path of type λ and weight μ whenever Kλμ 6= 0. This construction has additional consequences, it produces an explicit point in the building ...

Hopf Algebras, Symmetric Functions and Representations

1.1. Motivation. Much of representation theory can be unified by considering the representation theory of associative algebras. Specifically, the representation theory of Lie algebras may be studied via the representations of universal enveloping algebras; the representation theory of finite groups studied via the representation theory of the group algebra; the representation theory of quivers ...

The symmetric group - representations, combinatorial algorithms, and symmetric functions

Geometric Satake, Springer Correspondence, and Small Representations

For a simply-connected simple algebraic group G over C, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of G, generalizing a well-known fact about GLn. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number ...