University of Sydney Algebra Seminar

Michael C. Strayer

Friday 12 June, 12-1pm, in SMRI Seminar Room (Macleay Building A12 Room 301)

Minuscule Lie theory via colored posets

The well-known minuscule representations of simple Lie algebras are irreducible highest weight representations whose weights consist of a single Weyl group orbit.  In this talk we will discuss ways to generalize the essential "minuscule" properties of these representations to include highest weight and non-highest weight representations, Kac—Moody and positive Borel representations, and finite and infinite dimensional representations.  We will also discuss the dominant \(\lambda\)-minuscule Weyl group elements of D. Peterson.  Each setting uses a construction of a partially ordered set that has been "colored" by the nodes of the associated Dynkin diagram.  We will display examples in finite, affine, and indefinite types.  We also show how to produce a concrete combinatorial basis for a representation isomorphic to \(V(m\lambda)\), the irreducible highest weight representation of highest weight \(m\lambda\), where \(\lambda\) is one of the classic minuscule weights in finite type.  This basis is indexed by \(m\)-multichains in the weight lattice of \(V(\lambda)\) and is over the integers in a certain fashion.