Discrete Geometry and Combinatorics Seminar

Balázs ElekCornell University
Heaps, Crystals and Preprojective Algebra Modules

Monday, November 14, 2022 - 2:30pm
Malott 206

Abstract: Kashiwara crystals are combinatorial gadgets associated to a representation of a reductive algebraic group that enable us to understand the structure of the representation in purely combinatorial terms. We will describe a type-independent combinatorial construction of crystals of the form Bw(nλ), using the heap associated to a fully commutative element w in the Weyl group. Then we will discuss how we can also use the heap to define a module for the preprojective algebra of the underlying Dynkin quiver. If time permits, we will discuss how we can also realize the crystal Bw(nλ) via irreducible components of the quiver Grassmannians of n copies of this module, and we will describe an explicit crystal isomorphism between the two models. This is joint work with Anne Dranowski, Joel Kamnitzer and Calder Morton-Ferguson.