Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Fall 2022

Speaker:  Balázs Elek, Cornell
Title: Heaps, Crystals and Preprojective Algebra Modules
Time: 2:30 PM, Monday, November 14, 2022
Place:  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 $B_w(n\lambda)$, 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 $B_w(n\lambda)$ 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.

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