Heaps, crystals, and preprojective algebra modules

Anne Dranowski ^[1] ; Balázs Elek ^[2] ; Joel Kamnitzer ^[3] ; Calder Morton-Ferguson ^[4]
1. [1] University of Southern California
  
  University of Southern California
  
  Estados Unidos
2. [2] University of British Columbia
  
  University of British Columbia
  
  Canadá
3. [3] McGill University
  
  McGill University
  
  Canadá
4. [4] Stanford University
  
  Stanford University
  
  Estados Unidos
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
Idioma: inglés
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Resumen
- Fix a simply-laced semisimple Lie algebra. We study the crystal B(nλ), were λ is a dominant minuscule weight and n is a natural number. On one hand, B(nλ) can be realized combinatorially by height n reverse plane partitions on a heap associated to λ. On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage–Tingley, we realize B(nλ) via irreducible components of the quiver Grassmannian of n copies of this module. In this paper, we describe an explicit bijection between these two models for B(nλ) and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima’s tensor product quiver varieties.