Heaps, crystals, and preprojective algebra modules
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arXiv
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| Format: | Preprint |
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2022
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| author | Dranowski, Anne Elek, Balazs Kamnitzer, Joel Morton-Ferguson, Calder |
| author_facet | Dranowski, Anne Elek, Balazs Kamnitzer, Joel Morton-Ferguson, Calder |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_02490 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Heaps, crystals, and preprojective algebra modules Dranowski, Anne Elek, Balazs Kamnitzer, Joel Morton-Ferguson, Calder Representation Theory Quantum Algebra 17B10 (Primary), 22E57, 17B37 (Secondary) 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. |
| title | Heaps, crystals, and preprojective algebra modules |
| topic | Representation Theory Quantum Algebra 17B10 (Primary), 22E57, 17B37 (Secondary) |
| url | https://arxiv.org/abs/2202.02490 |