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Cluster structures on schemes of bands
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abstract
We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting.
Forward citations
Cited by 2 Pith papers
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Category $\mathcal{O}$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety
The Grothendieck ring of integral category O for shifted Yangians equals the Cox ring of a new open bi-infinite Bott-Samelson pro-variety, proving the Hernandez-Zhang and Frenkel-Hernandez/GHL conjectures in simply-la...
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An introduction to $(G,c)$-bands
A discrete Miura transformation built from (G,c)-bands is shown to reproduce the q-characters of quantum affine algebras of types A, D, E, verifying a conjecture of Frenkel and Reshetikhin.
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