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Structural Folding and Multi-Highest-Weight Subcrystals of \(B(\infty )\)

  • John M. DuselEmail author
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Abstract

We introduce a procedure to fold the structure of a crystal B of simply-laced Cartan type \({\mathscr{C}}\) by the action of an automorphism σ. This produces a crystal Bσ for the folded Langlands dual datum \({\mathscr{C}}^{\sigma \vee }\) which properly contains the well-studied \({\mathscr{C}}^{\sigma \vee }\) crystal of σ-invariant points. Our construction preserves normality and the Weyl group action, and is compatible with Kashiwara’s tensor product rule. Combinatorial properties of \(B(\infty )_{\sigma }\) reflect the structure of a subalgebra of \(U_{q}^{-}({\mathscr{C}})\), which is naturally a module over the graded-σ-fixed-point subalgebra of \(U_{q}^{-}({\mathscr{C}})\) via Berenstein and Greenstein’s quantum folding procedure. We find that \(B(\infty )_{\sigma }\) is generated by a set of highest-weight elements over the monoid of root operators. Through the Kashiwara-Nakashima-Zelevinsky polyhedral realization, the highest-weight set identifies with a commutative monoid which admits a Hilbert basis in finite type. A subset of the Weyl group called the balanced parabolic quotient is in one-to-one correspondence with the Hilbert basis for the pair \(\left ({\mathscr{C}}, {\mathscr{C}}^{\sigma \vee } \right ) = \left (D_{r}, C_{r-1} \right )\), and identifies with a proper subset of the Hilbert basis in other finite types. We obtain an explicit combinatorial description of the highest-weight set of \(B(\infty )_{\sigma }\) by establishing a connection between the action of root operators on \(B(\infty )\) and the semigroup structure in the polyhedral realization.

Keywords

Crystal bases Diagram automorphisms Weyl groups Hilbert bases 

Mathematics Subject Classification (2010)

17B37 17B10 

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Metron, Inc.RestonUSA

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