## Abstract

In this paper, we introduce twisted and folded AR-quivers of type A_{2n+1}, D_{n+1}, E_{6} and D_{4} associated to (triply) twisted Coxeter elements. Using the quivers of type A_{2n+1} and D_{n+1}, we describe the denominator formulas and Dorey's rule for quantum affine algebras U_{q} ^{′}(B_{n+1} ^{(1)}) and U_{q} ^{′}(C^{(1)} _{n}), which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for U_{q} ^{′}(B_{n+1} ^{(1)}) (resp. U_{q} ^{′}(C_{n} ^{(1)})) using certain statistics on any folded AR-quiver of type A_{2n+1} (resp. D_{n+1}) and Dorey's rule for U_{q} ^{′}(B_{n+1} ^{(1)}) (resp. U_{q} ^{′}(C_{n} ^{(1)})) applying the notion of minimal pairs in a twisted AR-quiver. By adopting the same arguments, we propose the conjectural denominator formulas and Dorey's rule for U_{q} ^{′}(F_{4} ^{(1)}) and U_{q} ^{′}(G_{2} ^{(1)}).

Original language | English |
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Pages (from-to) | 53-132 |

Number of pages | 80 |

Journal | Journal of Algebra |

Volume | 535 |

DOIs | |

State | Published - 1 Oct 2019 |

### Bibliographical note

Funding Information:This work was supported by the New Faculty Startup Fund from Seoul National University, NRF Grant # 2016R1C1B1010721 and # 2019R1F1A1059363.This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647).

Publisher Copyright:

© 2019 Elsevier Inc.

## Keywords

- Denominator formulas
- Folded AR-quivers
- Folded distance polynomials
- Longest element
- Twisted AR-quivers
- Twisted Coxeter elements
- r-cluster point