Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

Uhi Rinn Suh; Se jin Oh

doi:10.1016/j.jalgebra.2019.06.013

Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

Uhi Rinn Suh, Se jin Oh

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

In this paper, we introduce twisted and folded AR-quivers of type A_2n+1, D_n+1, E₆ and D₄ 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 ^′(F₄ ⁽¹⁾) and U_q ^′(G₂ ⁽¹⁾).

Original language	English
Pages (from-to)	53-132
Number of pages	80
Journal	Journal of Algebra
Volume	535
DOIs	https://doi.org/10.1016/j.jalgebra.2019.06.013
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

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10.1016/j.jalgebra.2019.06.013

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title = "Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras",

abstract = "In this paper, we introduce twisted and folded AR-quivers of type A2n+1, Dn+1, E6 and D4 associated to (triply) twisted Coxeter elements. Using the quivers of type A2n+1 and Dn+1, we describe the denominator formulas and Dorey's rule for quantum affine algebras Uq ′(Bn+1 (1)) and Uq ′(C(1) n), which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (1))) using certain statistics on any folded AR-quiver of type A2n+1 (resp. Dn+1) and Dorey's rule for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (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 Uq ′(F4 (1)) and Uq ′(G2 (1)).",

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author = "Suh, {Uhi Rinn} and Oh, {Se jin}",

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: {\textcopyright} 2019 Elsevier Inc.",

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N2 - In this paper, we introduce twisted and folded AR-quivers of type A2n+1, Dn+1, E6 and D4 associated to (triply) twisted Coxeter elements. Using the quivers of type A2n+1 and Dn+1, we describe the denominator formulas and Dorey's rule for quantum affine algebras Uq ′(Bn+1 (1)) and Uq ′(C(1) n), which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (1))) using certain statistics on any folded AR-quiver of type A2n+1 (resp. Dn+1) and Dorey's rule for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (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 Uq ′(F4 (1)) and Uq ′(G2 (1)).

AB - In this paper, we introduce twisted and folded AR-quivers of type A2n+1, Dn+1, E6 and D4 associated to (triply) twisted Coxeter elements. Using the quivers of type A2n+1 and Dn+1, we describe the denominator formulas and Dorey's rule for quantum affine algebras Uq ′(Bn+1 (1)) and Uq ′(C(1) n), which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (1))) using certain statistics on any folded AR-quiver of type A2n+1 (resp. Dn+1) and Dorey's rule for Uq ′(Bn+1 (1)) (resp. Uq ′(Cn (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 Uq ′(F4 (1)) and Uq ′(G2 (1)).

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KW - r-cluster point

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Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

Abstract

Bibliographical note

Keywords

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