Q-data and Representation Theory of Untwisted Quantum Affine Algebras

Fujita, Ryo; Oh, Se‐jin

doi:10.1007/s00220-021-04028-8

Cited by 23 publications

(34 citation statements)

References 41 publications

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“…We shall describe and by using - data [FO21]. A -datum generalizes a Dynkin quiver with a height function, which provides a uniform way of describing the Hernandez–Leclerc category .…”

Section: Preliminariesmentioning

confidence: 99%

“…A -datum generalizes a Dynkin quiver with a height function, which provides a uniform way of describing the Hernandez–Leclerc category . Our brief explanation follows [FO21, § 3] (see also [FHOO21, § 4] and [KKOP21, § 6]). Let be an affine Kac–Moody algebra and let be the simply laced finite-type Lie algebra corresponding to the affine type of in table (4.5).…”

Section: Preliminariesmentioning

confidence: 99%

“…We define an Dynkin diagram automorphism of as follows. For or type () we set , and for the remaining types is defined as follows (see [FO21, § 3.1]). …”

Section: Preliminariesmentioning

confidence: 99%

“…For a -datum associated to , let The generalized - Coxeter element associated with is defined in [FO21, Definition 3.33] and can be understood as a generalization of a Coxeter element. Here is the Weyl group of .…”

Section: Preliminariesmentioning

confidence: 99%

“…In this paper, we assume further that the height function satisfies Let be a total order of satisfying . Then we have (see [FO21, § 3.6] and also [FHOO21, Proposition 4.4] for more details).…”

Section: Preliminariesmentioning

confidence: 99%

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Simply laced root systems arising from quantum affine algebras

Kashiwara

Oh²,

Park³

2022

Compositio Math.

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Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$ , and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$ -modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$ ) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$ ) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ , respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.

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“…We shall describe and by using - data [FO21]. A -datum generalizes a Dynkin quiver with a height function, which provides a uniform way of describing the Hernandez–Leclerc category .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…We define an Dynkin diagram automorphism of as follows. For or type () we set , and for the remaining types is defined as follows (see [FO21, § 3.1]). …”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Simply laced root systems arising from quantum affine algebras

Kashiwara

Oh²,

Park³

2022

Compositio Math.

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Equivariant multiplicities via representations of quantum affine algebras

Casbi

Lǐ

2022

Sel. Math. New Ser.

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For any simply-laced type simple Lie algebra $$\mathfrak {g}$$ g and any height function $$\xi $$ ξ adapted to an orientation Q of the Dynkin diagram of $$\mathfrak {g}$$ g , Hernandez–Leclerc introduced a certain category $$\mathcal {C}^{\le \xi }$$ C ≤ ξ of representations of the quantum affine algebra $$U_q(\widehat{\mathfrak {g}})$$ U q ( g ^ ) , as well as a subcategory $$\mathcal {C}_Q$$ C Q of $$\mathcal {C}^{\le \xi }$$ C ≤ ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring $$\mathbb {C}[\textbf{N}]$$ C [ N ] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $${\widetilde{D}}_{\xi }$$ D ~ ξ on a torus $$\mathcal {Y}^{\le \xi }$$ Y ≤ ξ containing the image of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) under the truncated q-character morphism. We prove that the restriction of $${\widetilde{D}}_{\xi }$$ D ~ ξ to $$K_0(\mathcal {C}_Q)$$ K 0 ( C Q ) coincides with the morphism $$\overline{D}$$ D ¯ recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in $$\mathcal {C}_Q$$ C Q , as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of $$\overline{D}$$ D ¯ allows us to prove a conjecture by the first author on the distinguished values of $$\overline{D}$$ D ¯ on the flag minors of $$\mathbb {C}[\textbf{N}]$$ C [ N ] . We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra $$\overline{\mathcal {A}}_Q$$ A ¯ Q as a subquotient of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) naturally containing $$\mathbb {C}[\textbf{N}]$$ C [ N ] , and suggest the existence of an analogue of the Mirković–Vilonen basis in $$\overline{\mathcal {A}}_Q$$ A ¯ Q on which the values of $${\widetilde{D}}_{\xi }$$ D ~ ξ may be interpreted as certain equivariant multiplicities.

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Poles of finite-dimensional representations of Yangians

Gautam

Wendlandt

2022

Sel. Math. New Ser.

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Q-data and Representation Theory of Untwisted Quantum Affine Algebras

Cited by 23 publications

References 41 publications

Simply laced root systems arising from quantum affine algebras

Simply laced root systems arising from quantum affine algebras

Equivariant multiplicities via representations of quantum affine algebras

Poles of finite-dimensional representations of Yangians

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