Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

Naoi, Katsuyuki

doi:10.1093/imrn/rnw183

Cited by 19 publications

(43 citation statements)

References 51 publications

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“…As far as the X = M conjecture is concerned, there is an alternative proof by Naoi [Nao12] for type A (1) n and D (1) n using representation theory of Kirillov-Reshetikhin modules for a current algebra.…”

mentioning

confidence: 99%

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

2018

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We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov-Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types A (1) n or D (1) n , whose bijections have already been established. As a consequence we settle the X = M conjecture in full generality for nonexceptional types. Furthermore, the bijection extends to a classical crystal isomorphism and sends the combinatorial R-matrix to the identity map on rigged configurations.

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confidence: 99%

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

2018

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“…By Proposition 4.1, Φ is a well-defined injection. By [FK08] or [Nao12] we have |P(B, λ)| = |RC(L, λ)|, which proves that Φ is a bijection.…”

Section: Rc(l)mentioning

confidence: 79%

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

et al. 2017

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Abstract. We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov-Reshetikhin crystals of type D (1) n in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type D (1) n . In addition, we establish that the bijection is a classical crystal isomorphism.

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“…These modules are the graded limits of the KR modules for quantum affine algebras (see [8,25,26] and references therein). It follows from [4] (see also [8,17]) that…”

Section: Fusion Productsmentioning

confidence: 99%

“…They can also be seen as minimal affinizations, in the sense of [3], having a multiple of a fundamental weight as highest weight. The graded KR modules are the so called graded limits of the original KR modules, in the sense of [25]. The proof of Theorem 2.3.2 also relies on a result from [26] which gives a presentation in terms of generators and relations for fusion products of graded KR modules.…”

Section: Introductionmentioning

confidence: 99%

On truncated Weyl modules

Fourier

Martins

Moura

2019

Communications in Algebra

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We study structural properties of truncated Weyl modules. A truncated Weyl module WN (λ) is a local Weyl module for g [t], where g is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced g. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that g = sl2 related to Demazure flags and chains of inclusions of truncated Weyl modules.Part of this work was developed while the second author was a visiting Ph.D. student at the University of Cologne. He thanks Peter Littelmann and Deniz Kus for their guidance and support during that period and the University of Cologne for hospitality. He also thanks CAPES and CNPq (SWE 203324/2014-5) for the financial support.A. M. was partially supported by CNPq grant 304477/2014-1 and Fapesp grant 2014/09310-5.

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Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

Cited by 19 publications

References 51 publications

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

On truncated Weyl modules

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Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

Cited by 19 publications

References 51 publications

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

On truncated Weyl modules

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Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types

Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affine types