Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

Francesco, P. Di; Kedem, Rinat

doi:10.1093/imrn/rnn006

Cited by 29 publications

(33 citation statements)

References 21 publications

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“…In the case g = A n , the same recursion relation also occurs in other contexts: for instance in the study of Toda flows in Poisson geometry [21] and in preprojective algebras [20]. In [12] the combinatorial Kirillov-Reshetikhin conjecture which are the completeness conjectures for the generalized Heisenberg spin chains is established.…”

Section: Introductionmentioning

confidence: 66%

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Demazure Modules, Fusion Products and Q-Systems

Chari

Venkatesh

2014

Commun. Math. Phys.

113

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In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by an |R + |-tuple of partitions ξ = (ξ α ), where α varies over a set R + of positive roots of g and we assume that they satisfy a natural compatibility condition. In the case when the ξ α are all rectangular, for instance, we prove that these modules are Demazure modules in various levels. As a consequence we see that the defining relations of Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products, defined in [15], of representations of the current algebra. We prove that the Q-system of [22] extends to a canonical short exact sequence of fusion products of representations associated to certain special partitions ξ. Finally, in the last section we deal with the case of sl2 and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra and give monomial bases for these modules.

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Section: Introductionmentioning

confidence: 66%

“…The subject also has many connections with problems arising in mathematical physics, for instance the X = M conjectures, see [1], [12], [34]. To explain this further, recall that for any simple Lie algebra there exists an associated system of equations known as a Q-system.…”

Section: Introductionmentioning

confidence: 99%

Demazure Modules, Fusion Products and Q-Systems

Chari

Venkatesh

2014

Commun. Math. Phys.

113

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“…Moreover, graded limits are important as well in view of the theory of finitedimensional graded g[t]-modules, since they provide nontrivial and probably interesting such modules. (Though the original motivation to study finite-dimensional graded g[t]modules was mainly an application to the theory of U q (Lg)-modules, they are now also of independent interest, since they have connections with problems arising in mathematical physics such as the X = M conjecture [AK07, DFK08,Nao12a], and theory of symmetric functions such as Macdonald polynomials [CI15].) In fact, in [CV15] the authors have constructed a short exact sequence of g[t]-modules as a graded limit analog of the T -system, which is a distinguished exact sequence of U q (Lg)-modules (see [Her06]).…”

Section: Introductionmentioning

confidence: 99%

Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

Naoi

2016

Int Math Res Notices

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“…Equivalently, it is just the Lie algebra of polynomial maps C → g. The study of the category of finite dimensional graded representations of the current algebra has been the subject of many articles in the recent years. See for example [1], [2], [5], [11], [15], [16], [22], [23]. One of the original motivations for the study of this category was that it is closely related to the representation theory of the corresponding quantum affine algebra.…”

Section: Introductionmentioning

confidence: 99%

Fusion Product Structure of Demazure Modules

Venkatesh

2014

Algebr Represent Theor

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Let g be a finite-dimensional complex simple Lie algebra. Given a non-negative integer ℓ, we define P + ℓ to be the set of dominant weights λ of g such that ℓΛ0 +λ is a dominant weight for the corresponding untwisted affine Kac-Moody algebra g. For the current algebra g[t] associated to g, we show that the fusion product of an irreducible g-module V (λ) such that λ ∈ P + ℓ and a finite number of special family of g-stable Demazure modules of level ℓ (considered in [15] and [16]) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the g[t]-module structure of the irreducible g-module V (ℓΛ0 + λ) as a semi-infinite fusion product of finite dimensional g[t]-modules as conjectured in [16]. As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see [7]).

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Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

Cited by 29 publications

References 21 publications

Demazure Modules, Fusion Products and Q-Systems

Demazure Modules, Fusion Products and Q-Systems

Tensor Products of Kirillov–Reshetikhin Modules and Fusion Products

Fusion Product Structure of Demazure Modules

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