On the fermionic formula and the Kirillov-Reshetikhin conjecture

By Vyjayanthi Chari. 2000

146 citations recorded by the source corpus.

The fermionic formula conjectured by Kirillov and Reshetikhin describes the decomposition (as a module for $U_q(\frak g)$) of a tensor product of multiples of of fundamental representations $W(m\lambda_i)$ of the corresponding quantum affine algebras. In this paper, we show that the conjecture is true for the modules W(m\lambda_i), if $i$ is such that the corresponding simple root occurs in the highest root of the simple Lie algebra with multiplicity at most 2. In particular, the conjecture is established for all but a few nodes for the exceptional algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3681468c2f17c535ee8c1fe23dd7146db126a561

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Citations in the corpus

Level Zero Fundamental Representations over Quantized Affine Algebras and Demazure Modules (2003, background): “[3] Vyjayanthi Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, Internat.”
Existence of Kirillov–Reshetikhin crystals for nonexceptional types (2007, result): “[4] V. Chari and A. Pressley, Quantum affine algebras and their representations, in Representations of groups, CMS Conf.”
Tensor Product Structure of Affine Demazure Modules and Limit Constructions (2004, methodology): “Chari and the Kyoto school have calculated for classical Lie-algebras and some fundamental weights for nonclassical Lie-algebras the decomposition of the Kirillov-Reshetikhin module KR(lck ωk) into irreducible Uq(g)-modules [2].”
The Kirillov-Reshetikhin conjecture and solutions of T-systems (2005, unknown): “There are many results for these conjectures and related problems (see [27] for an historic and a guide through the huge literature on this subject; to name a few of very interesting recent ones see [23, 16, 25, 3, 31, 32]), but no proof for all types.”
A categorical approach to Weyl modules (2009, background): “Moreover, since the theorem is proved for k = 1 in [ C2 ], [CM], we shall assume that k > 1. In addition, we may assume that S is the maximal ideal generated by t1; ;tk.”
Demazure Modules, Fusion Products and Q-Systems (2013, methodology): “The Kirillov–Reshetikhin modules for the current algebra were proved in [5], [25] to be the specializations to q = 1 of Kirillov–Reshetikhin modules for the quantized enveloping algebra of the loop algebra.”
Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions (2005, background): “By [18], D(1,ω∨ 1 ) has dimension 11, and by [5], KR(ω1) has dimension 4.”
BAXTER'S RELATIONS AND SPECTRA OF QUANTUM INTEGRABLE MODELS (2013, methodology): “tions). A representation in Cis said to be thin if its ℓ-weight spaces are of dimension 1. Remark 7.2. Let i ∈ I and a ∈ C∗. If Ni = 1, then L(Yi,a) is simple as a Uq(g˙)-module, as shown by V. Chari [C2] (see also [HJ, Remark 7.7]). In particular, if this Uq(g˙) fundamental representation is minusc”
T-systems and Y-systems in integrable systems (2010, background): “[82, 65, 86, 252], such formulas have been established for all Q (a) m ’s for Ar, Br, Cr, Dr and many ones from E6,7,8, F4 and G2.”
On Minimal Affinizations of Representations of Quantum Groups (2006, background): “For example the minimal affinizations of simple Uq (g)-modules of highest weight a multiple of a fundamental weight are the Kirillov-Reshetikhin modules which have been intensively studied in recent years (for example see [KOS, KNH, Kl, HKOTY, KN, Cha2 , Nak4, Nak5, H4, CM3, FL] and references ther”
Kirillov–Reshetikhin crystals for nonexceptional types (2008, background): “1 These decompositions were proven by Chari [2] in the untwisted cases.”