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On the fermionic formula and the Kirillov-Reshetikhin conjecture. (English) Zbl 0982.17004
Let $$g$$ be a finite dimensional simple Lie algebra, $$\widehat{g}$$ its corresponding affine Lie algebra and $$U_q(\widehat{g})$$ the related quantum affine algebra. A. N. Kirillov and N. Yu. Reshetikhin conjectured the existence of a finite dimensional $$U_q(\widehat{g})$$-module $$V_q^{\text{aff}} (m\lambda_i)$$, where $$\lambda_i$$ is a fundamental weight of $$g$$ and $$m$$ is a positive integer, satisfying the following property: the decomposition of the tensor product of $$N$$ copies of $$V_q^{\text{aff}} (m\lambda_i)$$, as $$U_q(g)$$-module, is given by the so-called fermionic formula [see J. Sov. Math. 52, No. 3, 3156–3164 (1990); translation from Zap. Nauchn. Semin. LOMI 160, 211–221 (1987; Zbl 0637.16007)]. The author proves the conjecture in the case when $$g$$ is classical. She follows the combinatorial interpretation of the conjecture given in [M. Kleber, Int. Math. Res. Not. 1997, No. 4, 187–201 (1997; Zbl 0897.17022)], see also G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada [Contemp. Math. 248, 243–291 (1999; Zbl 1032.81015)]. The proof is based on previous work in [V. Chari and A. Pressley [Represent. Theory 5, 191–223 (2001; Zbl 0989.17019)], and uses a result from M. Kashiwara [Duke Math. J. 112, No. 1, 117–195 (2002; Zbl 1033.17017), preprint \urlmath.QA/0010293], see also M. Varagnolo and E. Vasserot [Duke Math. J. 111, No. 3, 509–533 (2002; Zbl 1011.17012)].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
##### Keywords:
quantum affine algebra; fermionic formula
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