On the fermionic formula and the Kirillov-Reshetikhin conjecture (Q2729309)

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. \textit{A. N. Kirillov} and \textit{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 [\textit{M. Kleber}, Int. Math. Res. Not. 1997, No. 4, 187--201 (1997; Zbl 0897.17022)], see also \textit{G. Hatayama, A. Kuniba, M. Okado, T. Takagi,} and \textit{Y. Yamada} [Contemp. Math. 248, 243--291 (1999; Zbl 1032.81015)]. The proof is based on previous work in [\textit{V. Chari} and \textit{A. Pressley} [Represent. Theory 5, 191--223 (2001; Zbl 0989.17019)], and uses a result from \textit{M. Kashiwara} [Duke Math. J. 112, No. 1, 117--195 (2002; Zbl 1033.17017), preprint \url{math.QA/0010293}], see also \textit{M. Varagnolo} and \textit{E. Vasserot} [Duke Math. J. 111, No. 3, 509--533 (2002; Zbl 1011.17012)].