Some canonical basis vectors in the basic \(U_q(\widehat{{\mathfrak sl}_n})\)-module
Let \(U_q(\widehat{sl_n})\) denote the quantized enveloping algebra associated to the affine Kac-Moody Lie algebra \(\widehat{sl_n}\). The basic \(U_q(\widehat{sl_n})\)-module has a canonical basis with favourable properties. \textit{A. Lascoux, B. Leclerc} and \textit{J.-Y. Thibon} [Commun. Math. Phys. 181, 205-263 (1996; Zbl 0874.17009)] gave a fast algorithm to compute this basis. In this paper, the authors give an explicit formula for a certain class of canonical basis vectors. This includes the entire canonical basis in a family of weight spaces of arbitrarily large dimension. This formula is computed via the wreath products of symmetric groups which arise in the proof of Broue's conjecture for certain blocks of the symmetric groups, see \textit{J. Chuang} and \textit{R. Kessar} [Bull. Lond. Math. Soc. 34, 174-184 (2002; Zbl 1033.20009)]. See also \textit{B. Leclerc} and \textit{H. Miyachi} [Represent. Theory 6, 290-312 (2002; Zbl 1030.17014)] for another formula for canonical basis elements in the basic \(U_q(\widehat{sl_n})\)-module.
scientific article; zbMATH DE number 1749651
scientific article