Cluster variables on certain double Bruhat cells of type \((u,e)\) and monomial realizations of crystal bases of type A (Q2353213)

Cluster algebras were introduced by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] in order to develop a combinatorial approach to study the problem of dual canonical bases in quantum groups. In [Duke Math. J. 126, No. 1, 1--52 (2005; Zbl 1135.16013)], \textit{A. Berenstein}, \textit{S. Fomin} and \textit{A. Zelevinsky} showed that the coordinate ring of every double Bruhat cell in a semisimple complex Lie group is isomorphic to an upper cluster algebra in which the initial cluster variables are given as certain generalized minors. In this paper under review, the authors focus on the following case: the simple complex Lie group is assumed to be \(\mathrm{SL}_{r+1}(\mathbb{C})\), the Weyl group elements \(v=e\)(the identity element) and certain \(u\) whose reduced word can be written as a left factor of \((1, 2, 3, \cdots, r, 1, 2, 3, \cdots, r-1,\cdots,1,2,1)\). In these cases, the authors prove that the generalized minors are represented in terms of summations over certain monomial realizations of Demazure crystals.