  We introduce a braid group action on $l$-tuple of rational functions for the
finite-dimensional representations of Yangians $Y(\mathfrak{g})$, where
$\mathfrak{g}$ is a complex simple Lie algebra. It provides an efficient way to
compute certain polynomials which allows us to provide a finite set of numbers
at which the tensor product of Kirillov-Reshetikhin modules of Yangians may
fail to be cyclic.
