  Structures of symmetries of transformations for Holman-Biedenharn-Louck $A_n$
hypergeometric series: $A_n$ terminating balanced ${}_4 F_3$ series and $A_n$
elliptic ${}_{10} E_9$ series are discussed. Namely the description of the
invariance groups and the classification all of possible transformations for
each types of $A_n$ hypergeometric series are given. Among them, a "periodic"
affine Coxeter group which seems to be new in the literature arises as an
invariance group for a class of $A_n$ ${}_4 F_3$ series.
