Q-tensor square of finitely generated nilpotent groups, q >= 0

The authors extend to the $q-$tensor square $G{\otimes }^{q}G$ of a group $G$, $q$ a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square $G\otimes G$ ($q=0$). The results are applied to the computation of $G{\otimes }^{q}G$ for finitely generated nilpotent groups $G$, specially for free nilpotent groups of finite rank. They also generalize to all $q\ge 0$ results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square $G\otimes G$ when $G$ is a $n-$generator nilpotent group of class 2. The paper ends with the computation of the $q-$tensor squares of the free $n-$generator nilpotent group of class 2, $n\ge 2$, for all $q\ge 0.$ This shows that the above mentioned upper bound is also achieved for these groups when $q>1.$