  Invariant tensors are states in the SU(2) tensor product representation that
are invariant under the SU(2) action. They play an important role in the study
of loop quantum gravity. On the other hand, perfect tensors are highly
entangled many-body quantum states with local density matrices maximally mixed.
Recently, the notion of perfect tensors recently has attracted a lot of
attention in the fields of quantum information theory, condensed matter theory,
and quantum gravity. In this work, we introduce the concept of an invariant
perfect tensor (IPT), which is a $n$-valent tensor that is both invariant and
perfect. We discuss the existence and construction of IPT. For bivalent
tensors, the invariant perfect tensor is the unique singlet state for each
local dimension. The trivalent invariant perfect tensor also exists and is
uniquely given by Wigner's $3j$ symbol. However, we show that, surprisingly,
there does not exist four-valent invariant perfect tensors for any dimension.
On the contrary, when the dimension is large, almost all invariant tensors are
perfect asymptotically, which is a consequence of the phenomenon of
concentration of measure for multipartite quantum states.
