Restrictions for $n$-Point Vertices in Higher-Spin Theories

We give a simple classification of the independent $n$-point interaction vertices for bosonic higher-spin gauge fields in $d$-dimensional Minkowski space-times. We first give a characterisation of such vertices for large dimensions, $d \geq 2n - 1$, where one does not have to consider Schouten identities due to over-antisymmetrisation of space-time indices. When the dimension is lowered, such identities have to be considered, but their appearance only leads to equivalences of large-$d$ vertices and does not lead to new types of vertices. We consider the case of low dimensions, $d<n$, in detail, where the large number of Schouten identities leads to strong restrictions on independent vertices. We also comment on the generalisation of our results to the intermediate case $n \leq d \leq 2n - 2$. In all cases, the independent vertices are expressed in terms of elementary manifestly gauge-invariant quantities, suggesting that no deformations of the gauge transformations are induced.