A Filippov algebra is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $S_m$, $m=(n-1)k+1$, acts naturally on the multilinear component of the free Filippov algebra corresponding to $k$-brackets. We will discuss some recent results on the irreducible decomposition of this representation in characteristic zero. This talk is based on joint work with Dimitra-Dionysia Stergiopoulou.