Invariants of linear combinations of transpositions

It is shown that for a one-parameter set H _N of linear combinations of N(N–1)/2 elementary transpositions {P _jk } (1jN) at arbitrary natural N3, one can construct a variety {I _m } (3mN) of operators which commute with H _N . Being applied to SU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction.