A LAnKe (also known as a Filippov algebra or a Lie algebra of the n-th kind) is a vector space equipped with a skew-symmetric n-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear part of the free LAnKe on 2n-1 generators as an irreducible representation. They announced that the multilinear component on 3n-2 generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.

A LAnKe (also known as a Lie algebra of the th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric -linear form that satisfies the generalized Jacobi identity. The symmetric group  acts on the multilinear part of the free LAnKe on  generators, where  is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for , its irreducible decomposition contains no summand whose Young diagram has at most  columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$

Trades are important objects in combinatorial design theory that may be realized as certain elements of kernels of inclusion matrices. Total trades were introduced recently by Ghorbani, Kamali and Khosravshahi, who showed that over a field of characteristic zero the vector space of trades decomposes into a direct sum of spaces of total trades. In this paper, we show that the vector space spanned by the permutations of a total trade is an irreducible representation of the symmetric group. As a corollary, the previous decomposition theorem is recovered. Also, a basis is obtained for the module of total trades in the spirit of Specht polynomials. More generally, in the second part of the paper we consider intersection matrices and determine the irreducible decompositions of their images. This generalizes previously known results concerning ranks of special cases.