Representations of the Symmetric Group in Deformations of the Free Lie Algebra.

We consider, for a given complex parameter alpha, the non-associative product defined on the tensor algebra of k-dimensional complex vector space by (x, y) = x tensor product y - alpha y tensor product x. For n symbols x sub 1,...., x sub n, left-normed bracketing is defined recursively to be the bracketing sequence b sub n, where b sub 1 = x sub 1, b sub 2 = (x sub 1, x sub 2), and b sub n = (b sub (n-1), x sub n). The linear subspace spanned by all left-normed bracketings of homogeneous degree n, in the basis vectors v sub 1,...., v sub k of C sup k, is then an S sub n - module V sub n (alpha).