Free Lie Superalgebras, Trees and Chains of Partitions

Several representations of the symmetric group, arising from different combinatorial, algebraic and geometric constructions, have lead to the same character, up to multiplication by the sign character: the homology of partition lattice (cf. [5, 7, 13]), the top component of a special quotient of the Stanley-Reisner ring of this same lattice [4], the top component of the cohomology algebra of the variety {x ~ C n [ x~ ~ x j if i # j } computed by Arnold [9], the free Lie algebra [7, 8]. Barcelo [1] and Bergeron and Barcelo [2] have also proved this equality of characters by showing that the matrices of these representations in the classical bases (Lyndon basis of the free Lie algebra, Garsia-Stanton basis, NBC basis of Bj6rner [3]) are equal, up to sign-character and transposition. The latter work has been the starting motivation of the present paper. It turns out that the character of Sn acting on the (multilinear part of the) free Lie algebra is the product by the sign character by its character on the (oddly generated) free Lie superalgebra R(X). This fact is already implicit in Ree's paper [12]. In the present paper, we give several combinatorial/algebraic constructions (analytic functors [7], or polynomial functors [10]) which are variant of the classical construction of T~(X) by trees (representing the brackets) or of its dual; these different functors coincide in their multilinear part with the previously mentioned constructions on the partition lattice, so that the equality of characters and matrices becomes natural. A striking fact in all these constructions is that they are obtained by introducing relations which in all cases are of two kinds: one of length 2, and one of length 3 (antisymmetry and Jacobi identity for the free Lie superalgebra, cohomology or Garsia-Stanton relations in the Stanley-Reisner ring, antisymmetry and cyclicity in the Arnold algebra). At first glance,