### On the support of the free Lie algebra: the Schützenberger problems

*Ioannis Michos*

#### Abstract

M.-P. Schützenberger asked to determine the support of the free
Lie algebra L

_{ℤm}(A) on a finite alphabet A over the ring ℤ_{m}of integers &bmod;m and all pairs of*twin*and*anti-twin*words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L_{ℤm}(A) in A^{*}as the set of all words w such that m divides all the coefficients appearing in the monomials of l^{*}(w), where l^{*}is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l^{*}(w) via the*shuffle product*, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = v (resp. if n is even and u = v), where v denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l_{n}on λ-tabloids, where λ is a*partition*of n. Representing a word w in two letters by the subset I of [n] = { 1, 2, …, n } that consists of all positions that one of the letters occurs in w, the computation of l^{*}(w) leads us to the notion of the*Pascal descent polynomial*p_{n}(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when |I| = 1. We provide a recursion formula for p_{n}(I) and show that if m ∤ Σ_{i ∈I}(-1)^{i-1}n-1, i-1, then w lies in the support of L_{ℤm}(A).Full Text: PDF PostScript