On the support of the free Lie algebra: the Schutzenberger problemsReport as inadecuate




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1 Department of Mathematics Ioannina

Abstract : M.-P. Schutzenberger asked to determine the support of the free Lie algebra LZm A on a finite alphabet A over the ring Zm of integers mod 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 LZm 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 over tilde resp. if n is even and u = v over tilde, where v over tilde 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 ln on lambda-tabloids, where lambda 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 pnI, a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for pn I and show that if m inverted iota Sigmai is an element of I1i-1 n - 1 i - 1, then w lies in the support of LZm A.

Mots-clés : Free Lie algebras Pascal triangle mod m shuffle product set partitions lambda-tabloids





Author: Ioannis C. Michos -

Source: https://hal.archives-ouvertes.fr/



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