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 Lℤm(A) on a finite alphabet
A
over the ring ℤm 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 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 (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 ln 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
pn(I),
a particular commutative multi-linear polynomial which is equal to the
signed
binomial coefficient when |I| = 1. We provide a recursion
formula
for pn(I) and show that if
m ∤ Σi ∈I (-1)i-1 n-1,
i-1, then
w lies in the support of Lℤm(A).
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