On the support of the free Lie algebra: the Schützenberger problems
| dc.creator | Michos, Ioannis | |
| dc.date | 2008-07-22 | |
| dc.date.accessioned | 2026-07-07T09:52:08Z | |
| dc.date.available | 2026-07-07T09:52:08Z | |
| dc.description | M.-P. Schützenberger asked to determine the support of the free Lie algebra ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We study these problems using the adjoint endomorphism $l^{*}$ of the left normed Lie bracketing $l$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. Calculating $l^{*}(w)$ via all factors of a given word $w$ of fixed length and the shuffle product, we recover the result of Duchamp and Thibon $(1989)$ for the support of the free Lie ring in a much more natural way. We rephrase these problems, for words of length $n$, in terms of the action of the left normed multi-linear Lie bracketing $l_{n}$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ - viewed as an element of the group ring of the symmetric group ${\mathcal S}_{n}$ - on $λ$-tabloids, where $λ$ is a partition of $n$. For words $w$ in two letters, represented by a subset $I$ of $[n] = \{1, 2, ..., n \}$, this leads us to the {\em Pascal descent polynomial} $p_{n}(I)$, a particular commutative multi-linear polynomial which equals to a signed binomial coefficient when $|I| = 1$ and allows us to obtain a sufficient condition on $n$ and $I$ in order that $w$ lies in ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. We also have a particular conjecture for twin and anti-twin words for the free Lie ring and show that it is enough to be checked for $|A| = 2$. | |
| dc.description | 22 pages (10pt) Latex file | |
| dc.identifier | https://arxiv.org/abs/0807.3519 | |
| dc.identifier | http://arxiv.org/abs/0807.3519 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/165485 | |
| dc.subject | Combinatorics | |
| dc.subject | 17B01 (Primary) 68R15, 05E10 (Secondary) | |
| dc.title | On the support of the free Lie algebra: the Schützenberger problems | |
| dc.type | text |