{
  "id": "0807.3519",
  "version": "v1",
  "published": "2008-07-22T17:02:29.000Z",
  "updated": "2008-07-22T17:02:29.000Z",
  "title": "On the support of the free Lie algebra: the Schützenberger problems",
  "authors": [
    "Ioannis Michos"
  ],
  "comment": "22 pages (10pt) Latex file",
  "categories": [
    "math.CO"
  ],
  "abstract": "M.-P. Sch\\\"utzenberger 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 $\\lambda$-tabloids, where $\\lambda$ 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 $|A| = 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-22T17:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B01",
      "68R15",
      "05E10"
    ],
    "keywords": [
      "free lie algebra",
      "schützenberger problems",
      "normed multi-linear lie bracketing",
      "anti-twin words",
      "free lie ring"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.3519M"
    }
  }
}