关于自由李代数的支撑：Schützenberger问题
On the support of the free Lie algebra: the Schützenberger problems
摘要：
M.- P. Schützenberger要求确定自由李代数${\mathcal L}_{{\mathbb Z}_{m}}(A)$在整数环${\mathbb Z}_{m}$上的有限字母表$A$上的支集$\bmod m$和所有相应的孪生和反孪生字对，即，以相等（resp.相反）的系数在每个李多项式。我们研究这些问题使用的伴随自同态$l^{*}$的左赋范李括号$l$的${\mathcal L}_{{\mathbb Z}_{m}}(A)$。通过给定的定长字$w$的所有因子和混洗积计算$l^{*}(w)$，我们以更自然的方式恢复了Duchamp和Thibon $(1989)$关于自由Lie环支撑的结果.我们重新措辞这些问题，对于长度为$n$的单词，根据${\mathcal L}_{{\mathbb Z}_{m}}(A)$的左赋范多线性李括号$l_{n}$的作用-被视为对称群${\mathcal S}_{n}$的群环的元素-在$λ$ -小报上，其中$λ$是$n$的分区。对于两个字母的单词$w$，由$[n] = \{1, 2, ..., n \}$的子集$I$表示，这将我们引向{ \em Pascal下降多项式} $p_{n}(I)$，一个特定的交换多线性多项式，当$|I| = 1$时，它等于一个有符号的二项式系数，并允许我们获得关于$n$和$I$的充分条件，以便$w$位于${\mathcal L}_{{\mathbb Z}_{m}}(A)$中。我们也有一个特殊的猜想，孪生和反孪生字的自由李环，并表明它是足够的检查$|A| = 2$。
英文摘要：
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 $|A| = 2$.
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