Noncommutative Symmetric Systems over Associative Algebras

This paper is the first of a sequence papers ([Z4]--[Z7]) on the {\it ${\mathcal N}$CS $(\text{noncommutative symmetric})$ systems} over differential operator algebras in commutative or noncommutative variables ([Z4]); the ${\mathcal N}$CS systems over the Grossman-Larson Hopf algebras ([GL],[F]) of labeled rooted trees ([Z6]); as well as their connections and applications to the inversion problem ([BCW],[E4]) and specializations of NCSFs ([Z5],[Z7]). In this paper, inspired by the seminal work [GKLLRT] on NCSFs (noncommutative symmetric functions), we first formulate the notion {\it ${\mathcal N}$CS systems} over associative $\mathbb Q$-algebras. We then prove some results for ${\mathcal N}$CS systems in general; the ${\mathcal N}$CS systems over bialgebras or Hopf algebras; and the universal ${\mathcal N}$CS system formed by the generating functions of certain NCSFs in [GKLLRT]. Finally, we review some of the main results that will be proved in the followed papers [Z4], [Z6] and [Z7] as some supporting examples for the general discussions given in this paper.Comment: A connection of NCS systems with combinatorial Hopf algebras of M. Aguiar, N. Bergeron and F. Sottile has been added in Remark 2.17. Latex, 32 page

Differential Operator Specializations of Noncommutative Symmetric Functions

Let $K$ be any unital commutative $\mathbb Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by \kzz and \kttzz the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $\alpha \geq 1$, let \cDazz be the unital algebra generated by the differential operators of \kzz which increase the degree in $z$ by at least $\alpha-1$ and \ataz the group of automorphisms $F_t(z)=z-H_t(z)$ of \kttzz with $o(H_t(z))\geq \alpha$ and $H_{t=0}(z)=0$. First, for any fixed $\alpha \geq 1$ and F_t\in \ataz, we introduce five sequences of differential operators of \kzz and show that their generating functions form a $\mathcal N$CS (noncommutative symmetric) system [Z4] over the differential algebra \cDazz. Consequently, by the universal property of the $\mathcal N$CS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms \cS_{F_t}: {\mathcal N}Sym \to \cDazz (F_t\in \ataz), which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms \cS_{F_t} above can also be viewed as specializations of NCSFs by the differential operators of \kzz. Secondly, we show that, in both commutative and noncommutative cases, this family \cS_{F_t} (with all $n\geq 1$ and F_t\in \ataz) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.Comment: Latex, 33 pages. Some mistakes and misprints have been correcte

Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps

Let $F=(F_1, F_2, ... F_n)$ be an $n$-tuple of formal power series in $n$ variables of the form $F(z)=z+ O(|z|^2)$. It is known that there exists a unique formal differential operator A=\sum_{i=1}^n a_i(z)\frac {\p}{\p z_i} such that $F(z)=exp (A)z$ as formal series. In this article, we show the Jacobian ${\cal J}(F)$ and the Jacobian matrix $J(F)$ of $F$ can also be given by some exponential formulas. Namely, ${\cal J}(F)=\exp (A+\triangledown A)\cdot 1$, where \triangledown A(z)= \sum_{i=1}^n \frac {\p a_i}{\p z_i}(z), and $J(F)=\exp(A+R_{Ja})\cdot I_{n\times n}$, where $I_{n\times n}$ is the identity matrix and $R_{Ja}$ is the multiplication operator by $Ja$ for the right. As an immediate consequence, we get an elementary proof for the known result that ${\cal J}(F)\equiv 1$ if and only if $\triangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed.Comment: Latex, 17 page

Inversion Problem, Legendre Transform and Inviscid Burgers' Equations

Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from \bC^n to \bC^n and $G(z)$ the formal inverse map of $F(z)$. We first study the deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse $G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps $F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for some P(z)\in \bC[[z]] of order $o(P(z))\geq 2$. We show that, under the gradient condition, $N_t(z)=\nabla Q_t(z)$ for some Q_t(z)\in \bC[[z, t]] and the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers' equation, from which a recurrent formula for $Q_t(z)$ also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers' equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers' equations and Legendre transforms of the power series $f(z)$ of $o(f(z))\geq 2$.Comment: Latex, 21 pages. Some misprints have been correcte

A Family of Invariants of Rooted Forests

Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for finite rooted forests by a recurrent algorithm using the operator $\Xi$ and show that the invariant $\Psi$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $\alpha (T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with U_n =\sum_{T\in \bT_n} \frac 1{\alpha (T)} \Psi (T), where \bT_n is the set of rooted trees with $n$ vertices. We show that the generating function $U(q)$ satisfies the equation $\Xi \exp U(q)= q^{-1} U(q)$. Consequently, we get a recurrent formula for $U_n$ $(n\geq 1)$, namely, $U_1=\Xi(1)$ and $U_n =\Xi S_{n-1}(U_1, U_2, >..., U_{n-1})$ for any $n\geq 2$, where $S_n(x_1, x_2, ...)$ (n\in \bN) are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well known quasi-symmetric function invariants of rooted forests are in the family of invariants $\Psi$ and derive some consequences about these well-known invariants from our general results on $\Psi$. Finally, we generalize the invariant $\Psi$ to labeled planar forests and discuss its certain relations with the Hopf algebra $\mathcal H_{P, R}^D$ in \cite{F} spanned by labeled planar forests.Comment: Ams-Latex, 19 pages. One section has been added. Appearing in {\it J. Pure Appl. Alg.