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Noncommutative Symmetric Functions and the Inversion Problem | | Wenhua Zhao
; | | Date: |
7 Sep 2005 | | Subject: | Complex Variables; Combinatorics MSC-class: 05E05, 14R10, 14C15 | math.CV math.CO | | Abstract: | Let $K$ be any unital commutative $Q$-algebra and $z=(z_1, z_2, ..., z_n)$ commutative or noncommutative variables. Let $t$ be a formal central parameter and $kttzz$ the formal power series algebra of $z$ over $K[[t]]$. In cite{GTS-II}, for each automorphism $F_t(z)=z-H_t(z)$ of $kttzz$ with $H_{t=0}(z)=0$ and $o(H(z))geq 1$, a cNcs (noncommutative symmetric) system (cite{GTS-I}) $Oft$ has been constructed. Consequently, we get a Hopf algebra homomorphism $cSft: cNsf o cDzz$ from the Hopf algebra $cNsf$ (cite{G-T}) of NCSF’s (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the cNcs system $Oft$ by using some identities of NCSF’s derived in cite{G-T} and the homomorphism $cSft$. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system $Oft$ for the Taylor series expansions of $u(F_t)$ and $u(F_t^{-1})$ $(u(z)in kttzz)$; the D-Log and the formal flow of $F_t$ and inversion formulas for the inverse map of $F_t$. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSF’s. | | Source: | arXiv, math.CV/0509135 | | Services: | Forum | Review | PDF | Favorites | |
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