Noncommutative Symmetric Functions and the Inversion Problem

Abstract. Let K be any unital commutative Q-algebra and z = (z1, z2, · · · , zn) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]〈〈z〉〉 the formal power series algebra of z over K[[t]]. In [Z6], for each automorphism Ft(z) = z−Ht(z) of K[[t]]〈〈z〉〉 with Ht=0(z) = 0 and o(H(z)) ≥ 1, a NCS (noncommutative symmetric) system ([Z5]) ΩFt has been constructed. Consequently, we get a Hopf algebra homomorphism SFt : NSym → D〈〈z〉〉 from the Hopf algebra NSym ([GKLLRT]) 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 NCS system ΩFt by using some identities of NCSF’s derived in [GKLLRT] and the homomorphism SFt . Secondly, we apply these identities to derive some formulas in terms of differential operator in the system ΩFt for the Taylor series expansions of u(Ft) and u(F −1 t ) (u(z) ∈ K[[t]]〈〈z〉〉); the D-Log and the formal flow of Ft and inversion formulas for the inverse map of Ft. Finally, we discuss a connection of the wellknown Jacobian conjecture with NCSF’s.