Differential Operator Specializations of Noncommutative Symmetric Functions

Let K be any unital commutative Q-algebra and z = (z1, · · · , zn) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K〈〈z〉〉 and K[[t]]〈〈z〉〉 the formal power series algebras of z over K and K[[t]], respectively. For any α ≥ 1, let D〈〈z〉〉 be the unital algebra generated by the differential operators of K〈〈z〉〉 which increase the degree in z by at least α− 1 and A [α] t 〈〈z〉〉 the group of automorphisms Ft(z) = z − Ht(z) of K[[t]]〈〈z〉〉 with o(Ht(z)) ≥ α and Ht=0(z) = 0. First, for any fixed α ≥ 1 and Ft ∈ A [α] t 〈〈z〉〉, we introduce five sequences of differential operators of K〈〈z〉〉 and show that their generating functions form a NCS (noncommutative symmetric) system ([Z4]) over the differential algebra D〈〈z〉〉. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSF’s (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms SFt : NSym → D 〈〈z〉〉 (Ft ∈ A [α] t 〈〈z〉〉), which are also grading-preserving when Ft satisfies certain conditions. Note that, the homomorphisms SFt above can also be viewed as specializations of NCSF’s by the differential operators of K〈〈z〉〉. Secondly, we show that, in both commutative and noncommutative cases, this family SFt (with all n ≥ 1 and Ft ∈ A [α] t 〈〈z〉〉) of differential operator specializations can distinguish any two different NCSF’s. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.