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[α]《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 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 , we introduce five sequences of differential operators of K《z》 and show that their generating functions form an NCS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363–382] over the differential algebra D[α]《z》. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218–348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms , which are also grading-preserving when Ft satisfies certain conditions. Note that the homomorphisms SFt above can also be viewed as specializations of NCSFs 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 ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289–301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967–982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.