Differential operator specializations of noncommutative symmetric functions

Abstract

Let K be any unital commutative 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 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 A t [ α ] 《 z 》 the group of automorphisms F t ( z ) = z − H t ( z ) of K 〚 t 〛 《 z 》 with o ( H t ( z ) ) ⩾ α and H t = 0 ( z ) = 0 . First, for any fixed α ⩾ 1 and F t ∈ A t [ α ] 《 z 》 , we introduce five sequences of differential operators of K 《 z 》 and show that their generating functions form an N CS (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 N CS 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 S F t : N Sym → D [ α ] 《 z 》 ( F t ∈ A t [ α ] 《 z 》 ) , which are also grading-preserving when F t satisfies certain conditions. Note that the homomorphisms S F t 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 S F t (with all n ⩾ 1 and F t ∈ A t [ α ] 《 z 》 ) 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.