Noncommutative symmetric functions of type B

Abstract

The noncommutative symmetric functions Sym of Gelfand et al. give not only a lifting of the well-developed commutative theory of symmetric functions to the non-commutative level, but also relate the descent algebras of Solomon and the quasi-symmetric functions, where the latter are dual to the noncommutative symmetric functions equipped with the internal product, which are anti-isomorphic to the descent algebras. Using this anti-isomorphism, properties of both noncommutative symmetric functions and of descent algebras can be studied. Generalizations of the above theory are made in the present work. The starting point is the quasi-symmetric functions of type B, BQSym, which are shown to have an algebra, a comodule, and a coalgebra structures. The noncommutative symmetric functions BSym are then introduced as a module over Sym dual to the comodule structure of BQSym. It is then made into a coalgebra dual to the algebra structure of BQSym, and into an algebra dual to the coalgebra structure of BQSym. The latter duality defines the internal product *B on BSym, which makes (BSym, *B) anti-isomorphic to the descent algebra [Sigma]Bn of the hyperoctahedral groups Bn, studied by Bergeron and Bergeron.

(cont.) Lie idempotents of both BSym and [Sigma]Bn are then studied via the anti-isomorphism. In particular, a one-parameter family of Lie idempotents, which is a q-analog of a known idempotent, is found. A specialization of this family gives, in the descent algebra [Sigma]B, a Dynkin-like idempotent whose action on words is a signed left bracketing. Natural noncommutative generalizations of the Eulerian numbers and of the Euler numbers of type B are given. By a specialization, formulas for some refinements of the Euler numbers of type B are also derived.