A One-Parameter Deformation of the Noncommutative Lagrange Inversion Formula

Abstract

Abstract. We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy. IntroductionThe existence of combinatorial interpretations of the Lagrange inversion formula [12] can be traced back to the existence of noncommutative generalizations [4,11]. In its simplest form, the classical version gives the compositional inverse of an invertible formal power series. In other words, it expresses the antipode of the Hopf algebra of polynomial functions on the group of formal diffeomorphisms of the real line, also known as the Faá di Bruno algebra [5].Formal power series in one variable with coefficients in a noncommutative algebra can be composed (by substitution of the variable). This operation is not associative, so that they do not form a group. However, the analogue of the Faá di Bruno algebra still exists in this context. It is investigated in [1] in view of applications in quantum field theory. In [1], one finds in particular a combinatorial formula for its antipode. This formula is rederived by Novelli and Thibon [10], who also show that it is equivalent to the noncommutative Lagrange formula of Gessel and Pak-Postnikov-Retakh. They obtain it from the Brouder-Frabetti-Krattenthaler formula by a simple application of the antipode of the Hopf algebra of noncommutative symmetric functions.In [2], Foissy obtains, as a byproduct of his investigation of combinatorial SchwingerDyson equations, one-parameter families of Hopf algebras. They interpolate respectively between symmetric functions and Faá di Bruno, and between noncommutative symmetric functions and the noncommutative Faá di Bruno algebra.The main result of this paper is a closed formula for the antipode of the noncommutative family. As we shall see, this is a natural deformation of the Brouder-FrabettiKrattenthaler formula. As in the original version, we obtain a closed formula for the antipode of the simple complete noncommutative symmetric functions S n . Namely, we compute explicitely the coefficients α I (I n) in their expansion in the basis of complete noncommutative symmetric functions S I . As in the original version, we Date: May 17, 2011.

Combinatorial properties of the noncommutative Faà di Bruno algebra

“…In the last three sections, we are interested in the noncommutative family. We give in [2] a closed formula for the corresponding antipode, which is a natural deformation of the BrouderFrabetti-Krattenthaler formula.…”

“…The multiplicity m k (I) of k in I is the number of parts in I equal to k. We represent I by a ribbon in which the lengths of the lines, read from the left to the right and from the bottom to the top, are the values of the parts of I. For example, (222) and (311) correspond respectively to (2) and…”