Noncommutative Symmetric Functions VII: Free Quasi-Symmetric Functions Revisited
نویسنده
چکیده
This article is essentially an appendix to [4]. We gather here some useful properties of the algebra FQSym of free quasi-symmetric functions which were overlooked in [4]. Recall that FQSym is a subalgebra of the algebra of noncommutative polynomials in infinitely many variables ai which is mapped onto Gessel’s algebra of quasi-symmetric functions QSym by the commutative image ai 7→ xi of K〈A〉. As an abstract algebra, it is isomorphic to the convolution algebra of permutations introduced by Reutenauer and his school [21, 16, 20], and further studied in [4, 14, 15, 1]. However, the realization in terms of the variables ai provides one with a better understanding of several aspects of the theory. For example, it becomes possible, and sometimes straigthforward, to imitate various constructions of the theory of symmetric (or quasi-symmetric) functions. An illustration is provided by the construction of the coproduct given in [4]: a free quasi-symmetric function F can be regarded as a “function” of a linearly ordered alphabet A, and the obvious analog of the coproduct of QSym, that is, F (A) 7→ F (A⊕A), where A and A are two mutually commuting copies of A, and ⊕ is the ordered sum, gives back the coproduct of [16]. In the present text, we further investigate the rôle of the auxiliary variables ai. We start with an alternative definition of the standard basis of FQSym, as resulting from a noncommutative lift of a Weyl-type character formula. Next, we formulate a free Cauchy identity in FQSym, and investigate its implications. In the classical theory of symmetric functions,
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