Noncommutative Symmetric Functions Vi: Free Quasi-Symmetric Functions and Related Algebras

Abstract

This article is devoted to the study of several algebras which are related to symmetric functions, and which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasisymmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable H n (0)-modules are discussed, and the homological properties of H n (0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra. expansionare the multiplicities of the simple H n (0) modules S I (parametrized by compositions I of n) as composition factors of the specialized module V λ (0). This interpretation leads to a q-analogue of QSym: the algebra QSym q of quantum quasisymmetric functions, defined in [29]. Here, the indeterminate q is introduced to record a certain filtration on H n (0)-modules. For generic complex values of q, QSym q is non commutative, and in fact isomorphic to Sym, but for q = 1 one recovers the commutative algebra of quasi-symmetric functions QSym.This construction can be somewhat clarified by the introduction of the larger algebra FQSym, a subalgebra of the free associative algebra C A (whence the name free quasisymmetric functions) which admits Sym as a subalgebra, and is mapped onto QSym q when one imposes the q-commutation relations of the quantum affine space (a j a i = qa i a j for j > i) on the letters of A.This algebra turns out to be isomorphic to the convolution algebra of symmetric groups studied by Malvenuto and Reutenauer [21]. It contains a subalgebra whose bases are naturally labelled by standard Young tableaux, which provides a concrete realization of the algebras of tableaux of Poirier and Reutenauer [25]. We call it FSym, the algebra of free symmetric functions. To illustrate the relevance of the realization of FSym as an algebra of noncommutative polynomials, we use it to present a complete proof of the Littlewood-Richardson rule within a dozen of lines (the idea of the proof is not new, but the formalism makes it quite compact and transparent). In the same vein, we show that the use of FQSym allows one to give simple presentations of Stanley's QS-distribution [28] and of the Hopf algebra of planar binary trees of Loday and Ronco [19]. The next step is to look for a representation theoretical interpretation of FQSym. It turns out that FQSym n can be interpreted as a kind of Grothendieck group for a certain category N n of H n (0)-modules, which contains in particular simple, projective, and skew Specht modules. However, this is far from exhausting all the H n (0)-modules, since we prove that for n ≥ 4, H n (0) is not representation finite. As a step towards a more exhaustive study of the 0-Hecke algebras, we determine their quivers for all n, and discuss their homological properties for small values...

Shuffle bialgebras

“…-Basis of primitive elements for the MalvenutoReutenauer bialgebra. In [5] and [2] the authors describe different basis for the subspace of primitive elements of the Malvenuto-Reutenauer bialgebra. We construct another one using our description of primitive elements of a preshuffle bialgebra.…”

Hopf algebras and dendriform structures arising from parking functions

“…Our notations for ordinary symmetric functions will be those of [23]. Other undefined notations can be found in [7,4], although the essential ones will be recalled when needed.…”

“…For such algebras to be useful in this context, it is necessary that their elements can be realized as polynomials in some auxiliary infinite set of variables (commutative or not), so as to recover ordinary symmetric functions after a chain of standard manipulations (such as imposing commutation relations among the variables or taking sums to reestablish complete symmetry). The best illustration of this approach is provided by the algebra FQSym of free quasi-symmetric functions [4]. This is an algebra of noncommutative polynomials F σ (A) labelled by permutations.…”

Higher Order Peak Algebras

“…Various applications are discussed in the series [5,7,4,8,9,3]. Let σ(t) be the generating series of the (S k ) k≥0 , with the convention S 0 = 1,…”