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...
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.
The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multistep DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.
Abstract.This paper is devoted to the study of the lower central series of the free partially commutative group F(A, ~) in connection with the associated free partially commutative Lie algebra. Using a convenient Magnus transformation, we show that the quotients of the lower central series of F(A, ~) are free abelian groups and that F(A, ~) can be fully ordered.
We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor M in the differential equation dS = M S) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.
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