NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND W-POLYNOMIALS

Let K, S, D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Viète formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analyzed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper.

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NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND THE INVERSION PROBLEM

International Journal of Algebra and Computation ◽

10.1142/s0218196708004615 ◽

2008 ◽

Vol 18 (05) ◽

pp. 869-899 ◽

Cited By ~ 1

Author(s):

WENHUA ZHAO

Keyword(s):

Hopf Algebra ◽

Symmetric Functions ◽

Differential Operators ◽

Symmetric System ◽

Algebra Homomorphism ◽

Inversion Problem ◽

Noncommutative Symmetric Functions ◽

Taylor Series Expansions ◽

Formal Power ◽

And Inversion

Let K be any unital commutative ℚ-algebra and z = (z1, z2, …, zn) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]〈〈z〉〉 the formal power series algebra of z over K[[t]]. In [29], for each automorphism Ft(z) = z - Ht(z) of K[[t]]〈〈z〉〉 with Ht=0(z) = 0 and o(H(z)) ≥ 1, a [Formula: see text] (noncommutative symmetric) system [28] ΩFt has been constructed. Consequently, we get a Hopf algebra homomorphism [Formula: see text] from the Hopf algebra [Formula: see text] [9] of NCSFs (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the [Formula: see text] system ΩFt by using some identities of NCSFs derived in [9] and the homomorphism [Formula: see text]. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system ΩFt for the Taylor series expansions of u(Ft) and [Formula: see text]; the D-Log and the formal flow of Ft and inversion formulas for the inverse map of Ft. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.

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Noncommutative Symmetric Bessel Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-2008-043-3 ◽

2008 ◽

Vol 51 (3) ◽

pp. 424-438 ◽

Cited By ~ 1

Author(s):

Jean-Christophe Novelli ◽

Jean-Yves Thibon

Keyword(s):

Bessel Functions ◽

Symmetric Functions ◽

Tensor Products ◽

Simple Explanation ◽

Noncommutative Symmetric Functions ◽

Combinatorial Properties

AbstractThe consideration of tensor products of 0-Hecke algebramodules leads to natural analogs of the BesselJ-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

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Colored Trees and Noncommutative Symmetric Functions

The Electronic Journal of Combinatorics ◽

10.37236/468 ◽

2010 ◽

Vol 17 (1) ◽

Author(s):

Matt Szczesny

Keyword(s):

Hopf Algebra ◽

Symmetric Functions ◽

Hall Algebra ◽

Quasisymmetric Functions ◽

Noncommutative Symmetric Functions

Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

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Antipode Formulas for some Combinatorial Hopf Algebras

The Electronic Journal of Combinatorics ◽

10.37236/5949 ◽

2016 ◽

Vol 23 (4) ◽

Cited By ~ 2

Author(s):

Rebecca Patrias

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Symmetric Functions ◽

Explicit Formulas ◽

Quasisymmetric Functions ◽

Noncommutative Symmetric Functions ◽

Combinatorial Hopf Algebras ◽

Combinatorial Formulas ◽

K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

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