Noncommutative Symmetric Functions, Lie Series and Descent Algebras

2021 ◽  
pp. 119-178
Author(s):  
Jean‐Yves Thibon
Keyword(s):  
Symmetric Functions ◽  
Lie Series ◽  
Noncommutative Symmetric Functions ◽  
Descent Algebras

scholarly journals Noncommutative Symmetric Functions II: Transformations of Alphabets

1997 ◽  
Vol 07 (02) ◽  
pp. 181-264 ◽  
Author(s):  
D. Krob ◽  
B. Leclerc ◽  
J.-Y. Thibon
Keyword(s):  
Symmetric Functions ◽  
Descent Algebra ◽  
Noncommutative Symmetric Functions ◽  
Complete Set ◽  
Lie Idempotents ◽  
Descent Algebras

Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. It is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various deformations of the classical sequences of Lie idempotents are obtained. In particular, we obtain several q-analogues of the Eulerian idempotents and of the Garsia-Reutenauer idempotents.

Noncommutative Symmetric Bessel Functions

2008 ◽  
Vol 51 (3) ◽  
pp. 424-438 ◽  
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.

scholarly journals Colored Trees and Noncommutative Symmetric Functions

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.

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
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.

scholarly journals Noncommutative symmetric functions and an amazing matrix

2012 ◽  
Vol 48 (3) ◽  
pp. 528-534 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Symmetric Functions ◽  
Noncommutative Symmetric Functions

scholarly journals NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND W-POLYNOMIALS

2007 ◽  
Vol 06 (05) ◽  
pp. 815-837 ◽  
Author(s):  
JONATHAN DELENCLOS ◽  
ANDRÉ LEROY
Keyword(s):  
Division Ring ◽  
Symmetric Functions ◽  
Differential Operators ◽  
Noncommutative Symmetric Functions ◽  
Linear 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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