Dual bases for noncommutative symmetric and quasi-symmetric functions via monoidal factorization

Bui, Van Chiên; Duchamp, Gérard; Minh, Vincel Hoang Ngoc; Kané, Ladji; Tollu, Christophe

doi:10.1016/j.jsc.2015.11.007

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…where A rat fin Y = ∪ F⊂ f inite Y A rat F , the algebra of series over finite subalphabets 30 . (ii) (Kronecker's theorem [1,51]) One has A rat x = {P(1 − xQ) −1 } P,Q∈A[x] (for x ∈ X ) and if A = K is an algebraically closed field of characteristic zero one 30 The last inclusion is strict as shows the example of the following identity [6] (ty also has K rat x = span K {(ax) * ⊔⊔ K x |a ∈ K}. (iii) The rational series (∑ x∈X α x x) * are conc-characters and any conccharacter is of this form.…”

Section: Syntactically Exchangeable Rational Seriesmentioning

confidence: 99%

“…It culminates with the fact that the coefficients of any suitable 5 solution is group-like, i.e. satisfies 6 , for any u, v ∈ X * and x i ∈ X ,…”

Section: Introductionmentioning

confidence: 99%

“…i.e. group-like at one -interior or frontier -point 6. In the first identity, also called Friedrichs criterion, is involved the shuffle product (⊔⊔ )[10,30,46] 7.…”

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confidence: 99%

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Towards a noncommutative Picard-Vessiot theory

Duchamp,

Minh,

Dinh

et al. 2020

Preprint

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A Chen generating series, along a path and with respect to m differential forms, is a noncommutative series on m letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) coefficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal differential equation, i.e. universality can be seen by replacing each letter by constant matrices (resp. analytic vector fields) and then solving a system of linear (resp. nonlinear) differential equations.Via rational series, on noncommutative indeterminates and with coefficients in rings, and their non-trivial combinatorial Hopf algebras, we give the first step of a noncommutative Picard-Vessiot theory and we illustrate it with the case of linear differential equations with singular regular singularities thanks to the universal equation previously mentioned. 194.1 Noncommutative differential equations 19 4.2 First step of a noncommutative Picard-Vessiot theory 22 5 Conclusion 24 References 24

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Section: Syntactically Exchangeable Rational Seriesmentioning

confidence: 99%

“…It culminates with the fact that the coefficients of any suitable 5 solution is group-like, i.e. satisfies 6 , for any u, v ∈ X * and x i ∈ X ,…”

Section: Introductionmentioning

confidence: 99%

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Towards a noncommutative Picard-Vessiot theory

Duchamp,

Minh,

Dinh

et al. 2020

Preprint

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“…Let X denotes either the alphabets X := {x 0 , x 1 } or Y := {y k } k≥1 and X * denotes the monoid freely generated by X (its unit is denoted by 1 X * ). In the sequel, we will consider, for any commutative ring A, the Hopf algebras (A X , conc, ∆ ⊔⊔ , 1 X * , ǫ) and (A Y , conc, ∆ , 1 Y * , ǫ) 5 Once equipped with a total ordering <, (X , <) a totally ordered alphabet for which, we can consider LynX ⊂ X * , its set of Lyndon words [21] and {P l } l∈LynX the basis of Lie C X with which the PBW-Lyndon basis {P w } w∈X * of noncommutative polynomials (A X , conc, 1 X * ) is constructed. Its graded dual basis is denoted by {S w } w∈X * , containing the pure transcendence basis {S l } l∈LynX of the shuffle algebra (A X , ⊔⊔ , 1 X * )) [24].…”

Section: Combinatorial Frameworkmentioning

confidence: 99%

Families of eulerian functions involved in regularization of divergent polyzetas

Bui,

Minh,

Dinh

et al. 2020

Preprint

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Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are interested in the ratios of ζ(2k)/π 2k and their multiindexed generalization, we will obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of noncommutative rational series. The same combinatorial frameworks also allow to study the independence of a family of eulerian functions.

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“…> . (2.5)4 Here, e denotes the counit defined by e( ) = ⟨ | 1 * ⟩ (for any ∈ ⟨ ⟩) 5. The dual family, i.e.…”

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confidence: 99%

On the solutions of the universal differential equation with three regular singularities (On solutions ofKZ3)

Minh¹

2020

Confluentes Mathematici

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On the solutions of the universal differential equation with three regular singularities (On solutions of 3) Tome 11, n o 2 (2019), p. 25-64. © Les auteurs et Confluentes Mathematici, 2019. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence Licence Internationale d'Attibution Creative Commons BY 4.0 https://creativecommons.org/licenses/by/4.0 L'accès aux articles de la revue « Confluentes Mathematici » (http://cml.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://cml.centre-mersenne. org/legal/). Confluentes Mathematici est membre du Centre Mersenne pour l'édition scientifique ouverte http://www.centre-mersenne.org/

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Dual bases for noncommutative symmetric and quasi-symmetric functions via monoidal factorization

Abstract: Abstract. In this work, an effective construction, via Schützenberger's monoidal factorization, of dual bases for the non commutative symmetric and quasi-symmetric functions is proposed.

Cited by 4 publications

References 16 publications

Towards a noncommutative Picard-Vessiot theory

Towards a noncommutative Picard-Vessiot theory

Families of eulerian functions involved in regularization of divergent polyzetas

On the solutions of the universal differential equation with three regular singularities (On solutions ofKZ3)

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