Primitive elements in a free dendriform algebra

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

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Sparse polynomials, redundant bases, gauss periods, and efficient exponentiation of primitive elements for small characteristic finite fields

Designs Codes and Cryptography ◽

10.1007/s10623-006-9016-7 ◽

2006 ◽

Vol 41 (3) ◽

pp. 299-306

Author(s):

Soonhak Kwon ◽

Chang Hoon Kim ◽

Chun Pyo Hong

Keyword(s):

Finite Fields ◽

Primitive Elements ◽

Sparse Polynomials ◽

Gauss Periods

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Linear combinations of primitive elements of a finite field

Finite Fields and Their Applications ◽

10.1016/j.ffa.2018.02.009 ◽

2018 ◽

Vol 51 ◽

pp. 388-406 ◽

Cited By ~ 1

Author(s):

Stephen D. Cohen ◽

Tomás Oliveira e Silva ◽

Nicole Sutherland ◽

Tim Trudgian

Keyword(s):

Finite Field ◽

Primitive Elements ◽

Linear Combinations

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Comultiplication Rules for the Double Schur Functions and Cauchy Identities

The Electronic Journal of Combinatorics ◽

10.37236/102 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 8

Author(s):

A. I. Molev

Keyword(s):

Symmetric Functions ◽

Schur Functions ◽

Primitive Elements ◽

Expansion Formula ◽

New Family ◽

Dual Object ◽

Multiplication Rule ◽

Distinguished Basis ◽

Natural Way

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.

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Perspectives pour la linguistique : de la linguistique descriptive à la linguistique explicative

Neophilologica 2019 ◽

10.31261/neo.2021.33.14 ◽

2021 ◽

Vol 33 ◽

pp. 1-29

Author(s):

Wiesław Banyś

Keyword(s):

General Theory ◽

Point Of View ◽

Primitive Elements ◽

Aim Of Science ◽

The Difference ◽

Main Components ◽

General Linguistic ◽

Object Of Study ◽

Comprehensive Study

The text deals with one of the challenges of linguistics, which is to effectively combine description and explanation in linguistics.It is necessary that linguistic theories are not only capable of adequately describing their object of study within their framework, but they must also have a suitable explanatory power.Linguistics centred around the explanation of the why of the system is called here ‘explanatory’ or ‘non-autonomous’, in contrast to ‘descriptive’ or ‘autonomous’ linguistics, which is focused on the description of the system, the distinction being based on the difference in the objects of study, the goals and the descriptive and explanatory possibilities of the theories.From the point of view presented here, a comprehensive study of language has three main components: a general theory of what language is, a resulting theory and description, which is a function of this theory, of how language is organised, functions and has evolved in the human brain, and an explanation of the properties of language found.The explanatory value of a general linguistic theory is a function of various elements, among others, the quantity of the primitive elements of the theory adopted and the effectiveness of Ockham’s razor principle of simplicity. It is also a function of the quality of those elements which can be drawn not only from within the system, but also from outside the system becoming in this situation logically prior to the object under study.In science, in linguistics, one naturally needs two types of approach, two types of linguistics, descriptive/autonomous and explanatory/non-autonomous, one must first describe reality in order to explain it. But it is also certain that since the aim of science is to explain in order to reach that higher level of scientificity above pure description, it is necessary that this aim be realized in different linguistic theories within different research programs, uniting descriptivist and explanatory approaches.

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