A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions

Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike

doi:10.4153/cjm-2013-013-0

Cited by 58 publications

(122 citation statements)

References 29 publications

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“…One can then consider the quasi-symmetric functions C I which are the commutative images of the coefficients of the monomials Y I . It turns out that they coincide with the dual immaculate basis of [2], up to mirror image of compositions.…”

Section: Introductionmentioning

confidence: 83%

Noncommutative Bell polynomials and the dual immaculate basis

Novelli¹,

Thibon²,

Toumazet³

2018

Algebraic Combinatorics

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We define a new family of noncommutative Bell polynomials in the algebra of free quasi-symmetric functions and relate it to the dual immaculate basis of quasi-symmetric functions. We obtain noncommutative versions of Grinberg's results [Canad. J. Math. 69 (2017), 21-53], and interpret these in terms of the tridendriform structure of WQSym. We then present a variant of Rey's self-dual Hopf algebra of set partitions [FPSAC'07, Tianjin] adapted to our noncommutative Bell polynomials and give a complete description of the Bell equivalence classes as linear extensions of explicit posets.

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Section: Introductionmentioning

confidence: 83%

Noncommutative Bell polynomials and the dual immaculate basis

Novelli¹,

Thibon²,

Toumazet³

2018

Algebraic Combinatorics

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“…(Note that since we are using French notation our definition varies slightly from the definition in [12] but produces the same diagrams modulo a horizontal flip.) The weight of an immaculate tableau U , denoted x U , is the product over all i of x #(i) i , where #(i) is the number of times i appears in U .…”

Section: 1mentioning

confidence: 99%

“…The 0-Hecke action defined above cannot move a j + 1 to the right of a j, so the subspace N α of M α spanned by all words which are not Y-words is a submodule of M α . [12] and proved in [18]) for the product of a fundamental quasisymmetric function and a dual immaculate quasisymmetric function, and much is known about the multiplication of the immaculate basis. However, multiplication rules in full generality for the dual immaculate quasisymmetric functions are still largely unknown.…”

Section: Definition 41 [12]mentioning

confidence: 99%

Recent Trends in Quasisymmetric Functions

Mason

2019

Association for Women in Mathematics Series

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This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncommutative symmetric functions, appearances of quasisymmetric functions within the theory of Macdonald polynomials, and analogues of symmetric functions. Topics include the significance of quasisymmetric functions in representation theory (such as representations of the 0-Hecke algebra), recently discovered bases (including analogues of well-studied symmetric function bases), and applications to open problems in symmetric function theory.

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“…Composition tableaux were introduced in [19] to define the basis of quasisymmetric Schur functions for the Hopf algebra of quasisymmetric functions. These functions are analogues of the ubiquitous Schur functions [41], have been studied in substantial detail recently [8,19,20,30,31,45], and have consequently been the genesis of an active new branch of algebraic combinatorics discovering Schur-like bases in quasisymmetric functions [1,5], type B quasisymmetric Schur functions [27,37], quasi-key polynomials [2,42] and quasisymmetric Grothendieck polynomials [35]. Just as Young tableaux play a crucial role in the combinatorics of Schur functions [40,43], composition tableaux are key to understanding the combinatorics of quasisymmetric Schur functions.…”

Section: Introductionmentioning

confidence: 99%

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Tewari

Willigenburg

2019

Journal of Combinatorial Theory, Series A

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We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

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A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions

Cited by 58 publications

References 29 publications

Noncommutative Bell polynomials and the dual immaculate basis

Noncommutative Bell polynomials and the dual immaculate basis

Recent Trends in Quasisymmetric Functions

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

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