An Introduction to Quasisymmetric Schur Functions

Luoto, Kurt; Mykytiuk, Stefan; Willigenburg, Stephanie van

doi:10.1007/978-1-4614-7300-8

Cited by 54 publications

(61 citation statements)

References 68 publications

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“…We begin by recalling some basic facts about the rings of quasisymmetric and noncommutative symmetric functions. We refer the reader to [24] for further details.…”

Section: -Hecke Algebrasmentioning

confidence: 99%

Categorification and Heisenberg doubles arising from towers of algebras

Savage

Yacobi

2015

Journal of Combinatorial Theory, Series A

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The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras. As an application of a generalized Stone--von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions.Comment: 30 pages. v2: Minor changes. References adde

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“…We begin by recalling some basic facts about the rings of quasisymmetric and noncommutative symmetric functions. We refer the reader to [24] for further details.…”

Section: -Hecke Algebrasmentioning

confidence: 99%

Categorification and Heisenberg doubles arising from towers of algebras

Savage

Yacobi

2015

Journal of Combinatorial Theory, Series A

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“…Proof Expand S α in terms of the fundamental quasisymmetric functions and apply the omega operator to obtain: Here, the first equation is derived from the expansion of the Young quasisymmetric Schur functions in terms of the fundamental quasisymmetric Schur functions [14]. The second equation is obtained by applying the quasisymmetric extension of the omega operator to the fundamental quasisymmetric functions appearing in the expansion of the Young quasisymmetric Schur functions.…”

Section: Theorem 12mentioning

confidence: 99%

Skew row-strict quasisymmetric Schur functions

Mason

Niese

2015

J Algebr Comb

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Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel's fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function.

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“…Combinatorial Hopf algebras. The following definitions, leading to the description of a combinatorial Hopf algebra, closely follow the expositions in [46] and [85]. Let R be a commutative ring with an identity element.…”

Section: 1mentioning

confidence: 99%

“…The product of a quasisymmetric Schur function and a Schur function expands into the quasisymmetric Schur function basis through a rule which refines the Littlewood-Richardson Rule [56] but a formula for the coefficients appearing in the product of arbitrary quasisymmetric Schur functions is unknown. See [85] for a thorough introduction to quasisymmetric Schur functions and their closely related counterpart, the Young quasisymmetric Schur functions. The Young quasisymmetric Schur functions are obtained from the quasisymmetric Schur functions by a simple reversal of the indexing composition and the variables, but at times the Young quasisymmetric functions are easier to work with due to their compatibility with semi-standard Young tableaux (rather than reverse semi-standard Young tableaux).…”

Section: Quasisymmetric Schur Functionsmentioning

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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An Introduction to Quasisymmetric Schur Functions

Cited by 54 publications

References 68 publications

Categorification and Heisenberg doubles arising from towers of algebras

Categorification and Heisenberg doubles arising from towers of algebras

Skew row-strict quasisymmetric Schur functions

Recent Trends in Quasisymmetric Functions

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