Indecomposable modules for the dual immaculate basis of quasi-symmetric functions

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

doi:10.1090/s0002-9939-2014-12298-2

Cited by 34 publications

(83 citation statements)

References 11 publications

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“…It is intimately related with the Hopf algebras of quasisymmetric functions and noncommutative symmetric functions respectively [13], in the same way as symmetric group representation theory is intimately connected with the Hopf algebra of symmetric functions. More information about H n (0) and its representations can be found in [10,34], and contemporary results can be found in [6,23,24,25,26,29]. Our interest in the 0-Hecke algebra stems from the authors' previous work in the context of providing a representation-theoretic interpretation for quasisymmetric Schur functions [45].…”

Section: 2mentioning

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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Section: 2mentioning

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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“…Like the quasisymmetric Schur functions, dual immaculate quasisymmetric functions correspond to characteristics of certain representations of the 0-Hecke algebra [13], but for the dual immaculate quasisymmetric functions these representations are indecomposable. In particular, let M α be the vector space spanned by all words on the letters {1, 2, .…”

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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“…For example, let (n, k) = (9, 5), w = 254689137 ∈ S 9 and i = (2, 2, 1, 1, 0, 0, 0, 0, 0). We have Des(w) = {2, 6}, so that the composition α |= 9 with Des(α) = Des(w) is α = (2,4,3). It follows that…”

Section: Garsia-stanton Type Basesmentioning

confidence: 99%

“…Proving 0-Hecke analogs of module theoretic results concerning the symmetric group has received a great deal of recent study in algebraic combinatorics [4,17,18,26]; let us recall the 0-Hecke analog of the variable permutation action of S n on a polynomial ring.…”

Section: Introductionmentioning

confidence: 99%

Ordered set partitions and the $0$-Hecke algebra

Huang¹,

Rhoades²

2018

Algebraic Combinatorics

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Let the symmetric group Sn act on the polynomial ring Q[xn] = Q[x1, . . . , xn] by variable permutation. The coinvariant algebra is the graded Sn-module Rn := Q[xn]/In, where In is the ideal in Q[xn] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient R n,k of the polynomial ring Q[xn] depending on two positive integers k ≤ n which reduces to the classical coinvariant algebra of the symmetric group Sn when k = n. The quotient R n,k carries the structure of a graded Sn-module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient S n,k of F[xn] which carries a graded action of the 0-Hecke algebra Hn (0), where F is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case k = n, we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.

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Indecomposable modules for the dual immaculate basis of quasi-symmetric functions

Abstract: We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.

Cited by 34 publications

References 11 publications

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

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

Recent Trends in Quasisymmetric Functions

Ordered set partitions and the $0$-Hecke algebra

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