Quasisymmetric expansions of Schur-function plethysms

Loehr, Nicholas A.; Warrington, Gregory S.

doi:10.1090/s0002-9939-2011-10999-7

Cited by 15 publications

(10 citation statements)

References 22 publications

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“…Examples include plethysms of Schur functions and k-Schur functions. The former is described in [Loehr and Warrington, 2012], while the latter has already seen some progress in [Assaf and Billey, 2012]. Though currently unproven, the shuffle conjecture provides another example.…”

Section: Resultsmentioning

confidence: 99%

“…Currently there are a number of functions that are easily expressed in terms of Gessel's fundamental quasisymmetric functions that are not easily expressed in terms of Schur functions. For example, such an expansion for plethysms is described in [Loehr and Warrington, 2012], for Lascoux-Leclerc-Thibon (LLT) polynomials in [Haglund et al, 2005b], for Macdonald polynomials in [Haglund et al, 2005a], and conjecturally for the composition of the nabla operator with an elementary symmetric function in [Haglund et al, 2005b]. An expressed goal of developing the theory of dual equivalence graphs is to create a tool for turning such quasisymmetric expansions into explicit Schur expansions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials

Roberts

2013

J Algebr Comb

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In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We further advance the theory of dual equivalence graphs by describing a broader class of graphs that correspond to an explicit Schur expansion in terms of Yamanouchi words. Along the way, we demonstrate several symmetries in the structure of dual equivalence graphs. We then apply these techniques to give explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with at most three cells in the first row and at most two cells in the second row, providing an extension to the combinatorial description of the two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials

Roberts

2013

J Algebr Comb

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“…In fact, for any family of functions expressible in terms of fundamental quasisymmetric functions which are conjectured to be Schur positive, one can look for a signed-colored graph structure or analog of dual equivalence on the indexing set which refines the Schur positivity question to connected components. Another example of this phenomena arises in the work of Loehr and Warrington [25]. They showed that the plethysm of two Schur functions has a nice expansion in terms of fundamental quasisymmetric functions indexed by a family of objects in analogy with standard tableaux.…”

Section: Connections With Llt Macdonald Polynomials and Beyondmentioning

confidence: 95%

Affine dual equivalence and $k$-Schur functions

Assaf

Billey

2012

Journal of Combinatorics

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The k-Schur functions were first introduced by Lapointe, Lascoux and Morse [18] in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono [17] as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A [15] and at t = 1 it is equivalent to the k-tableaux characterization of Lapointe and Morse [22]. In this paper, we extend Haiman's dual equivalence relation on standard Young tableaux [12] to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf's dual equivalence graphs. These graphs display much of the complexity of working with k-Schur functions and the interval structure on S n /S n . We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu de taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we state some open problems on other ways to generalize dual equivalence.

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“…See [Sagan, 2001] or [Stanley, 1999]) for a treatment. In many cases, such as Macdonald polynomials or plethysms of Schur functions, a symmetric function has a known expansion in terms of the fundamental quasisymmetric functions while an explicit expansion over the Schur functions remains elusive (see [Haglund et al, 2005] and [Loehr and Warrington, 2012]).…”

Section: Introductionmentioning

confidence: 99%

From symmetric fundamental expansions to Schur positivity

Roberts¹

2015

Preprint

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We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of f . We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisymmetric generating functions of shifted dual equivalence classes, as well as three new families of functions -one of which is conjectured to be a basis of the vector space of quasisymmetric functions. Each of the six families is realized as the fundamental quasisymmetric generating functions over the classes of some refinement of dual Knuth equivalence. Thus, we also produce a poset of refinements of dual Knuth equivalence. In doing so, we define quasi-dual equivalence to provide classes that generate quasisymmetric Schur functions.

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Quasisymmetric expansions of Schur-function plethysms

Cited by 15 publications

References 22 publications

Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials

Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials

Affine dual equivalence and $k$-Schur functions

From symmetric fundamental expansions to Schur positivity

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