Dual Equivalence Graphs Revisited

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Macdonald Polynomials ◽  
Quasisymmetric Function ◽  
Schur Function ◽  
Connected Component ◽  
Slight Improvement ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
International Audience ◽  
The Creation ◽  
Dual Equivalence Graphs

International audience 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 then demonstrate the utility of this result by giving 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 less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.

scholarly journals Quasisymmetric Schur functions

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
James Haglund ◽  
Sarah Mason ◽  
Kurt Luoto ◽  
Steph van Willigenburg
Keyword(s):  
Schur Functions ◽  
Quasisymmetric Function ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Quasisymmetric Schur Functions

International audience We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions. Nous étudions une nouvelle base des fonctions quasisymétriques, les fonctions de quasiSchur. Ces fonctions sont obtenues en spécialisant les fonctions de Macdonald dissymétrique. Nous décrivons les compositions que donne une simple fonction quasisymétriques. Nous décrivons aussi une règle par certaines fonctions de Schur.

scholarly journals Row-strict quasisymmetric Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sarah K Mason ◽  
Jeffrey Remmel
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Nous Obtenons ◽  
Schur Polynomials ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Quasisymmetric Schur Functions ◽  
The Relationship

International audience Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships. Haglund, Luoto, Mason, et van Willigenburg ont introduit une base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "column-strict'' (ordre strict sur les colonnes). Nous introduisons une nouvelle base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques "row-strict''}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "row-strict'' (ordre strict sur les lignes). Nous décrivons la relation entre cette nouvelle base et d'autres bases connues pour les fonctions quasi-symétriques, ainsi que ses relations avec les polynômes de Schur. Nous obtenons un raffinement de l'opérateur oméga comme conséquence de ces relations.

scholarly journals DUAL EQUIVALENCE GRAPHS I: A NEW PARADIGM FOR SCHUR POSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
SAMI H. ASSAF
Keyword(s):  
Systematic Study ◽  
Generating Functions ◽  
Universal Method ◽  
New Paradigm ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
Combinatorial Construction ◽  
Dual Equivalence Graphs ◽  
Schur Positive

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

scholarly journals A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
S. Assaf
Keyword(s):  
Graph Structure ◽  
Young Tableaux ◽  
Polynomial Representation ◽  
Crystal Graphs ◽  
Dual Equivalence ◽  
Weyl Duality ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Do So ◽  
Dual Equivalence Graphs

International audience For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

scholarly journals Quasisymmetric Schur functions and modules of the $0$-Hecke algebra

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Vasu Tewari ◽  
Stephanie van Willigenburg
Keyword(s):  
Schur Functions ◽  
Hecke Algebra ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Weak Bruhat Order ◽  
International Audience ◽  
Quasisymmetric Schur Functions ◽  
Hecke Modules ◽  
Composition Tableaux

International audience We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions. Nous définissons une action $0$-Hecke sur les tableaux de composition, et ensuite nous l’utilisons pour dériver les modules $0$-Hecke dont la caractéristique quasi-symétrique est une fonction de Schur quasi-symétrique. Nous mettons les modules en relation avec l’ordre de Bruhat faible et les utilisons pour dériver une nouvelle base pour les fonctions quasi-symétriques. Nous classons aussi ces modules qui sont tableau-cycliques et aussi indécomposable. Enfin, nous développons une règle de restriction qui reflète le coproduit des fonctions de Schur quasi-symétriques.

scholarly journals Hall-Littlewood Polynomials in terms of Yamanouchi words

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
International Audience ◽  
Leading Term ◽  
Definition Of

International audience This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

2017 ◽  
Vol 69 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Darij Grinberg
Keyword(s):  
Hopf Algebras ◽  
Young Tableau ◽  
Vertex Operators ◽  
Schur Function ◽  
Algebra Structure ◽  
Quasisymmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Dendriform Algebra ◽  
Natural Analogues ◽  
Creation Operators

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.

scholarly journals Macdonald polynomials at $t=q^k$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Gabriel Luque
Keyword(s):  
Macdonald Polynomials ◽  
Nous Montrons ◽  
International Audience

International audience We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$. Nous nous intéressons aux propriétés des polynômes de Macdonald symétriques $P_{\lambda} (\mathbb{X} ;q,t)$ pour la spécialisation $t=q^k$. En particulier nous montrons une égalité reliant les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$ et $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. Nous en déduisons la description d'un opérateur dont les valeurs propres caractérisent les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

scholarly journals Infinite limits and folding

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Anthony Bonato ◽  
Jeannette Janssen
Keyword(s):  
Biological Networks ◽  
Infinite Graphs ◽  
Isomorphism Type ◽  
Connected Component ◽  
Induced Subgraph ◽  
Pursuit Games ◽  
Graph Homomorphisms ◽  
International Audience

International audience We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.

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