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 On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Gamma Delta ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
Leading Term ◽  
Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for 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 $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ 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 $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals A Two Parameter Chromatic Symmetric Function

10.37236/940 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Ellison-Anne Williams
Keyword(s):  
Complete Graph ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Macdonald Polynomials ◽  
Simple Graph ◽  
Macdonald Polynomial ◽  
Chromatic Symmetric Function ◽  
Correlation Properties ◽  
Two Parameter

We introduce and develop a two-parameter chromatic symmetric function for a simple graph $G$ over the field of rational functions in $q$ and $t,\,{\Bbb Q}(q,t)$. We derive its expansion in terms of the monomial symmetric functions, $m_{\lambda}$, and present various correlation properties which exist between the two-parameter chromatic symmetric function and its corresponding graph. Additionally, for the complete graph $G$ of order $n$, its corresponding two-parameter chromatic symmetric function is the Macdonald polynomial $Q_{(n)}$. Using this, we develop graphical analogues for the expansion formulas of the two-row Macdonald polynomials and the two-row Jack symmetric functions. Finally, we introduce the "complement" of this new function and explore some of its properties.

scholarly journals Staircase Macdonald polynomials and the $q$-Discriminant

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Adrien Boussicault ◽  
Jean-Gabriel Luque
Keyword(s):  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Monomial Basis ◽  
Nous Montrons ◽  
International Audience ◽  
Nous Examinons

International audience We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis. Nous montrons qu’une $q$-déformation $\mathfrak{D}_k(\mathbb{X};q)$ des puissances du discriminant est égale, à un coefficient de normalisation près, à un polynôme de Macdonald indexé par une partition escalier pour une certaine spécialisation des paramètres. Nous examinons les développements de $\mathfrak{D}_k(\mathbb{X};q)$ dans différentes bases de fonctions symétriques. En particulier, nous montrons que son écriture dans la base des fonctions monomiales peut être explicitement décrite en terme de tableaux standard et nous généralisons un résultat de King-Toumazet-Wybourne sur le développement du $q$-discriminant dans la base de Schur.

scholarly journals How to get the weak order out of a digraph ?

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Francois Viard
Keyword(s):  
Wreath Product ◽  
Symmetric Function ◽  
Coxeter Groups ◽  
Symmetric Functions ◽  
Weak Order ◽  
Möbius Function ◽  
Mobius Function ◽  
Acyclic Digraph ◽  
International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, ainsi qu’une variante de l’ordre faible sur les produits en couronne &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$<sub>$n-1$</sub> où l’on obtient les séries de Stanley classiques.

scholarly journals Recursions and divisibility properties for combinatorial Macdonald polynomials

2011 ◽  
Vol Vol. 13 no. 1 (Combinatorics) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Elizabeth Niese
Keyword(s):  
Hilbert Series ◽  
Permutation Statistics ◽  
Macdonald Polynomial ◽  
Integer Partition ◽  
Theoretical Methods ◽  
Major Index ◽  
International Audience ◽  
Definition Of ◽  
Fermionic Formulas ◽  
Divisibility Properties

Combinatorics International audience For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

scholarly journals Crystal energy via charge

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Anne Schilling
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Recursive Definition ◽  
R Matrix ◽  
Single Column ◽  
Nous Montrons ◽  
International Audience ◽  
Definition Of

International audience The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix. La formule de Ram et Yip pour les polynômes de Macdonald (à t = 0) fournit une statistique que nous appelons la charge. Dans les types ${A}$ et ${C}$, elle peut être définie sur les produits tensoriels des cristaux pour les colonnes de Kashiwara–Nakashima. Dans ce papier, nous montrons que la charge est égale à (l'opposé de) la fonction d'énergie sur cristaux affines. L'algorithme pour calculer la charge est bien plus simple que la définition récursive de l'énergie en fonction de la ${R}$-matrice combinatoire.

scholarly journals Generalized Energy Statistics and Kostka―Macdonald Polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anatol N. Kirillov ◽  
Reiho Sakamoto
Keyword(s):  
Generating Function ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
International Audience

International audience We give an interpretation of the $t=1$ specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution. Nous donnons une intérprétation de la spécialisation à $t=1$ du polynôme de Macdonald modifié comme fonction génératrice des statistiques d'énergie définies sur l'ensemble des chemins qui apparaissent dans la théorie des Systèmes BBS (BBS-chemins). Nous présentons également des généralisations à un paramètre de la statistique d'énergie sur les chemins BBS qui toutes, conjecturalement, ont la même distribution.

scholarly journals Matrix product and sum rule for Macdonald polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Luigi Cantini ◽  
Jan De Gier ◽  
Michael Wheeler
Keyword(s):  
Exclusion Process ◽  
Macdonald Polynomials ◽  
Stationary Measure ◽  
Asymmetric Exclusion Process ◽  
Matrix Product ◽  
Macdonald Polynomial ◽  
Sum Rule ◽  
Infinite Dimensional ◽  
International Audience ◽  
Sum Formula

International audience We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.

scholarly journals Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

10.37236/1264 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Partially Ordered Sets ◽  
Schur Function ◽  
Ordered Sets ◽  
Graded Poset ◽  
Kostka Polynomials ◽  
Relative Version ◽  
Definition Of ◽  
Formal Power

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

scholarly journals The Murnaghan―Nakayama rule for k-Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Anne Schilling ◽  
Mike Zabrocki
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Schur Function ◽  
Noncommutative Symmetric Functions ◽  
Power Sum ◽  
International Audience

International audience We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

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