scholarly journals A variation on the tableau switching and a Pak-Vallejo's conjecture

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olga Azenhas
Keyword(s):  
Young Tableau ◽  
Jeu De Taquin ◽  
Fundamental Symmetry ◽  
Switching Algorithm ◽  
International Audience ◽  
Standard Operations

International audience Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the Benkart-Sottile-Stroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows $\textit{switching}$ as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence of tableaux whose sequence of partition shapes defines a Gelfand-Tsetlin pattern. This property leads to a $\textit{jeu de taquin-chain sliding}$ on Littlewood-Richardson tableaux. Pak et Vallejo ont défini la transformation de la symétrie fondamentale comme une bijection de tableaux de Young pour la comutativité des coefficients de Littlewood-Richardson $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. Ils ont considéré quatre bijections fondamentaux et ont conjecturé qu’elles sont équivalentes (2004). Les trois premières sont basées sur des opérations standard de la théorie des tableaux de Young et, dans ce cas, la conjecture a été confirmée par Danilov et Koshevoy (2005). La quatrième symétrie fondamentale, donnée par l’auteur (1999;2000) et reformulée par Pak et Vallejo, est définie par des opérations $\textit{nonstandard}$ dans la théorie des tableaux de Young. Cette bijection sera montrée équivalente à la première définie pour la propriété involutoire du $\textit{tableau switching}$ de Benkart-Sottile-Stroomer. La preuve de cette équivalence, dans le cas le premier tableau est de Yamanouchi, donne une variation du algorithme de $\textit{tableau switching}$ qui montre $\textit{switching}$ comme une opération qui prendre deux tableaux avec une même borde et meut un à travers de l’autre en décomposant le premier dans une séquence de tableaux dont la séquence des partitions des formats définit une diagramme de Gelfand-Tsetlin. Cette propriété conduit à un algorithme du type $\textit{jeu de taquin-glissements sur chaînes}$ pour les tableaux de Littlewood-Richardson.

scholarly journals Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber
Keyword(s):  
Young Tableau ◽  
Sorting Algorithm ◽  
Young Tableaux ◽  
Hook Length ◽  
Jeu De Taquin ◽  
Bijective Proof ◽  
Symmetry Properties ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

International audience The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.

scholarly journals Maximal increasing sequences in fillings of almost-moon polyominoes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović ◽  
Catherine H. Yan
Keyword(s):  
Pigeonhole Principle ◽  
Fixed Size ◽  
Jeu De Taquin ◽  
International Audience ◽  
Increasing Sequences

International audience It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it. Rubey a montré que le nombre de remplissages d’un polyomino lunaire donné par des zéros et des uns quine contiennent pas de chaîne nord-est d’une taille fixée ne dépend que de l’ensemble des longueurs des colonnesdu polyomino. La preuve de Rubey utilise une adaptation du jeu de taquin et de la promotion sur des remplissagesarbitraires de polyominos lunaires et déduit le résultat pour les remplissages 0/1 par inclusion-exclusion. Dans cetarticle, nous présentons la première preuve bijective de ce résultat en considérant des remplissages de polyominospresque lunaires, qui sont des polyominos lunaires dont on a supprimé une ligne. Plus précisément, nous construisonsune bijection simple qui préserve la taille de la plus longue chaîne nord-est des remplissages lorsque deux lignesadjacentes du polyomino sont échangées. Cette bijection préserve aussi la somme des colonnes des remplissages. Enoutre, nous présentons aussi une bijection simple qui préserve la taille de la plus longue chaîne nord-est, la sommedes lignes et la somme des colonnes si chaque ligne du remplissage contient au plus un 1. Nous fournissons donc nonseulement une preuve bijective du résultat de Rubey, mais aussi deux raffinements de celui-ci.

scholarly journals Monodromy and K-theory of Schubert curves via generalized jeu de taquin

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie ◽  
Jake Levinson
Keyword(s):  
Local Algorithm ◽  
Jeu De Taquin ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
The Real ◽  
Real Locus ◽  
Real Geometry ◽  
International Audience ◽  
K Theory ◽  
Rational Normal Curve

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

scholarly journals The Prism tableau model for Schubert polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Anna Weigandt ◽  
Alexander Yong
Keyword(s):  
Young Tableau ◽  
Schubert Varieties ◽  
Symmetric Polynomials ◽  
Schubert Polynomials ◽  
International Audience

International audience The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.

scholarly journals Cyclic Sieving, Promotion, and Representation Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Fixed Points ◽  
Canonical Basis ◽  
Dihedral Groups ◽  
Jeu De Taquin ◽  
Noncrossing Partitions ◽  
Dual Canonical Basis ◽  
International Audience ◽  
Cyclic Sieving

International audience We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.

scholarly journals Joint Burke's Theorem and RSK Representation for a Queue and a Store

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Moez Draief ◽  
Jean Mairesse ◽  
Neil O'Connell
Keyword(s):  
Young Tableau ◽  
Transient Behavior ◽  
Arrival Process ◽  
Single Server ◽  
Storage Model ◽  
Server Queue ◽  
Rsk Algorithm ◽  
International Audience ◽  
The Stability ◽  
Standard Young Tableau

International audience Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ which is an extension of the classical Burke Theorem. In fact, $r$ can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.

scholarly journals A uniform realization of the combinatorial $R$-matrix

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Arthur Lubovsky
Keyword(s):  
Tensor Product ◽  
Directed Graphs ◽  
Type A ◽  
Affine Lie Algebras ◽  
Jeu De Taquin ◽  
R Matrix ◽  
Finite Dimensional ◽  
Produit Tensoriel ◽  
International Audience ◽  
Alcove Model

International audience Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves Les cristaux de Kirillov–Reshetikhin (KR) sont des graphes orientés avec des arêtes étiquetées qui encodent certaines représentations de dimension finie des algèbres de Lie affines. Les produits tensoriels des cristaux KR de type colonne ont été récemment réalisés de manière uniforme, pour tous les types affines symétriques, en termes du modèle des alcôves quantique. Nous enrichissons ce modèle en l’utilisant pour donner une réalisation uniforme de la $R$-matrice combinatoire, c’est à dire, l’isomorphisme des cristaux affines unique quit permute les facteurs dans un produit tensoriel des cristaux KR. En d’autres termes, nous généralisons pour tous les types de Lie le jeu de taquin de Schützenberger sur les tableaux de Young, qui réalise la $R$-matrice combinatoire dans le type $A$. Nous montrons aussi que le modèle des alcôves quantique ne dépend pas du choix d’une suite d’alcôves.

scholarly journals Cyclic Sieving and Plethysm Coefficients

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
David B Rush
Keyword(s):  
Fixed Points ◽  
Schur Function ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Cyclic Action ◽  
Domino Tableaux ◽  
Ribbon Tableaux ◽  
International Audience ◽  
Cyclic Sieving ◽  
Element Set

International audience A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Une expression combinatoire pour le coefficient de la fonction de Schur $s_{\lambda}$ dans l’expansion du pléthysme $p_{n/d}^d \circ s_{\mu}$ est donné pour tous $d$ que disent $n$, dans les cas où $n=2$, ou $\lambda$ est rectangulaire. Dans ces cas, le coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ se montre à compter, où l’on ignore le signe, le nombre des point fixés d’un ensemble de $\langle s_{\mu}^n, s_{\lambda} \rangle$ éléments sous la puissance $d^e$ d’une action cyclique de l’ordre $n$. Si $n=2$, l’action est l’involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si $\lambda$ est rectangulaire, l’action est une certaine puissance de l’avancement jeu-de-taquin de Schützenberger et Shimozono.Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas $n=2$ , la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d’une fonction de Schur.

Electron Microscopy of Graphite Intercalation Compounds

1982 ◽  
Vol 40 ◽  
pp. 544-545
Author(s):  
G. Timp ◽  
L. Salamanca-Riba ◽  
L.W. Hobbs ◽  
G. Dresselhaus ◽  
M.S. Dresselhaus
Keyword(s):  
Electron Microscopy ◽  
Phase Transitions ◽  
Domain Wall ◽  
Intercalation Compounds ◽  
Lattice Plane ◽  
Wall Model ◽  
Graphite Intercalation Compounds ◽  
Fundamental Symmetry ◽  
Graphite Intercalation

Electron microscopy can be used to study structures and phase transitions occurring in graphite intercalations compounds. The fundamental symmetry in graphite intercalation compounds is the staging periodicity whereby each intercalate layer is separated by n graphite layers, n denoting the stage index. The currently accepted model for intercalation proposed by Herold and Daumas assumes that the sample contains equal amounts of intercalant between any two graphite layers and staged regions are confined to domains. Specifically, in a stage 2 compound, the Herold-Daumas domain wall model predicts a pleated lattice plane structure.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

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